New nonlinear adjustment approaches for
applications in Geodesy and related fields
vorgelegt von
M.Sc.
Georgios Malissiovas
von der Fakult¨at VI – Planen Bauen Umwelt
der Technischen Universit¨at Berlin
zur Erlangung des akademischen Grades
Doktor der Ingenieurwissenschaften
-Dr.-Ing.-
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr.-Ing. Martin Kada
Gutachter: Prof. Dr.-Ing. Frank Neitzel
Gutachter: Prof. Dr. techn. Wolf-Dieter Schuh
Gutachter: Prof. Dr. Andreas Wieser
Gutachter: Priv.-Doz. Dr. techn. habil. Svetozar Petrovi´c
Tag der wissenschaftlichen Aussprache: 10. Mai 2019
Berlin 2019
iii
Summary
This dissertation deals with a class of nonlinear adjustment problems that has a direct least squares solution
for certain weighting cases. In the literature of mathematical statistics these problems are expressed in a
nonlinear model called Errors-In-Variables (EIV) and their solution became popular as total least squares
(TLS). The TLS solution is direct and involves the use of singular value decomposition (SVD), presented in
most cases for adjustment problems with equally weighted and uncorrelated measurements. Additionally,
several weighted total least squares (WTLS) algorithms have been published in the last years for deriving
iterative solutions, when more general weighting cases have to be taken into account and without linearizing
the problem in any step of the solution process.
This research provides firstly a well defined mathematical relationship between TLS and direct least squares
solutions. As a by-product, a systematic approach for the direct solution of these adjustments is established,
using a consistent and complete mathematical formalization. By transforming the problem to the solution
of a quadratic or cubic algebraic equation, which is identical with those resulting from TLS, it will be shown
that TLS is an algorithmic approach already known to the geodetic community and not a new method.
A second contribution of this work is the clear overview of weighted least squares solutions for the discussed
class of problems, i.e. the WTLS solution in the terminology of the statistical community. It will be shown
that for certain weighting cases a direct solution still exists, for which two new solution strategies will be
proposed. Further, stochastic models with more general weight matrices are examined, including correlations
between the measurements or even singular cofactor matrices. New algorithms are developed and presented,
that provide iterative weighted least squares solutions without linearizing the original nonlinear problem.
The aim of this work is the popularization of the TLS approach, by presenting a complete framework for
obtaining a (weighted) least squares solution for the investigated class of nonlinear adjustment problems.
The proposed approaches and the implemented algorithms can be employed for obtaining direct solutions
in engineering tasks for which efficiency is important, while iterative solutions can be derived for stochastic
models with more general weights.
v
Zusammenfassung
Die vorliegende Dissertation besch¨aftigt sich mit einer Klasse von nichtlinearen Ausgleichungsproblemen,
die eine direkte L¨osung nach der Methode der kleinsten Quadrate unter spezifischen Gewichtungsf¨allen
aufweisen. In der Literatur der mathematischen Statistik werden derartige Probleme in einem nichtlinearen
Modell namens Errors-In-Variables (EIV) ausgedr¨uckt und deren L¨osung wurde als Total Least Squares
(TLS) popul¨ar. In den meisten F¨allen l¨asst sich f¨ur gleich gewichtete und unkorrelierte Messungen eine TLS
L¨osung direkt durch eine Singul¨arwertzerlegung (SVD) bestimmen. Dar¨uber hinaus wurden in den letzten
Jahren mehrere Weighted Total Least Squares (WTLS) Algorithmen zur Herleitung iterativer L¨osungen
ver¨offentlicht, bei denen allgemeinere Gewichtungsf¨alle ber¨ucksichtigt werden k¨onnen, ohne das Problem in
jedem Schritt des L¨osungsprozesses zu linearisieren.
Zun¨achst wird in dieser Arbeit eine klar definierte mathematische Beziehung zwischen TLS und direk-
ter L¨osungen nach der Methode der kleinsten Quadrate dargestellt. Des Weiteren wird ein systematis-
cher Ansatz zur direkten L¨osung derartiger Ausgleichungsproblemen unter Verwendung einer konsistenten
und vollst¨andigen mathematischen Formalisierung entwickelt. Durch die ¨
Uberf¨uhrung des Problems in die
L¨osung einer quadratischen oder kubischen algebraischen Gleichung wird gezeigt, dass TLS ein algorithmis-
cher Ansatz ist, der der geod¨atischen Gemeinschaft bereits bekannt ist und keine neue Methode darstellt.
Ein weiterer Beitrag dieser Arbeit besteht in einer klaren ¨
Ubersicht von gewichteten Kleinste-Quadrate
L¨osungen f¨ur die hier diskutierte Klasse von Problemen, wie z.B. der WTLS-L¨osung aus der Terminologie
der statistischen Gemeinschaft. Es wird gezeigt, dass f¨ur bestimmte Gewichtungsf¨alle noch eine direkte
L¨osung existiert, wof¨ur zwei neue L¨osungsstrategien vorgestellt werden. Weiterhin werden stochastische
Modelle mit allgemeineren Gewichtsmatrizen untersucht, einschließlich Korrelationen zwischen den Messun-
gen oder sogar singul¨aren Kofaktor-Matrizen. Es werden neue Algorithmen entwickelt und vorgestellt, die
gewichtete Kleinste-Quadrate L¨osungen iterativ berechnen, ohne das urspr¨ungliche nichtlineare Problem zu
linearisieren.
Das Ziel dieser Arbeit ist die Popularisierung des TLS-Ansatzes, indem eine umfassende Strategie zur
Berechnung einer (gewichteten) Kleinste-Quadrate L¨osung f¨ur die betrachtete Klasse an nichtlinearen Aus-
gleichungsproblemen bereitgestellt wird. Die vorgeschlagenen Ans¨atze und implementierten Algorithmen
k¨onnen zur Berechnung direkter L¨osungen in vielen Ingenieuraufgaben eingesetzt werden, bei denen Effizienz
wichtig ist, w¨ahrend f¨ur stochastische Modelle mit allgemeineren Gewichten auf die iterativen L¨osungsans¨atze
zur¨uckgegriffen werden kann.
vii
Contents
Titlepage i
Summary iii
Zusammenfassung v
Contents vii
List of Figures xi
List of Tables xiii
Abbreviations xv
1 Introduction and Motivation 1
1.1 Researchcontributions........................................ 3
1.2 Organizationofthisthesis...................................... 4
Part I - Fundamentals 7
2 Adjustment calculus 9
2.1 Mathematical modelling of adjustment problems . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.1 Thefunctionalmodel .................................... 10
2.1.2 Thestochasticmodel..................................... 13
2.1.3 Criteria for the solution of adjustment problems . . . . . . . . . . . . . . . . . . . . . 15
2.2 Adjustment of observations with the method of least squares . . . . . . . . . . . . . . . . . . 16
2.2.1 Statistical formulation of least squares problems . . . . . . . . . . . . . . . . . . . . . 17
2.2.2 Least squares parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.3 Definition of linear and nonlinear least squares problems . . . . . . . . . . . . . . . . . 22
2.3 Error estimation of adjustment results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Synopsis of the basics in adjustment calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 Solutions of nonlinear least squares problems 29
3.1 Traditional geodetic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1.1 Adjustment with observation equations and constraints . . . . . . . . . . . . . . . . . 31
3.1.1.1 Least squares parameter estimation within the GMM . . . . . . . . . . . . . 33
3.1.1.2 Error estimation within the GMM . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.1.3 Least squares parameter estimation within the GMM with constraints . . . . 37
3.1.1.4 Error estimation within the GMM with constraints . . . . . . . . . . . . . . 39
3.1.2 Adjustment with condition equations and constraints . . . . . . . . . . . . . . . . . . . 40
3.1.2.1 Least squares parameter estimation within the GHM . . . . . . . . . . . . . 42
3.1.2.2 Error estimation within the GHM . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.2.3 Least squares parameter estimation within the GHM with constraints . . . . 46
3.1.2.4 Error estimation within the GHM with constraints . . . . . . . . . . . . . . . 49
viii Contents
3.2 Totalleastsquares .......................................... 49
3.2.1 Nonlinear adjustments within the EIV model . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.1.1 Least squares parameter estimation using TLS . . . . . . . . . . . . . . . . . 53
3.2.1.2 Least squares parameter estimation using WTLS . . . . . . . . . . . . . . . . 54
3.3 Discussionandopenquestions.................................... 59
Part II - Methodological contributions 61
4 Direct solutions of nonlinear least squares problems with equal weights 63
4.1 Basic idea and general methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Fitting of a straight line in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2.1 Least squares adjustment with a direct solution . . . . . . . . . . . . . . . . . . . . . . 66
4.2.1.1 Definition of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.1.2 Simplification of the problem by substituting one unknown parameter . . . . 69
4.2.2 TLSsolutionwithSVD ................................... 71
4.2.2.1 TLS solution based on the minimum eigenvalue principle . . . . . . . . . . . 72
4.2.2.2 Solution by the eigenvalue/eigenvector decomposition . . . . . . . . . . . . . 72
4.3 Fitting of a straight line in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.1 Direct least squares solution for fitting a straight line in 3D . . . . . . . . . . . . . . . 75
4.3.2 TLS fitting of a straight line in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4 Fittingofaplanein3D ....................................... 78
4.4.1 Direct least squares solution for fitting a plane in 3D . . . . . . . . . . . . . . . . . . . 79
4.4.2 TLSfittingofaplanein3D................................. 81
4.5 2D similarity transformation of coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.5.1 Direct least squares solution for the 2D similarity transformation . . . . . . . . . . . . 83
4.5.2 TLS 2D similarity transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.6 General formulation and classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.7 Discussionandopenquestions.................................... 89
5 Direct and iterative solutions of weighted nonlinear least squares problems 91
5.1 Basic idea and general methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2 Fitting of a straight line in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.2.1 Weighting case 1 - Equally weighted observations in each direction . . . . . . . . . . . 93
5.2.1.1 Direct least squares solution in a scaled coordinate system . . . . . . . . . . 95
5.2.2 Weighting case 2 - Individually weighted points in 2D . . . . . . . . . . . . . . . . . . 97
5.2.2.1 Direct weighted least squares solution . . . . . . . . . . . . . . . . . . . . . . 97
5.2.3 Weighting case 3 - Individually weighted 2D coordinates . . . . . . . . . . . . . . . . . 100
5.2.3.1 Iterative least squares solution without linearization . . . . . . . . . . . . . . 102
5.2.4 Weighting case 4 - Individually weighted and correlated 2D coordinates . . . . . . . . 105
5.2.4.1 Iterative least squares solution without linearization . . . . . . . . . . . . . . 107
5.2.4.2 Solution for singular cofactor matrices . . . . . . . . . . . . . . . . . . . . . . 109
5.3 Fittingofaplanein3D .......................................113
5.3.1 Weighting case 1 - Equally weighted observations in each direction . . . . . . . . . . . 113
5.3.2 Weighting case 2 - Individually weighted points in 3D . . . . . . . . . . . . . . . . . . 115
5.3.3 Weighting case 3 - Individually weighted 3D coordinates . . . . . . . . . . . . . . . . . 118
5.3.4 Weighting case 4 - Individually weighted and correlated 3D coordinates . . . . . . . . 122
5.4 2D similarity transformation of coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.4.1 Weighting case 1 - Equally weighted observations in each coordinate system . . . . . . 130
5.4.2 Weighting case 2 - Individual weight for each pair of homologous points in both systems132
5.4.3 Weighting case 3 - Individually weighted coordinates . . . . . . . . . . . . . . . . . . . 136
5.4.4 Weighting case 4 - Individually weighted and correlated coordinates in each coordinate
system.............................................141
ix
5.5 Discussion of weighted nonlinear least squares solutions . . . . . . . . . . . . . . . . . . . . . 149
6 Numerical Investigations 151
6.1 Fitting of a straight line in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.2 2D similarity transformation of coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
7 Conclusion and outlook 171
7.1 Conclusion ..............................................171
7.2 Outlook ................................................172
Appendices 175
A Stochastic models for the numerical investigations 177
A.1 Singular cofactor matrix for fitting a straight line in 2D . . . . . . . . . . . . . . . . . . . . . 177
A.2 Singular cofactor matrix for the 2D similarity transformation . . . . . . . . . . . . . . . . . . 178
Bibliography 181
xi
List of Figures
2.1 Simple example of linear variance-covariance propagation . . . . . . . . . . . . . . . . . . . . 24
2.2 Simple example of a first order variance-covariance propagation . . . . . . . . . . . . . . . . . 26
3.1 Two optimization approaches for the solution of a class of nonlinear least squares problems. . 30
4.1 Flowchart for two possible direct solutions of a class of nonlinear least squares problems. . . . 64
4.2 Representation of a straight line in 2D using equation (4.1). . . . . . . . . . . . . . . . . . . . 65
4.3 Example of fitting a straight line to points in 2D with both xand ycoordinates subject to
measurementerrors. ......................................... 67
4.4 Example of fitting a straight line to points in 2D with coordinates reduced to the centre of
massofthemeasuredpoints. .................................... 70
5.1 Flowchart for possible direct and iterative solutions of a class of nonlinear weighted least
squaresproblems............................................ 92
5.2 Example of fitting a straight line to points in 2D, with observed xand ycoordinates and px,
pyindividual constant weights for each coordinate axis. . . . . . . . . . . . . . . . . . . . . . 94
5.3 Example of fitting a straight line to the scaled points in 2D. . . . . . . . . . . . . . . . . . . . 96
5.4 Example of fitting a straight line to points in 2D with xand ymeasured coordinates
and pxi,pyibeing equal weights for each point. . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.5 Example of fitting a straight line to points in 2D with observed xiand yicoordinates and
pxi,pyiindividual weights for the coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.1 Fitting a straight line to points in 2D, with observed xand ycoordinates of equal precision. . 153
6.2 Fitting a straight line to points in 2D, with observed xand ycoordinates and px,pyindividual
constant weights for each coordinate axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.3 Fitting a straight line to points in 2D, with individual weight for the coordinates of each point.156
6.4 Fitting a straight line to points in 2D, with individual weight for each measured coordinate. . 158
6.5 Fitting a straight line to points in 2D, with individually weighted and correlated coordinates
foreachpoint. ............................................161
6.6 Fitting a straight line to points in 2D, with a singular cofactor matrix. . . . . . . . . . . . . . 162
xiii
List of Tables
6.1 Example dataset of measured points in 2D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.2 Solution within the GHM using the algorithm of Neitzel and Petrovic (2008). . . . . . . . . . 152
6.3 Direct least squares solution (section 4.2.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.4 TLSsolution(section4.2.2)......................................153
6.5 Direct least squares solution (section 5.2.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.6 Individual weights for each point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.7 Direct least squares solution (section 5.2.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.8 Individual weights for each coordinate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.9 Iterative least squares solution using Algorithm 1 (section 5.2.3). . . . . . . . . . . . . . . . . 157
6.10 Individual weights for each coordinate and correlations for each point. . . . . . . . . . . . . . 159
6.11 Iterative least squares solution using Algorithm 2 (section 5.2.4.1). . . . . . . . . . . . . . . . 160
6.12 Iterative least squares solution using Algorithm 3 (section 5.2.4.2). . . . . . . . . . . . . . . . 162
6.13 Example dataset for the 2D similarity transformation . . . . . . . . . . . . . . . . . . . . . . 163
6.14ResultsfromNeitzel(2010)......................................163
6.15 Direct least squares solution for the 2D similarity transformation . . . . . . . . . . . . . . . . 164
6.16 Direct least squares solution (section 5.4.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.17 Individual weights for homologous points in both systems. . . . . . . . . . . . . . . . . . . . . 165
6.18 Direct least squares solution (section 5.4.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.19 Individual weight for each coordinate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.20 Iterative least squares solution (section 5.4.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.21 Weights and correlations for the coordinates of the points in the target system. . . . . . . . . 167
6.22 Weights and correlations for the coordinates of the points in the source system. . . . . . . . . 167
6.23 Iterative least squares solution (section 5.4.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.24 Example dataset from Neitzel and Schaffrin (2016). . . . . . . . . . . . . . . . . . . . . . . . . 168
6.25 Iterative least squares solution (section 5.4.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
xv
Abbreviations
CTLS Constrained Total Least Squares
EIV Errors In Variables
EVD Eigen Value Decomposition
GHM Gauss Helmert Model
GMM Gauss Markov Model
MCS Monte Carlo Simulation
NS Neitzel-Schaffrin
STLS Structured Total Least Squares
SVD Singular Value Decomposition
TLS Total Least Squares
UT Uncented Transformation
VC Variance Covariance
WTLS Weighted Total Least Squares
2D Two dimensionional
3D Three dimensional
1
1 Introduction and Motivation
In geodetic practice, engineers are engaged in performing measurements for the numerical description of
reality, including the characteristics of some physical phenomena or geometrical properties of real objects.
The desired values often cannot be measured directly, but they are linked to the measured values via a
mathematical model. The mathematical modelling of the measurement results, together with the errors that
influence them, results in an under-determined algebraic problem. The target is in most cases the “optimal”
estimation of some unknown parameters. For more than two centuries mathematicians and geodesists have
solved these adjustment problems using the method of least squares, according to the fundamental studies of
Gauss (1809) and Gauss (1823). A least squares estimate can be obtained by minimizing a defined objective
function (the sum of squared residuals) and thus by solving a system of normal equations, i.e. a system of
equations that follow from the partial derivatives of the objective function with respect to all unknowns.
Depending on the nature of the problem, a least squares adjustment can be linear or nonlinear. Obviously
linear least squares problems can be solved using the rules of linear algebra, whilst the nonlinear cases require
most of the time a numerical method for obtaining a solution. This thesis investigates only the second type
of problems.
The solution of nonlinear adjustment problems with the use of least squares has a long history and the
simplicity of the “recipe” of this method is recognized by its wide application in all scientific fields that
deal with redundant observations and seeking for an “optimal” solution. Helmert (1924), for instance,
proposed a least squares solution by linearizing the functional model of the nonlinear adjustment problem.
Deming (1931) and Deming (1934) tackled the same problem by developing algorithms based on the iterative
linearization of the functional model following the Gauss-Newton approach. Pope (1974) pointed out that a
least squares solution of a nonlinear adjustment problem can be obtained either by using the Gauss-Newton
approach, or by setting up and linearizing the nonlinear normal equations, according to the Newton-Raphson
approach. It is often in geodetic literature that the least squares principle is applied in the form of two
adjustment models, namely the Gauss-Markov Model (GMM), see (Niemeier 2008, p. 137 ff.), and the
Gauss-Helmert Model (GHM), see e.g. (Niemeier 2008, p. 172 ff.). In (Krakiwsky 1975, pp. 7-26) these
models can be found under the name parametric (case) adjustment and combined (case) adjustment, as
well. A least squares solution from both models is based on the Gauss-Newton approach.
Furthermore, a class of nonlinear least squares problems exists, for which a direct solution is possible for
the nonlinear normal equations, especially for such cases where the solving of the normal equations can be
converted into an eigenvalue problem. Thus, a solution is obtained by computing the roots of a polynomial
(i.e. the characteristic equation of the eigenvalues) and a direct solution can be possible depending on the
polynomial’s degree. Such adjustment problems have been discussed in geodetic literature since long time.
Linkwitz (1960), for instance, presented a least squares solution for two adjustment problems that belong to
this class, the fitting of a straight line in two dimensions (2D) and the fitting of a plane in three dimensions
2Chapter 1. Introduction and Motivation
(3D), while the coordinates of the points in all directions are considered as measurements. Another example
is the work of Joviˇci´c et al. (1982), who investigated the fitting of a straight line in 3D. Therefore, it can
be seen from past publications that the solution of such nonlinear least squares problems, using eigenvalue
decomposition (EVD), has been already a standard procedure for the members of the geodetic community.
Nevertheless, an alternative approach was developed during the last decades for the solution of this class of
nonlinear least squares problems by the mathematical community. This is called total least squares (TLS)
and has been firstly defined and presented by Golub and Van Loan (1980). Since then many researchers dealt
with the solution of adjustment problems using TLS and developed modern and sophisticated algorithms.
As it was defined in (Golub and Van Loan 1980) or (Van Huffel and Vandewalle 1991, p. 33 ff.), the TLS
solution is related only with this class of nonlinear adjustment problems which can be expressed within an
errors in variables (EIV) model and solved with the use of singular value decomposition (SVD), without
involving any kind of linearization of the functional model.
Within literature the work coming from the TLS community is often distinguished from the classical least
squares by stating that TLS functions differently. There are expectations that TLS might produce a “more
realistic” result than the classic least squares, as indicated for example in (Felus and Schaffrin 2005) or
(Schaffrin et al. 2006). Petrovic (2003) has already pointed out that this view has been caused possibly
by the work of Golub and Van Loan (1980), where the solution of TLS was compared with that of least
squares for fitting a straight line in 2D. In that study, for the least squares solution it was assumed that
only the y-coordinates are regarded as observed values and the x-coordinates as error free, which led to the
misleading conclusion that TLS functions differently from least squares. For geodesists it has already been
clear that the most important steps for the adjustment of observations is to formulate a correct model and
minimize the correct objective function. When these requirements are fulfilled, then for a linear problem
the solution will be unique, regardless of the solution strategy that has been followed.
Contrary to the belief that TLS is an additional method like least squares (or even a generalisation of it),
several scientists have shown that TLS describes just a particular algorithmic approach to find the (weighted)
least squares solution. One of the first critical views on TLS can be found in the appendix of (Petrovic 2003).
This author concluded that least squares and TLS are not individual methods, but applications of minimizing
the sum of squared residuals. Afterwards, Neitzel and Petrovic (2008) and Neitzel (2010) showed on two
practical examples that in fact TLS can be regarded as a special case of the least squares method within the
GHM. The iterative solution of the GHM has been proven to be numerically equivalent to the TLS solution
in both cases. Other contributions that follow this line of thinking are (Reinking 2008) and (Mihajlovic and
Cvijetinovic 2016). These studies provided the motivation for the first part of this research, which has been
partly published in (Malissiovas et al. 2016). In that article, a clear mathematical relationship between TLS
and direct least squares solutions has been presented.
A TLS solution has been mainly investigated when the measurements are equally weighted and uncorrelated.
Consequently, the following scientific challenges focused on the algorithmic development of TLS when indi-
vidual precisions or correlations are postulated for the measurements. The solutions from these algorithms
are iterative, they do not include a linearization of the functional model and have been published under the
name “weighted TLS” (WTLS). For instance, Schaffrin and Wieser (2008) developed a WTLS algorithm for
the weigthed least squares solution of fitting a straight line in 2D, which was the basis for further algorith-
mic developments from Shen et al. (2011), Fang (2011), Amiri-Simkooei and Jazaeri (2012) and Mahboub
(2012). Additionally, various names for the TLS solution emerged due to algorithmic complications caused
by the stochastic model of each problem. For instance, the term STLS (structured TLS) was presented in
1.1. Research contributions 3
(Schaffrin et al. 2012) for the solution of a 2D similarity transformation of coordinates, or the term CTLS
(constrained TLS) presented in (Abatzoglou et al. 1991) or (Schaffrin 2006). Despite the name TLS, in
all above cases the solution has been obtained iteratively and does not follow the definition of TLS that
was established by Golub and Van Loan (1980). A clear overview of the latest WTLS algorithms has been
presented in the dissertation of Snow (2012), who also covered special cases of singular cofactor matrices
being present in the model.
Nevertheless, York (1966) and Williamson (1968) had already presented weighted least squares solutions
for the problems that have been tackled in terms of WTLS, i.e. iterative solutions of the nonlinear normal
equations without linearizing the functional model. York (1968) and Petrovi´c et al. (1983) developed also
algorithms even for the case of correlated observations. These contributions provide the motivation for the
next research question of this dissertation. In this second part, the solution of the discussed class of nonlinear
least squares problems is investigated for various cases where the observations have different precisions or
correlations.
1.1 Research contributions
In the first place a mathematical relationship between direct least squares solutions and TLS is developed
by investigating four individual adjustment problems, namely: the fitting of a straight line in 2D, the fitting
of a straight line in 3D, the fitting of a plane in 3D and the 2D similarity transformation of coordinates. As
a consequence, a systematic approach is established for the direct least squares solution of these nonlinear
adjustment problems. This leads to a deeper understanding of the underlying principles of TLS adjustment
and to the class of problems that can be solved via SVD, i.e. the solving of nonlinear normal equations can
be transformed into an eigenvalue problem.
The second part of this study focuses on the solution of the class of nonlinear least squares problems, when
individual precisions and correlations between the measurements have to be taken into account. Novel
algorithms are implemented to provide direct and iterative weighted least squares solutions, depending on
the stochastic model of each problem. In contrast to the known WTLS algorithms, the presented solutions
do not always need iterations but can have in some weighted cases a direct solution. When iterations are
necessary, the developed algorithms can provide a weighted least squares solution without performing any
linearization of the problem under investigation. The cases of singular cofactor matrices in the stochastic
model are also covered (when the criterion of Neitzel and Schaffrin (2016) is fulfilled) by the developed
approaches, without the need of any special treatment of the problem.
From an engineering point of view, the proposed solutions and algorithms can be an asset for providing
the (weighted) least squares solution for this class of problems without any iteration or starting values for
the unknown parameters. Thus, this is an advantageous algorithmic option for researchers and engineers
in terms of efficiency. The adjustment approaches that were developed so far have a significant impact on
several applications in geodesy. For instance, 3D point clouds obtained by terrestrial laser scanners can
be easily handled, depending on the needs of the engineering task. The developed algorithms can handle
efficiently the vast amount of data, providing for example direct solutions for fitting planes or estimating
the transformation parameters between several data sets. On the other hand, the presented algorithms
that provide iterative weighted least squares solutions can be utilized also when correlations between the
observations have to be taken into account.
4Chapter 1. Introduction and Motivation
One of the objectives from the contributions of this dissertation is to present a simple explanation of the
TLS and WTLS solutions to scientists, researchers and engineers from all scientific fields that deal with
adjustment calculus. Thus, it can enable them to use appropriately the TLS/WTLS algorithms for the
solution of nonlinear adjustment problems. Furthermore, the proposed solution strategies, accompanied by
individual algorithms for each case, can be utilized for the solution of this class of problems in the future
without the need of TLS or WTLS theory and practice.
1.2 Organization of this thesis
This thesis is organized into two main parts, the part describing fundamental ideas of adjustment and the
methodological contributions, and involves six chapters. The content of each chapter is summarized in the
following.
Fundamentals
Chapter 2 describes fundamental requirements for the adjustment of observations and gives a definition of
the method of least squares and hence sets the basis for the definition of nonlinear least squares problems
in this dissertation. These requirements are deduced from a review of related work on adjustment calculus.
Since the mathematical formalization of nonlinear normal equations with a solution using EVD or SVD is
one of the main goals of this thesis, basic mathematical notions and concepts which serve as foundation for
this purpose are briefly discussed.
Chapter 3 presents a part of existing solution strategies and numerical methods for the solution of nonlinear
least squares problems, based on the requirements of chapter 2. In this core chapter of the thesis, both the
traditional geodetic and the most modern solution strategies are elaborated in depth. Three fundamental
models, namely the GMM, the GHM and the EIV model, are defined and mathematically described. Based
on these models, further concepts such as the Gauss-Newton, the TLS and the WTLS approach are defined
and formalized. Scientific questions emerge from the developed concepts and are outlined at the end of this
chapter.
Methodological contributions
Chapter 4 is dedicated to the development of a well defined mathematical relation between direct least squares
solutions and TLS solutions for a class of nonlinear adjustment problems. It involves the elaboration of both
solutions for four adjustment problems that often occur in practice. Based on this, a new strategy for the
direct solution of such problems is elaborated and presented in detail. Subsequently, common features are
identified between the investigated adjustment problems that can serve as classification factors for others
that belong to this group. The chapter concludes with a discussion on the nature of TLS adjustment and
raises further scientific questions.
Chapter 5 addresses different solutions of the same class of nonlinear least squares problems as in chapter 4,
by postulating individual stochastic models for each case. In contrast to the iterative algorithms known as
WTLS, various weighting cases are identified in this chapter that can still lead to a direct solution, based
on the same concepts as in chapter 4. Additionally, a standard solution strategy is presented for cases
1.2. Organization of this thesis 5
where iterations are necessary for a weighted least squares solution, without applying any linearization to
the investigated problems. From this, individual algorithms are designed that fit into the framework of the
proposed solution approach. The feasibility of the developed algorithms is also demonstrated on cases where
the cofactor matrix of the problem is singular.
Chapter 6 illustrates the application of the developed methodologies and algorithms for two adjustment
cases. The first case presents the proposed direct and iterative least squares solution for fitting a straight
line to a set of measured points in 2D, while postulating various weights and correlations between the
observations. In the same line of thinking the 2D similarity transformation of coordinates is examined as a
second example.
Chapter 7 summarizes this work and draws conclusions with respect to the stated research goals as well as the
scientific questions that emerged in each chapter. It reviews and evaluates the results of this investigation,
lists and discusses contributions to the field of adjustment calculus and identifies and outlines scientific
problems that could be tackled in a future research.
7
Part I - Fundamentals
9
2 Adjustment calculus
This chapter briefly summarizes the concept of adjustment calculus. This is necessary for understanding the
fundamentals of the adjustment of observations and the estimation of unknown parameters. Many parts of
this chapter have been adopted from the work of Pasioti (2015) and have been extended appropriately to
fit the needs of this dissertation.
It can be said that adjustment calculus is a part of geodesy and all scientific fields that deal with redundant
measurements. The measurement results are necessary for describing numerically the characteristics of some
physical phenomena or geometrical properties of real objects. The target is in most cases the “optimal”
estimation of some unknown parameters by means of an under-determined algebraic system that occurs
from the mathematical modelling of the measured quantities1. Additionally, the statistical properties of
the estimated parameters are of great importance for drawing conclusions concerning the precision and the
reliability of the adjustment results. Based on the adjustment phases presented by Neitzel and Petrovic
(2008), five basic steps can be identified for the gradual process of any adjustment problem. These can be
summarized with the following systematic stages:
1. Definition of the problem/task:
In this step it must be clearly defined what are the measurements that are subject to errors, which
parameters are fixed (error free) and which parameters are unknown.
2. Clear description of the mathematical model:
The mathematical model is the combination of an appropriate functional model, which is selected by
the user, with the stochastic model that incorporates the stochastic properties of the measurements.
3. Selection of the method for a solution of the adjustment problem:
A solution of an adjustment problem can be derived by selecting an appropriate criterion for the
unknown errors, depending on their distribution. The choice of such a criterion implies the method
that will be followed. For example, in case of normally distributed errors the least squares method
should be employed.
4. Calculation of the adjustment results:
A least squares solution for an adjustment problem can be obtained by solving a system of equations
(i.e. the normal equations). Various approaches exist for the solution of the normal equations in
numerical mathematics, that depend on the nature (linear or nonlinear) and the well-posedness of the
problem.
1As it will be explained in subsection 2.1.1, the functional model of an adjustment problem includes unknown parameters
to be estimated and residuals, which are also unknown, leading to an under-determined set of equations
10 Chapter 2. Adjustment calculus
5. Computation of stochastics:
The precision measures associated with the adjustment results can be computed after obtaining the
solution of the problem using, for example, the rules of error propagation.
These fundamental stages for the solution of adjustment problems serve as a basis for the explanation of
the theory of adjustment calculus that is presented in this chapter.
2.1 Mathematical modelling of adjustment problems
Under the theoretical assumption that measurements are contaminated by errors (Taylor 1982, p. 3), an
under-determined algebraic system emerges that is strongly related to the stochastic properties of the mea-
surements. Important for the solution of an adjustment problem is the clear description of the deterministic
part (the derived under-determined system of equations) and the stochastic part2(the measurement errors
and their disribution). The first is described by the functional model and the second by the stochastic
model, which are always combined to form the mathematical model of the problem, as it has been defined in
(Mikhail and Ackermann 1976, p. 5) or (Perovi´c 2005, p. 72).
For obtaining meaningful adjustment results, a correct formulation of the mathematical model is essential
in every case and should clearly answer the following questions:
- What are the observations that are subject to measurement errors?
- What is the precision of the observations?
- What are the unknown parameters to be estimated?
- Which parameters are error-free or fixed/constant?
The two fundamental parts of a mathematical model, i.e. the functional and stochastic model, are explained
in the following subsections.
2.1.1 The functional model
Observations li(with i= 1, ..., n) and unknown parameters xj(j= 1, ..., m) can be mathematically related
through a set of functions, which are often referred as the functional model. Three main categories of
functional relationships are disinguished in the geodetic literature, see for example (Wells and Krakiwsky
1971, pp. 102-104) or (Perovi´c 2005, p. 63):
1. Implicit functional relationships between the observations
f(li)≈0.
2see (Perovi´c 2005, p. 55) for a definition of the deterministic and stochastic model.
2.1. Mathematical modelling of adjustment problems 11
2. Explicit functional relationships between the observations and the unknown parameters
li≈f(xj).
3. Explicit or implicit functional relationships between the observations and the unknown parameters
f(li, xj)≈0.
The presented equations are approximately equal, as the observed quantities are subject to errors and have
to be corrected, as it is explained below. According to (Wells and Krakiwsky 1971, p. 104), the mathematical
relationships of categories 1 and 2 are only special cases of the most general case in 3. This study investigates
adjustment problems in which some unknown parameters need to be estimated 3.
Explicit functional relationships between observations and unknown parameters
Let a number of observations l1, ..., lnbe performed and x1, ..., xmunknown parameters to be estimated.
The mathematical relationship between the observations and the unknown parameters can be expressed by
the equation system
l1≈f1(x1, ..., xm),
l2≈f2(x1, ..., xm),
.
.
.
ln≈fn(x1, ..., xm).
(2.1)
Depending on the number of equations in (2.1), three individual cases can emerge (Helmert 1924, p. 47):
1. The number of observations is smaller than the number of the parameters to be estimated (n < m).
There is no unique solution (there are infinitely many solutions).
2. The number of observations is equal to the number of the parameters to be estimated (n=m). There
exists a unique solution, but the presence of blunders cannot be identified.
3. The number of observations is larger than the number of the parameters to be estimated (n>m).
This is an overdetermined system with no unique solution.
A usual geodetic problem consists of repeated observations, seeking for a solution to a set of unknown
parameters. Obviously, the third case presented above is what geodesists usually face. Furthermore, in
equation (2.1) the measured values lhave to be corrected for their random errors. According to (Bjerhammar
1973, p. 1), the true value ˜
lcan be obtained by excluding the error (e) from the measurement:
˜
l=l−e. (2.2)
3Category 1 leads to the famous adjustment of condition equations (when taking into account the necessary residuals).
However, this adjustment problem is out of the scope of this work. More information for this case can be found in most common
literature of adjustment calculations in geodetic science, see for example (Helmert 1924, p. 228 ff.).
12 Chapter 2. Adjustment calculus
However, because the true values are only a theoretical concept, the adjusted value of a measurement ˆ
lis
usually considered by adding a correction/residual v
ˆ
l=l+v. (2.3)
Thus, including the necessary residuals in the functional model yields the under-determined4system of
equations
l1+v1=f1(x1, ..., xm),
l2+v2=f2(x1, ..., xm),
.
.
.
ln+vn=fn(x1, ..., xm).
(2.4)
The developed equations (with included residuals) are the observation equations. As claimed in (Wells and
Krakiwsky 1971, p. 102), adjustment problems of this type are widely known as adjustment of observation
equations,parametric adjustment or adjustment of indirect observations5.
Implicit functional relationships between observations and unknown parameters
A number of observations has been collected and some unknown parameters need to be estimated. Assuming
that the mathematical relationship between the observations and the unknown parameters can be expressed
by
f1(l1, ..., ln, x1, ..., xm)≈0,
f2(l1, ..., ln, x1, ..., xm)≈0,
.
.
.
fr(l1, ..., ln, x1, ..., xm)≈0.
(2.5)
Depending on the number of equations rin (2.5), an adjustment problem exists when the number of equations
is larger than the number of the parameters to be estimated (r > m). Adding the necessary residuals to the
observations yields
f1(l1+v1, ..., ln+vn, x1, ..., xm)=0,
f2(l1+v1, ..., ln+vn, x1, ..., xm) = 0,
.
.
.
fr(l1+v1, ..., ln+vn, x1, ..., xm) = 0.
(2.6)
This system of equations is known as condition equations with unknowns. Adjustment problems of this
type can be found under the name combined adjustment or adjustment of condition equations and unknown
parameters (Wells and Krakiwsky 1971, p. 102) 6.
4The developed functional model is under-determined, because viare also unknown quantities.
5More information from the traditional German literature can be found in (Helmert 1924, p. 43) or (Linkwitz 1960, p. 156).
It is worth noticing that the German term “Vermittelnde Beobachtungen” for this type of adjustment cannot be translated
literally.
6In the German literature this type of adjustment problems is known as “Bedingte Beobachtungen mit Unbekannten”, see
for example (Helmert 1924, p. 52) or (Linkwitz 1960, p. 192).
2.1. Mathematical modelling of adjustment problems 13
Constraints between the unknown parameters
Additionally, part of the functional model can be a number of constraints that have to be enforced on the
unknown parameters. For example, such a constraint can be formulated as
u(x1, x2, ..., xm)=0,(2.7)
with udenoting here a function of the unknowns. This is just a special case of the condition equations (2.6).
Adjustment problems with imposed constraints can be divided into two cases:
- Adjustment of observation equations with constraints between the unknown parameters, see for ex-
ample (Helmert 1924, p. 262) or (Linkwitz 1960, p. 197).
- Adjustment of condition equations and unknowns with constraints between the unknown parameters,
as explaned in (Wells and Krakiwsky 1971, p. 142) and (Mikhail and Ackermann 1976, p. 213 ff.).
2.1.2 The stochastic model
In a complete mathematical model the stochastic properties of the obtained observations must be taken
into account. In the literature, the theoretical error that can influence an observation is denoted with the
term “standard deviation” of the measurement, as for example in (Niemeier 2008, p. 6). The measurement
standard deviations are often based on the physical characteristics of some instrument that has been used
for measurements and imply how precise the observations are. This a priori7information influences the
adjustment procedure in a sense of how much an observation contributes to the adjustment (i.e. a precise
observation is more valuable than another with a low precision).
Theoretical variances and covariances of a random variable
In statistics the precision of an observation can be expressed in terms of its variance. Based on the theory
of errors the theoretical variance of a discrete random variable is defined as
σ2= lim
N→∞
1
N
N
X
i=1
e2
i= lim
N→∞
1
N
N
X
i=1
(li−˜
l)2=E{(li−˜
l)2},(2.8)
see, e.g. (Bjerhammar, 1973, p. 22), (Niemeier, 2008, p. 23) or any other textbook on statistics. ˜
ldenotes
the true value of l,ethe observation errors and E{} the expectation of a variable. Nis the total number
of random errors with the same probability that belong to a universal set. The standard deviation can be
defined as the positive square root of the variance
σ= +√σ2.(2.9)
7In the literature the term “a priori” is often confused with the term “prior”. While the former defines some information
that has been obtained after theoretical deduction and has been traditionally used by scientists for expressing such arguments,
the latter refers only to some moment in time.
14 Chapter 2. Adjustment calculus
The theoretical covariance between two measurements, for example l1and l2, can be written as
σ1,2= lim
N→∞
1
N
N
X
i=1
(l1i−˜
l1)(l2i−˜
l2) = E{(l1i−˜
l1)(l2i−˜
l2)},(2.10)
with ˜
l1and ˜
l2denoting the true values of l1and l2, respectively.
Empirical variances and covariances of a random variable
In most cases the true value ˜
lis not known and thus it is replaced by the expected value E{l}(for least
squares problems the mean value ¯
lis regarded as the expected value E{l}), see for example (Montgomery
and Runger 2010, p. 74) or (Everitt and Skrondal 2010, p. 156) for a definition.
An estimate for the variance of lhas been given by (Bjerhammar, 1973, p. 37) or (Dekking et al., 2005, p.
292) as
s2=1
n−1
n
X
i=1
v2
i=1
n−1
n
X
i=1
(li−E{l})2=1
n−1
n
X
i=1
(li−¯
l)2,(2.11)
with ndenoting a finite number of residuals v, that belong to a subset of the universal set containing the
random errors of the problem. In (Linnik, 1961, p. 79) and (Bjerhammar, 1973, p. 127) it has been shown
that for a linear adjustment problem, the estimated variance s2is an unbiased estimate of σ2, i.e.
E{s2}=σ2.(2.12)
Equivalently, an estimate for the covariance between two measurements l1and l2can be derived by
s1,2=1
n−1
n
X
i=1
(l1i−E{l1})(l2i−E{l2)}=1
n−1
n
X
i=1
(l1i−¯
l1)(l2i−¯
l2).(2.13)
Storing all the observations in one vector
L= [l1, l2,··· , ln]T,(2.14)
the variances and covariances of all measurements can be expressed in matrix notation, like in (Wells and
Krakiwsky 1971, p. 89) or (Niemeier 2008, p. 29), stored in the variance-covariance (VC) matrix
ΣLL =1
n−1(L−E{L})(L−E{L})T=
σ2
1σ1,2··· ···
σ2,1σ2
2··· ···
.
.
..
.
.....
.
.
.
.
..
.
.. . . σ2
n
.(2.15)
(Ghilani 2010, p. 166) is one of the many textbooks where it was shown that the cofactor matrix of the
observations can be computed by
QLL =1
σ2
0
ΣLL,(2.16)
2.1. Mathematical modelling of adjustment problems 15
which for a regular cofactor matrix results in the weight matrix
P=Q−1
LL.(2.17)
Matrices ΣLL,QLL and Pbecome diagonal, when the observed quantities are uncorrelated. Nevertheless,
it is worth mentioning that for correlated observations the cofactor matrix QLL is not always regular, as it
was pointed out by (Ghilani, 2010, p. 160). In such cases the inverse weight matrix cannot be computed
and a specific strategy has to be employed for a solution that is able to deal with singular cofactor matrices,
like for example in the work of Neitzel and Schaffrin (2016).
The theoretical variance of the universal weight8σ2
0has been defined by (Mikhail and Ackermann, 1976, p.
65) as an arbitrary scalar value that influences the stochastic model. Following (Linnik 1961, p. 100), in
case of large measurement samples it can be estimated by
s2
0=1
rd
n
X
i=1
v2
i,redundancy : rd=n−m, (2.18)
with
Es2
0=σ2
0.(2.19)
According to (Niemeier 2008, p. 165), listing all residuals in one vector vand introducing the weight matrix
from equation (2.17), an estimate for the theoretical variance of the unit weight can be computed by
s2
0=vTPv
rd
.(2.20)
2.1.3 Criteria for the solution of adjustment problems
Both types of adjustment that were discussed in the previous subsection resulted in an under-determined
mathematical problem. Infinite possibilities exist for the solution of an adjustment problem, by choosing
a suitable criterion for the residuals. The most common of all is the least squares criterion, also known
as the method of least squares or the minimization of the L2-norm. A least squares estimate ˆxjof the
unknown parameters can be computed by minimizing the sum of squared residuals, formulated symbolically
for equally weighted and uncorrelated observations by the objective function
Ω(v1, v2, . . . , vn) =
n
X
i=1
v2
i→min.(2.21)
Taking into account the individual precisions of the observations, the objective function obtains the form
Ω(v1, v2, . . . , vn) =
n
X
i=1
piv2
i→min,(2.22)
8Also known as a priori variance of the unit weight or reference variance.
16 Chapter 2. Adjustment calculus
or with correlated observations
Ω(v1, v2, . . . , vn) =
n
X
i=1
n
X
j=1
pi,j vivj→min.(2.23)
The objective function can be expressed equivalently in matrix notation as
Ω(v) = vTPv →min.(2.24)
As already mentioned by many authors, for example by Gonin (1989) or Neitzel (2004), different criteria
can be employed for a solution of an adjustment problem, other than least squares. An example of such a
criterion is the minimization of L1norm, also known as the method of least absolute deviations. A solution
for the estimation of the unknown parameters for this case can be obtained by minimizing the sum of
absolute residuals, i.e. the objective function
Ω(v) =
n
X
i=1 |vi| → min.(2.25)
Algorithmic approaches for the solution of this problem can be distinguished between the rigorous solutions
via linear programming or using the Simplex-Algorithm, as presented for example in (Dantzig 1949), (Dantzig
1963) or (Fuchs 1980) and the simulated L1solutions via reweighted least squares adjustment as offered for
example by (Krarup et al., 1980). In general, the minimization of any norm can be employed in this way.
The minimization of Lpnorm, with p ∈Rand p ≥1, can be formulated similarly to the presented objective
functions, see for example (Marx, 2013), by
Ω(v1, v2, . . . , vn) =
n
X
i=1 |vi|p→min.(2.26)
Additionally, the M-estimators are based on the maximum likelihood method and can be employed for a
robust solution of an adjustment problem. Such solutions have been presented for example by Huber (1964)
or Hampel (1980).
2.2 Adjustment of observations with the method of least squares
Amongst all methods (e.g. L1, L2, Lpor M-estimators) for the solution of adjustment problems, the method
of least squares is the most popular. There is a rich literature that deals with adjustment solutions based
on the principles of least squares, for example (Helmert 1924), (Linkwitz 1960), (Linnik 1961), (Deming
1964), (Wells and Krakiwsky 1971), (Bjerhammar 1973), (Krakiwsky 1975), (Meissl 1982), (Perovi´c 2005),
(Niemeier 2008) or (Ghilani 2010).
The method of least squares has been utilized for around two centuries to provide “optimal” solutions
for under-determined algebraic problems that often occur in the mathematical modelling of measurement
results. From a historical perspective, the French mathematician Adrien-Marie Legendre (1752-1833) was
the first who applied this method to the astronomic problem of determining the orbits of comets. In his
book “Nouvelles m´ethodes pour la d´etermination des orbites des com`etes”, that was published in 1805, he
2.2. Adjustment of observations with the method of least squares 17
included an appendix under the title “Sur la m´ethode des moindres quarr´es”, which can be translated into
English as “On the method of least squares”.
Four years later, the famous German geodesist and mathematician Johann Carl Friedrich Gauss (1777-1855)
presented his theory for the calculation of the orbits of celestial bodies in (Gauss 1809). In his work, Gauss
made extensive use of the method of least squares and claimed that he has already been using it since
1795. Although the statement of Gauss that he was the first being using least squares can be judged on
its merits as contradictory (in 1795 Gauss was 18 years old), he is deservedly acknowledged as the pioneer
of the method of least squares, as it has been already discussed by Stigler (1981). Gauss not only used
least squares for the solution of his geodetic problem, but also introduced the statistical distribution of
the errors, as well as the precision of the observed quantities, as an important parameter for obtaining the
most probable estimate of the unknown parameters. In this first work the famous Gaussian (or normal)
distribution has been presented for the first time. Nevertheless, the most important outcome is that both
Legendre and Gauss contributed greatly to the scientific community, with a method for the adjustment of
observations that is widely used in various scientific fields.
A year after Gauss’s first publication, Pierre-Simon Laplace (1749–1827) investigated Gauss’s method of
least squares and the errors distribution utilizing the central limit theorem. The development of the method
of least squares continues with the second publication of Gauss (also known as the 2nd foundation of least
squares), see (Gauss 1823). In his second book Gauss argued that the method of least squares could be
employed also in case of not normally distributed errors. However, in this case the solution is not the most
probable but can be considered as the most appropriate or the most plausible. After that, the application of
least squares met success in various scientific fields and in geodetic science the work of Helmert (1872) became
a standard textbook for the application of this method to geodetic problems with redundant observations.
2.2.1 Statistical formulation of least squares problems
Since the work of Gauss, the method of least squares is related to the normal distribution of the mea-
surement errors, as it has been already pointed out in (Bjerhammar 1973, p. 80). For normally distributed
observations the method of least squares will result in the most probable solution for the unknown quantities
or equivalently in the solution of maximum likelihood. The relationship between least squares and the error
distribution can be put forward with the following simple example adopted from (Petrovi´c et al. 1983):
Example 2.2.1. least squares and normally distributed errors:
An adjustment problem is under investigation, where nobservations land their residuals vare related to
munknown parameters x. Suppose that the measurement errors originate from a universal set of normally
distributed errors9, expressed symbolically by
ei∼N(0, σi).(2.27)
In this line of thinking, the residuals vare also assumed to be normally distributed, vi∼N(0, σi), with
known expectations
E(vi) = 0 (2.28)
9In fact, errors of different measurements (or type of measurements) belong to different normal distributions.
18 Chapter 2. Adjustment calculus
and variances
Ev2
i=σ2
i.(2.29)
The probability density function (or Likelihood function, see Linnik 1961, p. 321 for a similar example) of
each individual residual is then
φ1(v1) = 1
σ1√2πexp −v2
1
2σ2
1,
φ2(v2) = 1
σ2√2πexp −v2
2
2σ2
2,
.
.
.
φn(vn) = 1
σn√2πexp −v2
n
2σ2
n.
(2.30)
Assuming that the observations are uncorrelated and their random errors independent, the vector of residuals
v= [v1, v2, . . . , vn]T,(2.31)
would have the probability density function
L(v1, v2, . . . , vn) = φ1φ2. . . φn=
n
Y
i=1
1
σi√2πexp −v2
i
2σ2
i=1
n
Y
i=1 σi√2πexp −
n
X
i=1
v2
i
2σ2
i!.(2.32)
Expressing the weights for the residuals by
pi=1
σ2
i
,(2.33)
function (2.32) can be reformulated to
L(v1, v2, . . . , vn) =
n
Y
i=1
√pi
√2πnexp −1
2
n
X
i=1
piv2
i!=K1exp −1
2
n
X
i=1
piv2
i!,(2.34)
with the introduced parameter K1being a positive constant. There are infinite solutions for the residuals v,
as well as the unknown parameters x. A solution is required that maximizes the probability density function
L(i.e. the maximum likelihood solution for the unknowns). Thus, for K1>0 it is obvious that
L(v1, v2, . . . , vn)→max ⇔exp −1
2
n
X
i=1
piv2
i!→max.(2.35)
According to (Bronshtein et al. 2005, pp. 49-50), function L(v1, v2, . . . , vn) is strictly monotonically decreas-
ing. It follows
exp −1
2
n
X
i=1
piv2
i!→max ⇔1
2
n
X
i=1
piv2
i→min,(2.36)
i.e.
L(v1, v2, . . . , vn)→max ⇔
n
X
i=1
piv2
i→min.(2.37)
2.2. Adjustment of observations with the method of least squares 19
Thus, it can be said that for uncorrelated observations with normally distributed errors the solution of
maximum likelihood (by maximizing the probability density function) is equivalent to the least squares
solution (by minimizing the sum of weighted squared residuals). Equivalent explanations can be found in
(Merimman 1877, p. 16), (Helmert 1924, pp. 94-98), (Wells and Krakiwsky 1971, p. 93) or (Petrovi´c
et al. 1983). The first of these authors states in a few sentences that “... the most probable values of
quantities, which are the object of measurements, are those which render the sum of the squares of the errors
a minimum.”.
It should be pointed out that the method of least squares can be applied in cases that the errors are not
normally distributed as well. However, the estimated unknown parameters would not be the most probable,
but in some cases can be acceptable or appropriate.
2.2.2 Least squares parameter estimation
A least squares adjustment problem can be seen as an optimization problem, due to the fact that always
the extreme values of an objective function are requested (i.e., for least squares problems the minimum of
the sum of squared residuals). To clearly demonstrate the procedure for obtaining a least squares solution,
a simple adjustment problem will be employed. For example, a linear functional model between a set of n
observations l, their residuals vand the unknown parameters x, which can be expressed by the observation
equations
l1+v1=f1(x1, x2, . . . , xm),
l2+v2=f2(x1, x2, . . . , xm),
.
.
.
ln+vn=fn(x1, x2, . . . , xm),
(2.38)
with fdenoting linear functions of the unknown parameters. Obviously an adjustment problem exists when
the number of observations is larger than the number of unknown parameters (n > m). Such a functional
model occurs in practice, for example when the length of a side of an object has been measured repeatedly
and the length needs to be estimated (in this case only one unknown parameter is requested). Another
adjustment example of such a linear functional model, often presented in geodetic and statistical literature,
assumes that the y-coordinates from a set of npoints in 2D have been measured, while the x-coordinates
are taken as error-free and the parameters of a straight line that fits best to the points are unknown and
need to be estimated. Nevertheless, the adjustment problem will be kept more general here. Solving (2.38)
for the residuals, yields the system of equations
v1=f1(x1, x2, . . . , xm)−l1,
v2=f2(x1, x2, . . . , xm)−l2,
.
.
.
vn=fn(x1, x2, . . . , xm)−ln.
(2.39)
Postulating uncorrelated observations of equal precision, the least squares method leads to the objective
function
Ω(v1, v2, . . . , vn) =
n
X
i=1
v2
i,(2.40)
20 Chapter 2. Adjustment calculus
which after substituting the residuals from (2.39) can be written as
Ω(x1, x2, . . . , xm) =
n
X
i=1
[fi(x1, x2, . . . , xm)−li]2.(2.41)
Estimates of the unknown parameters ˆx1,ˆx2,...,ˆxmare required, that minimize the objective function
(2.41). According to Fermat’s Theorem (Bronshtein et al. 2005, p. 388), the extreme values of a function
can be determined by setting the first derivative with respect to the unknown terms equal to zero. This is a
necessary condition to obtain the stationary points of the objective function Ω but it is not sufficient for the
least squares solution, in the sense that the derived stationary points can still be a maximum, a minimum or
even a saddle point (also called inflection point for cases of one variable). Stationary points of the objective
function (2.41) can be computed by
∂Ω(x1, x2, . . . , xm)
∂x1
=
n
X
i=1
2 [fi(x1, x2, . . . , xm)−li]∂fi(x1, x2, . . . , xm)
∂x1
= 0,
∂Ω(x1, x2, . . . , xm)
∂x2
=
n
X
i=1
2 [fi(x1, x2, . . . , xm)−li]∂fi(x1, x2, . . . , xm)
∂x2
= 0,
.
.
.
∂Ω(x1, x2, . . . , xm)
∂xm
=
n
X
i=1
2 [fi(x1, x2, . . . , xm)−li]∂fi(x1, x2, . . . , xm)
∂xm
= 0,
(2.42)
which is a system of mequations with munknown parameters, known as the system of normal equations10.
To ensure that the computed stationary point of an objective function of several variables is minimum, the
matrix including the second order partial derivatives should be built as
D=
∂2Ω
∂x2
1
∂2Ω
∂x1∂x2··· ···
∂2Ω
∂x2∂x1
∂2Ω
∂x2
2··· ···
.
.
..
.
.....
.
.
.
.
..
.
.. . . ∂2Ω
∂x2
m
=
d11 d12 ··· ···
d21 d22 ··· ···
.
.
..
.
.....
.
.
.
.
..
.
.. . . dmm
.(2.43)
According to (Bronshtein et al. 2005, p. 402), if all subdeterminants of matrix Dare positive (i.e. Dis a
positive definite matrix):
d11 >0,
d11d22 −d12d21 >0,
.
.
.
(2.44)
then it is guaranteed that the computed extreme value of Ω is a minimum. Furthermore, for functions
fi(x1, x2, . . . , xm) in equation (2.38) being linear, the partial derivatives with respect to the unknown pa-
rameters are constants, resulting in linear normal equations in (2.42). The least squares estimates of the
10In (Wells and Krakiwsky 1971, p. 87) an equivalent explanation of the normal equations is presented using matrix notation.
2.2. Adjustment of observations with the method of least squares 21
unknown parameters ˆx1,ˆx2,...,ˆxmcan be computed in this case straightforward. The estimated unknowns
can be used in equation (2.39) to determine the residuals
ˆv1=f1(ˆx1,ˆx2,...,ˆxm)−l1,
ˆv2=f2(ˆx1,ˆx2,...,ˆxm)−l2,
.
.
.
ˆvn=fn(ˆx1,ˆx2,...,ˆxm)−ln,
(2.45)
leading to the adjusted observations
ˆ
l1=l1+ ˆv1,
ˆ
l2=l2+ ˆv2,
.
.
.
ˆ
ln=ln+ ˆvn.
(2.46)
Least squares parameter estimation with constraints
In many cases there are additional constraints between the unknowns that have to be taken into account,
for example a constraint
u(x1, x2, . . . , xm) = 0,(2.47)
has to be enforced to the unknown parameters. The functional model includes not only the observation
equations (2.38), but also the constraint (2.47). Following (Bronshtein et al. 2005, p. 403), the Lagrange
multiplier method can be employed in this case for obtaining a least squares estimate for the unknown
parameters, that minimizes the objective function (2.41). Therefore, by combining Ω(x1, x2, . . . , xm) with
the constraint (2.47), leads to the formulation of the Lagrange function (also known as Lagrangian)
K(x1, x2, . . . , xm, k) =
n
X
i=1
[(fi(x1, x2, . . . , xm)−li)]2−2k(u(x1, x2, . . . , xm)) ,(2.48)
where parameter kdenotes the Lagrange multiplier. The stationary points of K can be obtained by taking
the partial derivatives with respect to all unknown parameters and setting them to zero, which yields the
22 Chapter 2. Adjustment calculus
normal equation system
∂K(x1, x2, . . . , xm, k)
∂x1
=
n
X
i=1
2 [fi(x1, x2, . . . , xm)−li]∂fi(x1, x2, . . . , xm)
x1−2k∂u(x1, x2, . . . , xm)
∂x1
= 0,
∂K(x1, x2, . . . , xm, k)
∂x2
=
n
X
i=1
2 [fi(x1, x2, . . . , xm)−li]∂fi(x1, x2, . . . , xm)
x2−2k∂u(x1, x2, . . . , xm)
∂x2
= 0,
.
.
.
∂K(x1, x2, . . . , xm, k)
∂xm
=
n
X
i=1
2 [fi(x1, x2, . . . , xm)−li]∂fi(x1, x2, . . . , xm)
xm−2k∂u(x1, x2, . . . , xm)
∂xm
= 0,
∂K(x1, x2, . . . , xm, k)
∂k =−2 (u(x1, x2, . . . , xm)) = 0.
(2.49)
The derived system of normal equations would be linear, as long as the functional model of the problem is
also linear (i.e. the observation equations and the constraint are linear). Thus, the least squares solution
for the unknown parameters and the Lagrange multiplier can be obtained straightforward by solving the
normal equations (2.49). Matrix notation can be easily utilized for such solutions of linear least squares
problems. Examples of this procedure using matrices can be found in most common adjustment literature,
like (Koch and Pope 1969), (Wolf 1979), (Wells and Krakiwsky 1971, p. 142ff.), (Mikhail and Ackermann
1976, p. 213ff.) or (Perovi´c, 2005, p. 189ff.).
2.2.3 Definition of linear and nonlinear least squares problems
Before discussing any strategy for solving nonlinear least squares problems, it is necessary to define what
is a linear and what a nonlinear least squares adjustment problem. A thorough analysis of the adjustment
theory can be be found in (Pasioti 2015), which makes clear the different interpretations that exist in geodetic
literature concerning the nature of least squares problems. For example, Teunissen and Knickmeyer (1988)
described the nature of an adjustment in terms of the solution space curvature. Nevertheless, two clear
definitions are given in (Pasioti 2015) regarding linear and nonlinear least squares, which are adopted here
and extended by taking into account the special cases with constraints between the unknown parameters.
Therefore, the following definitions can be stated:
Definition 2.1. Linear least squares problems
A least squares problem is linear, when the observation/condition equations and the additional constraints
are linear both with respect to the unknown parameters and the residuals.
Definition 2.2. Nonlinear least squares problems
A least squares problem is nonlinear, when the observation/condition equations or the additional constraints
are nonlinear with respect to the unknown parameters or the residuals.
Obviously, a linear functional model will lead to linear normal equations, respectively a nonlinear functional
model to nonlinear normal equations.
2.3. Error estimation of adjustment results 23
2.3 Error estimation of adjustment results
In addition to the parameter estimation and the adjustment of the observed quantities using least squares
or some other criterion, it is important to know the precision of the derived parameters in order to verify the
quality of the adjustment results and draw valid conclusions. The target is to calculate how an infinitesimal
change of the observed quantities would affect the unknown parameters. In other words, it is investigated
how the variances and covariances of the observations (l1, l2, . . . , ln) propagate to the unknown parameters
(x1, x2, . . . , xm) and the residuals (v1, v2, . . . , vn). This procedure can be found under the name “error
propagation” or “variance and covariance propagation” and has been presented in various publications, for
example in (Wells and Krakiwsky 1971, p. 20), (Mikhail and Ackermann 1976, p. 76ff.), (Cross 1994, p.
32) or (Niemeier 2008, p. 51ff.). A brief explanation of the error propagation is presented below for two
individual cases, regarding linear and nonlinear functional relationships.
Computation of stochastic parameters in linear functional relationships
Assuming for a moment a linear functional relationship between the unknown parameters and the observa-
tions
x1=f1(l1, l2, . . . , ln),
x2=f2(l1, l2, . . . , ln),
.
.
.
xm=fm(l1, l2, . . . , ln)
(2.50)
and taking into account that functions f1, f2, . . . , fmare linear, the latter equation system can be formulated
equivalently in matrix notation as
X=FL,(2.51)
with vector X= [x1, x2, . . . , xm]Tlisting all the unknown parameters and vector L= [l1, l2, . . . , ln]Tthe
observed quantities. The computation of the standard deviations of the unknown parameters can be ge-
ometrically interpreted as a projection of the error distribution from the measurements on the unknown
parameters. A naive example of such a projection is depicted in Figure 2.1.
Further, applying the expectation operator to equation (2.51) yields
E{X}=E{FL}.(2.52)
As long as matrix Fis deterministic and only Lcan be considered as stochastic, the last equation results in
E{X}=FE{L} ⇒ E{X}=F(L+v),(2.53)
24 Chapter 2. Adjustment calculus
u
n
k
n
o
w
n
p
a
ra
meters
mea
su
remen
ts
Figure 2.1: Simple example of linear variance-covariance propagation
with vector vholding the residuals of the observations in L. According to Koch and Pope (1969), the
theoretical VC matrix of the unknown parameters can be expressed by definition as
ΣXX =En[X−E{X}] [X−E{X}]To
⇒ΣXX =En[FL −(FL +v)] [FL −F(L+v)]To
⇒ΣXX =FEvvT
|{z }
ΣLL
FT
⇒ΣXX =F ΣLL FT.
(2.54)
The last equation is known as the propagation law of variances and covariances and can be found e.g. in
(Wells and Krakiwsky 1971, p. 20) or (Niemeier 2008, p. 56). Introducing the variance of the unit weight
σ2
0, the developed VC matrix can be written equivalently as
ΣXX =σ2
0F QLL FT.(2.55)
2.3. Error estimation of adjustment results 25
with the respective cofactor matrix being defined as
QXX =FQLLFT.(2.56)
Analogously, the variances and covariances of the adjustment results (estimated unknown parameters, resid-
uals and adjusted observations) can be derived by applying the law of error propagation that is presented
above. A detailed formulation of these VC matrices is given in chapter 3, covering various adjustment cases.
Computation of stochastic parameters in nonlinear functional relationships
In practice, nonlinear functional relationships between the unknown parameters and the observed quantities
occur more often than linear ones. For example, assume a nonlinear least squares problem with the estimates
of the unknown parameters being expressed by the nonlinear system of equations
x1=ψ1(l1, l2, . . . , ln),
x2=ψ2(l1, l2, . . . , ln),
.
.
.
xm=ψm(l1, l2, . . . , ln),
(2.57)
with ψ1, ψ2, . . . , ψmdenoting nonlinear functions of the observations. The propagation law of variances and
covariances, that was applied in the linear case, cannot be utilized here. Several solution strategies exist for
obtaining estimates for the variances and covariances of parameters in nonlinear problems. Following the
study of L¨osler et al. (2016), four main procedures (but not only these) can be distinguished:
- First order variance-covariance propagation.
- Second order variance-covariance propagation.
- Monte Carlo simulation (MCS).
- Unscented Transformation (UT).
A detailed explanation of the first procedure is presented in chapter 3 for various adjustment examples.
The analysis of the remaining procedures is out of the scope of this work. Nevertheless, it is important to
discuss some main characteristics of all these cases and point out their main drawbacks that could eventually
mislead to wrong interpretations of the adjustment results and unfortunately lead to wrong conclusions.
The first order variance-covariance propagation is a procedure based on the linear approximation of the
problem and it has been presented e.g. in (Wells and Krakiwsky 1971, p. 21), (Mikhail and Ackermann
1976, pp. 79-81), (Taylor 1982, p. 80), (J¨ager et al. 2005, p. 68), (Niemeier 2008, p. 63-64) or (Ghilani
2010, p. 89). The derived linearized expressions can be utilized for obtaining approximate solutions for the
variances and covariances of the unknown parameters, based on the law of variance-covariance propagation.
Geometrically, the application of the error propagation on a linearized problem can be seen as the projection
of the measurements’ error distribution on the unknown parameters, by a linear approximation of the
“original” nonlinear function. A simple example of this procedure is depicted in Figure 2.2.
26 Chapter 2. Adjustment calculus
u
n
k
n
o
w
n
p
a
ra
meters
mea
su
remen
ts
Figure 2.2: Simple example of a first order variance-covariance propagation
The solution of the error propagation from a linearized function cannot be always trusted, as it is only a
linear approximation of the “real” solution. Therefore, the propagation law of variances and covariances
should not be applied to an arbitrary nonlinear problem, but only in these cases where the linearized
solution is a “good” approximation of the original nonlinear one. This has been already noticed by (Mikhail
and Ackermann, 1976, p. 80), who stated “Although in practical applications linearized functions are used
regularly for the propagation of variances and covariances, it should be pointed out that this is permitted
if the range of dispersion in ˜x1,˜x2is small when linear approximation is compared to the curvature of
the function in the neighborhood of x0
1,x0
2. In other words the function should be approximated well by its
tangent within the region of interest - that is, the region of dispersion of the random variables.”. In the same
line of thinking, L¨osler et al. (2016) considered this approximate solution for the variances and covariances
as “distorted”.
A solution coming from a second order variance-covariance propagation can be also seen as an approximate
solution for the variances and covariances of the unknown parameters, by taking into account the first and
second order terms of the nonlinear functional relationships. The solution coming from this procedure can
be seen as a better approximation than the first order variances-covariances, however, it also needs to be
verified in the same manner as the first order approximation. A numerical example for this approach can be
found in (L¨osler et al. 2016).
2.4. Synopsis of the basics in adjustment calculus 27
The Monte Carlo method is a famous statistical approach that is based on the sequential generation of
statistically random data samples for performing simulations. Some of the authors that employed MCS for
estimating the variances and covariances of adjustment results are Alkhatib (2007), Alkhatib and Schuh
(2007) or L¨osler et al. (2016). The advantage of MCS is that the variances and covariances can be estimated
directly by using the nonlinear functional relationships, in contrast to VC propagation that is restricted to
approximate functional relationships. To demonstrate a solution for the variances and covariances using
MCS, equation (2.57) is expressed in vector form as
X= Ψ(L),(2.58)
with Ψ denoting a vector that lists the nonlinear functions. Following (L¨osler et al. 2016), the expectation
values and the variances and covariances of the unknown parameters Xcan be obtained by means of a MC
simulation, using Nindependent repetitions of a random experiment. The first step involves the generation
of Nrandom samples of the measurements
Lj=L+ ∆Lj,for j= 1, . . . , N (2.59)
with the error vector ∆Lbeing randomly distributed (∆L∼ N(0,ΣLL)), that results in
E{X}=1
N
N
X
j=1
Xj=1
N
N
X
j=1
Ψ(Lj),
ΣXX =1
N
N
X
j=1
En[Xj−E{Xj}] [Xj−E{Xj}]To.
(2.60)
Moreover, the Unscented Transformation is a statistical approach that has been developed in the last decades,
utilized also for estimating variances and covariances from nonlinear functional relationships. It has been
firstly presented by Julier et al. (1995) for filtering nonlinear systems, and later on has been extended in
(Julier and Uhlmann 1996) and (Julier and Uhlmann 2000) for the approximation of distribution functions
and variances-covariances of unknown parameters. This approach involves a sampling strategy of “sigma
points”11 based on the a priori statistical information of the measurements. The derived sigma points are
used in their turn to approximate the distribution functions or variances and covariances of the unknown
parameters, even in cases of nonlinear functional relationships. Julier and Uhlmann (2000) state that
this approximation is similar to a second order approximation, however, without the need of computing
derivatives. A solution from the UT approach has been discussed in the study of L¨osler et al. (2016), as
well.
2.4 Synopsis of the basics in adjustment calculus
It has been shown that the mathematical modelling of redundant measurement results, as well as the
statistical properties of the measurement errors, embody the fundamental parts of every adjustment problem.
11“Sigma points” can be seen as synthetic measurements, that have been generated by adding randomly distributed errors
to the original measurements.
28 Chapter 2. Adjustment calculus
Only a correct mathematical model can lead to meaningful conclusions regarding estimates of unknown
parameters and their standard deviations.
An under-determined system of equations is a consequence of every adjustment of measurements that are
contaminated by errors. Depending on the nature of the errors (e.g. random errors) an appropriate method
is employed by means of a criterion for the residuals, which can lead to “optimal” adjustment results.
A least squares solution for the unknown parameters, the residuals and the adjusted measurements can
be obtained by solving a system of normal equations, which is a result of the minimization of a clearly
defined objective function. The adjustment results can be evaluated in terms of precision and reliability by
computing and interpreting their stochastic parameters. These are statistical measures that can be obtained
by using the rules of error propagation or some other approach, depending on the nature of the problem,
i.e. the functional relationship between the estimated parameters and the measurements.
A definition of linear and nonlinear least squares problems has been presented in this chapter. Linear
functional models lead to linear normal equations and a straightforward solution of the adjustment problem
is possible. However, for nonlinear cases the solution can be more complicated and a specific strategy might
be necessarily followed. In the next chapter various approaches are discussed for the solution of nonlinear
least squares problems. Thus, traditional approaches are covered that have been utilized in geodetic science
for many years and can be found in most standard adjustment textbooks, as well as the most modern
approaches that have been presented by the mathematical/statistical community in the last decades.
29
3 Solutions of nonlinear least squares problems
This chapter summarizes two main strategies for solving nonlinear least squares problems. The first has
been presented in most common geodetic literature and used extensively by geodesists, while the second
has been developed recently by the mathematical and statistical scientific community. Nevertheless, before
any discussion about solving nonlinear adjustment problems with the method of least squares, it would
be advantageous to make clear the viewpoint on the various models and optimization approaches that are
considered in the following.
Nonlinear adjustments occur often not only in geodetic science, but also in geodetic practice. Depending on
the functional relationship between the observed quantities, the unknown parameters and the fixed/constant
parameters, two nonlinear cases are distinguished here:
- The adjustment of nonlinear observation equations (extended to the case with nonlinear constraints);
- The adjustment of nonlinear condition equations with unknown parameters (extended to the case with
nonlinear constraints).
The least squares solution of such adjustment problems can be obtained iteratively involving sometimes a
linearization, or in some specific cases directly. Various optimization approaches exist for the solution of such
problems and can be classified into global optimization or local optimization. Some of those approaches have
been extensively applied in geodetic science, for instance in (Pope 1974), (Madsen et al. 2004) or (Neitzel
and Petrovic 2008), as for example the
- Gauss-Newton,
- Newton-Raphson,
- Levenberg–Marquardt,
- heuristic optimization.
The numerical characteristics of a class of iterative algorithms for the solution of nonlinear least squares
problems, with particular focus on the Gauss-Newton approach, have been investigated in (Teunissen 1985)
and (Teunissen 1990) in terms of differential geometric concepts.
30 Chapter 3. Solutions of nonlinear least squares problems
A comparison between these approaches1is out of the scope of this work and only the Gauss-Newton
approach will be utilized in the following sections. The rest have been mentioned here for the sake of
completeness and will not be analysed further.
An additional algorithmic approach for a class of nonlinear least squares problems has been defined and
presented by (Golub and Van Loan 1980) under the name TLS. In the last three decades many authors dealt
with this approach and various modern algorithms have been developed since then. A thorough analysis of
TLS and its development is provided at a later point in this chapter.
Depending on the nature of the adjustment problem, as well as the chosen optimization strategy, three
individual adjustment models are identified and discussed:
•The Gauss-Markov model (GMM)
•The Gauss-Helmert model (GHM)
•The Errors in variables model (EIV)
Figure 3.1 depicts a diagram with the solutions of nonlinear least squares problems that are discussed in
this chapter.
Class of nonlinear least-
squares problems
Geodetic Solution
Linearized functional model
Gauss-Helmert
model
Iterative solution -
Local optimization
Mathematical Solution
Nonlinear functional model
Direct solution
TLS with SVD -
Global optimization
Iterative solution
WTLS algorithms -
Local optimization
Errors in variables
model
Gauss-Newton approach Total least-squares approach
Gauss-Markov
model
Figure 3.1: Two optimization approaches for the solution of a class of nonlinear least squares problems.
It must be clarified that Gauss-Newton is a general optimization approach and can be employed for finding
the minimum solution of any nonlinear problem. On the other hand the TLS approach can be utilized only
1It is important to mention that all presented approaches are iterative and can be employed for the solution of nonlinear
least squares problems. However, without guaranteeing a convergence necessarily. Thus, it can happen that a solution is not
always possible by a specific approach, or some can provide a solution and some not, depending on the individual adjustment
problem. More information about additional optimization approaches and a thorough analysis of the presented ones can be
found in (Madsen et al. 2004).
3.1. Traditional geodetic solutions 31
for a special class of nonlinear least squares problems. It is, thus, necessary for the scope of this work to
discuss only adjustment problems that have a solution using TLS.
3.1 Traditional geodetic solutions
Solutions of nonlinear least squares problems with the Gauss-Newton approach have been already discussed
in (Wells and Krakiwsky 1971), (Pope 1974), (Niemeier 2008), (Neitzel and Petrovic 2008) or (Neitzel
2010). This involves a linear approximation of the observation/condition equations (as well as the additional
constraint between the unknowns in special cases), thus initial values for the unknown parameters are
necessary. The derived objective function in this case can have one minimum and after an iterative process
reaches a local minimum of the “original” nonlinear problem. The estimated unknown parameters in each
iteration serve as corrections for the initial values of the next iteration. This iterative process continues
until a predefined threshold (or until a condition is met). Thus, the main characteristics of a least squares
solution of an adjustment problem using Gauss-Newton are:
- linear approximation of the nonlinear observation/condition equations,
- linear approximation of the nonlinear constraints between the unknown parameters,
- initial values for the unknown parameters,
- iterative procedure,
- local optimization.
Depending on the individual nonlinear adjustment problem (for example adjustment with observation equa-
tions or adjustment with condition equations), a least squares solution can be obtained within the GMM or
the GHM. Both adjustment models are thoroughly discussed and analysed in the next subsections.
3.1.1 Adjustment with observation equations and constraints
Point of beginning is the functional model that explicitly relates the observations li, the residuals vi(with
i= 1,2, ..., n) and the unknown parameters xj(with j= 1,2, ..., m) and can be expressed by the nonlinear
observation equations 2
l1+v1=φ1(x1, ..., xm),
l2+v2=φ2(x1, ..., xm),
.
.
.
ln+vn=φn(x1, ..., xm),
(3.1)
with φidenoting nonlinear differentiable functions of the unknown parameters xj. Storing the observations
liin a column-vector L, the residuals viin vector vand the nonlinear functions φi(x1, ..., xm) in the formal
vector Φ(X), it is possible to write the system of observation equations (3.1) in vector notation as
L+v=Φ(X).(3.2)
2The linear case of this problem has been discussed in section 2.2.2, expressed by the linear system of equations 2.38 with
fidenoting in that case linear functions.
32 Chapter 3. Solutions of nonlinear least squares problems
Linearization of the functional model
The first step in the Gauss-Newton approach is the linear approximation of the nonlinear functional model.
In case of observation equations, a linearization is performed on the nonlinear differentiable functions
φi(x1, ..., xm). For instance, the first order Taylor series expansion of the formal vector Φ(X) at the point
X0can be taken, which reads
L+v=Φ(X)≈ΦX0+∂Φ(X)
∂XX=X0X−X0,(3.3)
with X0denoting the vector of approximate unknown parameters x0
1, x0
2, ..., x0
m(i.e. initial values are
necessary for approximating X). The partial derivatives of the nonlinear functions in Φ(X) with respect to
the unknown parameters can be expressed equivalently by the Jacobian matrix
Jx=∂Φ(X)
∂XX=X0
=
∂φ1
∂x1
∂φ1
∂x2··· ∂φ1
∂xm
∂φ2
∂x1
∂φ2
∂x2··· ∂φ2
∂xm
.
.
..
.
.....
.
.
∂φn
∂x1
∂φn
∂x2··· ∂φn
∂xm
.(3.4)
A vector of reduced observations is introduced with
l=L−Φ(X0) (3.5)
and the vector of corrections, containing the differences between the unknown parameters (X) to be estimated
and the approximated ones (X0), written as
x=X−X0.(3.6)
Substituting the Jacobian matrix Jx, the vector of reduced observations land the vector of corrections xin
equation (3.3), results in the linearized observation equations
l+v=Jxx.(3.7)
Following (Pasioti 2015), the Jacobian matrix Jxcan be represented symbolically as the design matrix A,
whose elements are the partial derivatives of the nonlinear functions φi(x1, x2, ..., xm) with respect to all
unknown parameters. Thus, the linearized observation equations can be expressed equivalently by
l+v=Ax.(3.8)
3.1. Traditional geodetic solutions 33
Taking into account the precision of the observed quantities, the stochastic model of the problem can be
expressed by the weight matrix P. The mathematical model of this problem represents the well known
GMM, as explained in (Niemeier 2008, p. 137). Assuming normally distributed residuals (v∼N(0,ΣLL)),
the method of least squares will provide the most probable solution for the vector of unknown parameters
and the residuals.
3.1.1.1 Least squares parameter estimation within the GMM
For obtaining a least squares solution of an adjustment problem within the GMM, an appropriate objective
function has to be built and the sum of squared residuals needs to be minimized. In matrix notation the
objective function is
Ω(v) = vTPv →min.(3.9)
Solving for the residual vector in equation (3.8) yields
v=Ax −l,(3.10)
which can be used to reformulate the objective function to
Ω(x)=(Ax −l)TP(Ax −l).(3.11)
A solution for the unknown correction vector xis requested that minimizes Ω(x). Taking the partial deriva-
tives of the objective function with respect to the unknowns and setting the solution to zero yields the
normal equation system
∂Ω(x)
∂xT=∂Ω
∂x1
,∂Ω
∂x2
,··· ,∂Ω
∂xm= 2(ATPAx −ATPl) = 0.(3.12)
Therefore, the least squares estimate for the correction vector is
ˆ
x= (ATPA)−1(ATPl).(3.13)
However, it is worth mentioning that matrix inversion is prone to rounding errors and even the use of
double-precision floating-point format for computation is in many cases insufficient. An error analysis in
matrix computations and alternative direct and iterative solutions for linear systems can be found e.g. in
(Golub and Van Loan 1996) or (Bj¨orck 2015).
According to (Perovi´c 2005, p. 84), the second derivative of the objective function with respect to the
unknowns is ∂2Ω(x)
∂x2= 2(ATPA).(3.14)
As long as the product of matrices ATPA is positive semi-definite, the stationary point (3.13) of the objective
function will be a minimum.
34 Chapter 3. Solutions of nonlinear least squares problems
Iterative solution for the adjustment results
Madsen et al. (2004) explained that a solution for the unknown vector of parameters ˆ
Xcan be obtained
iteratively, by means of a local minimizer ˆ
x. The initial values given in the first iteration step will define the
area where the algorithm starts descending towards the local minimum of the objective function. From all
local minima the global minimum is requested, thus “good” initial values are necessary. In every iteration
step the estimated vector ˆ
Xis utilized as initial approximation for the next iteration
ˆ
X0
i+1 =ˆ
Xi,(3.15)
with idenoting the iteration step. After the termination of the iterative procedure, the final solution for
the vector of corrections ˆ
xis utilized for computing an estimate for the vector of unknown parameters
ˆ
X=ˆ
X0
final +ˆ
xfinal,(3.16)
with the subscript “final” denoting the last iteration step. The solution for the residuals can be obtained by
ˆ
v=Aˆ
xfinal −l(3.17)
and the vector of adjusted observations
ˆ
L=L+ˆ
v.(3.18)
The iterative process can be terminated when adequate break-off conditions (stopping criteria) are fulfilled.
According to (Pasioti 2015), suitable stopping criteria for the iterations can be the following:
1. Computation error:
The element of the vector of corrections ˆ
xwith the maximum absolute value, should become smaller
or at least equal to a predefined threshold :
max|ˆ
x| ≤ . (3.19)
In (Pasioti 2015) it is stated “in practice all computations are performed with software compilers and
the usage of decimal places is translated differently to the computer’s world. Instead significant places
are used. Therefore, a small value for is highly recommended to be chosen”. Moreover, it must be
pointed out that a meaningful choice for the threshold parameter depends on the significant digits
and their position inside the number system (in computer this is a binary system while humans usually
think in decimal system).
2. Linearization error:
The maximum absolute difference between the elements of the estimated vector ˆ
L(linearized problem)
and vector Φ(ˆ
X) (“original” nonlinear problem), should become smaller or equal to a predefined value
δ:
max|ˆ
L−Φ(ˆ
X)| ≤ δ. (3.20)
3.1. Traditional geodetic solutions 35
For the second criterion, a value for δshould be chosen as close to zero as possible. This threshold
value ensures that the linear approximation of the functional model has been performed correctly, as
explained in (Pasioti 2015) in a few sentences: “The linearisation error is a safeguard against wrong
linearisation so that the solution of the linearised problem is also the solution of the original nonlinear
problem. If the tolerance criterion is not met then the linearisation is inconsistent.”
3.1.1.2 Error estimation within the GMM
The precision of the estimated unknown parameters in ˆ
xcan be expressed by the VC matrix Σˆ
Xˆ
X, as ex-
plained in (Niemeier 2008, p. 272). Approximate solutions for the stochastic properties of the unknown
parameters in a nonlinear adjustment problem can be obtained in terms of a first order variance-covariance
propagation. This procedure is based on the utilization of the linearized functional model and the employ-
ment of the propagation law of variances and covariances that was discussed in section 2.3.
Stochastic properties of the estimated unknown parameters
Point of beginning is the linear functional relationship between the vector of corrections xand the vector of
reduced observations l:
ˆ
x= (ATPA)−1ATP l.
This equation can be equivalently formulated as
ˆ
x=F l,(3.21)
after introducing matrix
F= (ATPA)−1ATP.(3.22)
Following (Niemeier 2008, p. 140), only lis a stochastic parameter, while Fcan be taken as “fixed” or
as deterministic. Utilizing the propagation law of variances and covariances from section 2.3, the cofactor
matrix of ˆ
xcan be computed here by
Qˆxˆx =F Qll FT
⇒Qˆxˆx =ATPA−1ATP Qll ATPA−1ATPT
⇒Qˆxˆx =ATPA−1ATP Qll
|{z}
In
PA ATPA−1
⇒Qˆxˆx =ATPA−1ATPA ATPA−1
|{z }
Im
⇒Qˆxˆx = (ATPA)−1.
(3.23)
36 Chapter 3. Solutions of nonlinear least squares problems
In case of large measurement samples the a posteriori variance of the unit weight
s2
0=vTPv
rd
,with redundancy : rd=n−m, (3.24)
converges stochastically to σ2
0, with
Es2
0=σ2
0(3.25)
and the VC matrix for the estimated unknown parameters can be computed by
Σˆxˆx =s2
0Qˆxˆx.(3.26)
Finally, a first order approximate solution for the variances and covariances of the estimated unknown
parameters ˆ
Xcan be derived as
Σˆ
Xˆ
X=Σˆxˆx and Qˆ
Xˆ
X=Qˆxˆx.(3.27)
Stochastic properties of the residuals and the adjusted observations
Approximate cofactor and VC matrices for the adjusted observations and the computed residuals can be
derived following the same line of thinking as in (Niemeier 2008, p. 141). Making use of the linearized
functional model (3.8) it is possible to express the adjusted reduced observations by
ˆ
l=l+ˆ
v=Aˆ
x,(3.28)
with the respective cofactor matrix being
Qˆ
lˆ
l=AQˆxˆxAT.(3.29)
Reformulating appropriately equation (3.28) results in
ˆ
v=Aˆ
x−l
⇒ˆ
v=AQˆxˆxATPl −l
⇒ˆ
v=AQˆxˆxATP−Inl.
(3.30)
Thus, the cofactor matrix of the residuals can be computed by
Qˆvˆv =AQˆxˆxATP−InQll AQˆxˆxATP−InT,(3.31)
which after some examination can be simplified to
Qˆvˆv =Qll −Qˆ
lˆ
l.(3.32)
3.1. Traditional geodetic solutions 37
3.1.1.3 Least squares parameter estimation within the GMM with constraints
In this case a set of constraints between the unknown parameters has to be taken into account in the
functional model. If these constraints are represented by nonlinear functional relationships, then they have
to be linearized together with the observation equations for a solution using the Gauss-Newton approach.
For example, a number of ncconstraints are enforced in the adjustment problem, which can be expressed
by the system of nonlinear equations
ψ1(x1, x2, ..., xm)=0,
ψ2(x1, x2, ..., xm)=0,
.
.
.
ψnc(x1, x2, ..., xm)=0.
(3.33)
Listing the nonlinear functions ψ1, ψ2, . . . , ψncin the formal vector Ψ(X), it is possible to write the system
of constraints in vector notation
Ψ(X) = 0.(3.34)
The first order Taylor series expansion of Ψ(X) at the point X0reads
Ψ(X)≈Ψ(X0) + ∂Ψ(X)
∂XX=X0
(X−X0).(3.35)
The partial derivatives of the constraint functions ψwith respect to the unknown parameters can be repre-
sented by a Jacobian matrix
Jc=∂Ψ(X)
∂XX=X0
=
∂ψ1
∂x1
∂ψ1
∂x2··· ∂ψ1
∂xm
∂ψ2
∂x1
∂ψ2
∂x2··· ∂ψ2
∂xm
.
.
..
.
.....
.
.
∂ψnc
∂x1
∂ψnc
∂x2··· ∂ψnc
∂xm
.(3.36)
Introducing the vector of corrections x=X−X0from equation (3.6), the Jacobian matrix Jcand the
vector of misclosures
w=Ψ(X0) (3.37)
into equation (3.34), yields the system of linearized constraints
Jcx+w= 0.(3.38)
The Jacobian Jccan be regarded as a design matrix, with its elements being the linear approximations of
the nonlinear functions in (3.33) with respect to all unknown parameters. Denoting Jcby C, equation (3.38)
38 Chapter 3. Solutions of nonlinear least squares problems
can be written as
C x +w= 0.(3.39)
The mathematical model in this case can be regarded as a GMM with constraints between the unknown
parameters. A definition can be found also in (Perovi´c 2005, p. 189).
In the special case that constraints are imposed on the unknown parameters, a least squares estimate can be
acquired with the method of Lagrange multipliers, as it was explained in subsection 2.2.2. Thus, combining
the objective function (3.9) and the constraints (3.39) yields the Lagrangian
K(v,x,k) = vTPv + 2kT(C x +w)→min,(3.40)
which can be expressed equivalently as
K(x,k)=(Ax −l)TP(Ax −l)+2kT(C x +w).(3.41)
The auxiliary vector kholds the Lagrange multipliers. A least squares solution for vectors xand kis required
that minimizes the developed Lagrange function. Taking the partial derivatives of K(x,k) with respect to
all unknowns and setting the solution to zero yields the system of normal equations
∂K
∂xT= 2(ATPAx −ATPl +CTk)=0
⇒ATPAx +CTk=ATPl,
(3.42)
∂K
∂kT=−2(Cx +w) = 0
⇒Cx =−w.
(3.43)
Equations (3.42) and (3.43) can be combined with the block matrices
"ATPA CT
C 0 #" x
k#="ATPl
−w#.(3.44)
The least squares estimate for the vector of corrections and the vector of Lagrange multipliers is
"ˆ
x
ˆ
k#="ATPA CT
C 0 #−1"ATPl
−w#.(3.45)
According to (Niemeier 2008, p. 265), an equivalent solution can be obtained in case of nonsingular products
[(ATPA)] and [CT(ATPA)−1C] by
"ˆ
x
ˆ
k#="Q11 Q12
Q21 Q22 #" ATPl
−w#,(3.46)
3.1. Traditional geodetic solutions 39
with the respective quantities
Q22 =−C(ATPA)−1CT−1,
Q12 =−Q22C(ATPA)−1,
Q21 =QT
12,
Q11 = (ATPA)−1(Im−CTQ12).
(3.47)
Imis an identity matrix, with the subscript mdenoting the number of unknown parameters and specifies the
dimensions of the identity matrix. The least squares estimate for the vector of corrections can be explicitly
expressed by
ˆ
x=Q11(ATPl)−Q12w(3.48)
and the vector of Lagrange multipliers
ˆ
k=Q21(ATPl)−Q22w.(3.49)
A local minimizer of the Lagrange function (3.41) can be obtained iteratively. Equations (3.16), (3.17) and
(3.18) can be further employed to compute the unknown parameters ˆ
X, the residuals ˆ
vand the vector of
adjusted observations ˆ
L.
3.1.1.4 Error estimation within the GMM with constraints
Approximate error estimates for the parameters in ˆ
xcan be derived by making use of the linearized functional
relationship
ˆ
x=Q11(ATPl)−Q12w.(3.50)
Applying the propagation law of variances and covariances to the last equation, the cofactor matrix for the
unknown parameters can be found after some investigation in
Qˆxˆx =Q11.(3.51)
In this case the number of constraint equations must be taken into account when computing the redundancy
of the problem, with the estimated variance of the unit weight
s2
0=vTPv
rd
,with redundancy : rd=n−m+nc.(3.52)
Assuming that s2
0converges stochastically to σ2
0, the VC matrix for ˆ
xis
Σˆxˆx =s2
0Qˆxˆx,(3.53)
Furthermore, the VC and cofactor matrices of the estimated unknown parameters ˆ
X, the adjusted observa-
tions ˆ
Land the residuals ˆv are equivalent to those of section 3.1.1.2.
40 Chapter 3. Solutions of nonlinear least squares problems
3.1.2 Adjustment with condition equations and constraints
In this section a nonlinear functional model is under consideration, that implicitly relates the observations
li, their residuals vi(with i= 1, ..., n) and the unknown parameters xj(j= 1, ..., m) with the condition
equations
φ1(l1+v1, ..., ln+vn, x1, ..., xm) = 0,
φ2(l1+v1, ..., ln+vn, x1, ..., xm) = 0,
.
.
.
φr(l1+v1, ..., ln+vn, x1, ..., xm)=0.
(3.54)
while φ1, φ2, ..., φrare nonlinear differentiable functions of the unknown parameters and the residuals. This
system of condition equations is expressed equivalently in matrix notation as
Φ(X,L+v) = 0,(3.55)
with the formal vector Φholding the nonlinear functional relationship between the vector of observations
L, the vector of residuals vand the vector of unknown parameters X.
Following the Gauss-Newton approach, a linear approximation of the nonlinear condition equations has to
be introduced. Here it is important to mention that a correct linearization involves an approximation of
both the unknown parameters xj0and the unknown residuals vi0. This type of linearization leads to the
rigorous solution of the nonlinear adjustment problem, as it has been already examined in (Pope 1972),
(Lenzmann and Lenzmann 2004) and demonstrated on a practical example by Neitzel and Petrovic (2008).
In the latter contributions has been shown very clearly which terms when neglected will produce merely
approximate formulas for the linearized problem3that yield and unusuable solution. Unfortunately, these
approximate formulas can be found in many popular textbooks on adjustment calculus. For more details
please refer to (Neitzel 2010).
A rigorous solution within the GHM is presented here for a combined adjustment problem, according to the
remarks of (Lenzmann and Lenzmann 2004) and (Neitzel 2010). The first order Taylor series approximation
of the formal vector Φ(X,L+v) at the point X0and v0reads
Φ(X,L+v)≈Φ0(X0,L+v0) + ∂Φ(X,L+v)
∂XX=X0,v=v0
(X−X0)
+∂Φ(X,L+v)
∂vX=X0,v=v0
(v−v0).
(3.56)
A linear approximation of the condition equations (3.55) can be expressed by
Φ0(X0,L+v0) + ∂Φ(X,L+v)
∂XX=X0,v=v0
(X−X0) + ∂Φ(X,L+v)
∂vX=X0,v=v0
(v−v0) = 0.(3.57)
3This approximate linearization has been introduced in standard adjustment textbooks in the past, for example in (Helmert,
1924, pp. 171-174) and could lead to simpler algebraic equations for the approximate solution of the nonlinear adjustment
problem without iterating (i.e. the approximate solution of the problem was obtained after one iteration).
3.1. Traditional geodetic solutions 41
Forming a first Jacobian matrix that contains the partial derivatives of the condition equations with respect
to the unknown parameters
Jx=∂Φ(X,L+v)
∂XX=X0,v=v0
=
∂φ1
∂x1
∂φ1
∂x2··· ∂φ1
∂xm
∂φ2
∂x1
∂φ2
∂x2··· ∂φ2
∂xm
.
.
..
.
.....
.
.
∂φr
∂x1
∂φr
∂x2··· ∂φr
∂xm
(3.58)
and a second that contains the partial derivatives of the condition equations with respect to the residuals
Jv=∂Φ(X,L+v)
∂vX=X0,v=v0
=
∂φ1
∂v1
∂φ1
∂v2··· ∂φ1
∂vn
∂φ2
∂v1
∂φ2
∂v2··· ∂φ2
∂vn
.
.
..
.
.....
.
.
∂φr
∂v1
∂φr
∂v2··· ∂φr
∂vn
(3.59)
and introducing them into equation (3.57) results in
Φ0(X0,L+v0) + Jx(X−X0) + Jv(v−v0) = 0.(3.60)
The Jacobians Jxand Jvcan be regarded as design matrices. Denoting Jxby Aand Jvby B, this linearized
equation system can be equivalently written as
Φ0(X0,L+v0) + A(X−X0) + B(v−v0) = 0.(3.61)
Introducing the vector of misclosures
w=Φ0(X0,L+v0)−Bv0,(3.62)
and the vector of corrections x=X−X0into equation (3.61), yields the linearized functional model
Bv +Ax +w=0.(3.63)
The combination of the developed linearized functional model together with the stochastic model for the
observed quantities results in the famous Gauss-Helmert model. An equivalent definition of this model can
42 Chapter 3. Solutions of nonlinear least squares problems
be found in various textbooks and publications in geodetic literature, like (Wolf 1978), (Lenzmann and
Lenzmann 2004), (Perovi´c 2005, p. 203), (Neitzel and Petrovic 2008) or (Neitzel 2010).
3.1.2.1 Least squares parameter estimation within the GHM
A least squares solution for the unknown corrections xcan be estimated by minimizing the objective function
Ω(v) = vTPv.(3.64)
Due to the implicit functional relationship between the parameters of this adjustment case, a Lagrangian
K(x,v,k) = vTPv −2kT(Bv +Ax +w) (3.65)
can be formed. Vector kis the vector of Lagrange multipliers. Computing the partial derivatives of K with
respect to all unknown parameters and setting the solution to zero yields the stationary points
∂K
∂vT= 2Pv −2BTk=0
⇒v=QllBTk,(3.66)
∂K
∂xT=−2kTA=0
⇒ATk=0,(3.67)
∂K
∂kT=−2 (Bv +Ax +w) = 0.(3.68)
Inserting the residual vector from equation (3.66) into (3.68) gives
BQllBTk+Ax +w=0.(3.69)
Combining equations (3.67) and (3.69) with the block matrices
BQllBTA
AT0
k
x
=
−w
0
,(3.70)
the solution for the unknown parameters is obtained by
ˆ
k
ˆ
x
=
BQllBTA
AT0
−1
−w
0
.(3.71)
3.1. Traditional geodetic solutions 43
Under the condition that the product [BQllBT] is not singular, the last equation can be expressed by
ˆ
k
ˆ
x
=
Q11 Q12
Q21 Q22
−w
0
,(3.72)
as it has been presented in (Niemeier 2008, p. 177), with the respective quantities
Q22 =−AT(BQllBT)−1A−1,
Q12 =−(BQllBT)−1AQ22,
Q21 =QT
12,
Q11 = (BQllBT)−1(In−AQ21).
(3.73)
Explicit expressions for the vector of corrections and the vector of Lagrange multipliers are
ˆ
x=−Q21w=−hATBQllBT−1Ai−1ATBQllBT−1w(3.74)
and
ˆ
k=−Q11w=−BQllBT−1(Aˆ
x+w).(3.75)
A least squares solution for the vector of unknown parameters can be computed iteratively by
ˆ
Xi=ˆ
xi+X0(3.76)
and approximate estimates for the residuals by
ˆ
vi=QllBTˆ
ki,(3.77)
with iindicating the iteration step. Due to the iterative procedure of the Gauss-Newton approach, a local
minimizer ˆ
xfor the Lagrange function K(x,v,k) will be estimated by a series of adjustments within the
GHM. Two vectors have to be updated in each iteration step in this case. The solution for the vector
containing the unknown parameters ˆ
Xwill be introduced as the vector of initial values for the unknown
parameters in the next iteration
ˆ
X0
i+1 =ˆ
Xi,
and the computed residual vector will be used as an initial residual vector
v0
i+1 =ˆ
vi.(3.78)
44 Chapter 3. Solutions of nonlinear least squares problems
A solution can be obtained after the fulfillment of the two stopping criteria (break-off conditions), according
to equations (3.19) and (3.20). The final estimated vector of corrections can be utilized for computing the
vector of unknown parameters
ˆ
X=ˆ
X0
final +ˆ
xfinal,(3.79)
the residuals
ˆ
vfinal =QllBTˆ
kfinal,(3.80)
and the adjusted observations
ˆ
L=L+ˆ
vfinal,(3.81)
with the subscript “final” denoting the last iteration step.
3.1.2.2 Error estimation within the GHM
Point of beginning for the error estimates in the case of a GHM is the linearized functional relationship
between the estimated parameters ˆ
xand the vector of misclosures wof equation (3.74), written as
ˆ
x=−hATBQllBT−1Ai−1ATBQllBT−1w.
Following (Niemeier 2008, p. 178), it is necessary to derive an expression for the vector of misclosures was
a linear function of the vector of the observations. Therefore, taking a first order Taylor approximation of
wfrom equation (3.62) with respect to the vector of observations L, results in
w=Φ0(X0,L0+v0)
|{z }
0
+∂w
∂LL=L0
(L−L0).(3.82)
Forming a Jacobian matrix that contains the partial derivatives of wwith respect to the observations
Jw=∂w
∂LL=L0
(3.83)
and by introducing a vector of reduced observations
l=L−L0(3.84)
in equation (3.82), returns
w=Jwl.(3.85)
At the last step of the iterative procedure, i.e. the final results for the unknown parameters ˆ
xfinal, it can be
shown that the elements of the Jacobian matrix Jwwill be equal to the elements of matrix Bfrom equation
3.1. Traditional geodetic solutions 45
(3.63). The vector of misclosures can be related linearly with the reduced vector of observations, with
w=B l,(3.86)
which can be introduced in (3.74) to derive
ˆ
x=−hATBQllBT−1Ai−1ATBQllBT−1B l,
ˆ
x=F l.
(3.87)
The auxiliary matrix Fcan be defined in this case as
F=−hATBQllBT−1Ai−1ATBQllBT−1B.(3.88)
Assuming that lhas the same stochastic properties as L, the former can be regarded as the only stochastic
parameter in equation (3.87).The cofactor matrix for the vector of corrections is
⇒Qˆxˆx =F Qll FT
⇒Qˆxˆx =hATBQllBT−1Ai−1ATBQllBT−1B Qll BT
|{z }
InBQllBT−1AhATBQllBT−1Ai−1
⇒Qˆxˆx =hATBQllBT−1Ai−1ATBQllBT−1A
|{z }
ImhATBQllBT−1Ai−1
⇒Qˆxˆx =hATBQllBT−1Ai−1=Q22.
(3.89)
The estimated variance of the unit weight can be computed in this case by
s2
0=vTPv
rd
,with redundancy : rd=r−m, (3.90)
or using the expression for the residuals from equation (3.66), by
s2
0=kTBQllPv
rd
=kTBv
rd
=−kT(Aˆ
x+w)
rd
.(3.91)
The VC matrix for the vector of corrections is
Σˆxˆx =s2
0Qˆxˆx.(3.92)
46 Chapter 3. Solutions of nonlinear least squares problems
The residual vector and the vector of adjusted observations can be expressed as functions of the observed
quantities. Introducing kand wfrom equations (3.75) and (3.86) into equation (3.66) results in
ˆ
v=QllBTk=−QllBTQ11w=−QllBTQ11Bl (3.93)
and
ˆ
l=l+ˆ
v=In−QllBTQ11Bl.(3.94)
Following the same line of reasoning as (Niemeier 2008, p. 179), the required cofactor of the residuals can
be found from
Qˆvˆv =QllBTQ11BQll (3.95)
and the cofactor for the adjusted observations from
Qˆ
lˆ
l=Qll −Qˆvˆv =Qll In−BTQ11BQll.(3.96)
Finally, the necessary VC and cofactor matrices of the adjustment results can be computed as
Σˆ
Xˆ
X=Σˆxˆx ,Qˆ
Xˆ
X=Qˆxˆx
and
Σˆ
Lˆ
L=Σˆ
lˆ
l,Qˆ
Lˆ
L=Qˆ
lˆ
l.
3.1.2.3 Least squares parameter estimation within the GHM with constraints
An adjustment with condition equations and constraints between the unknown parameters can be treated
like the one presented in subsection 3.1.1.4. For constraints that are represented by nonlinear functional
relationships, a linear approximation will result in the linearized constraint equations (3.39), which are
expressed in this subsection by
Cx +w2=0,(3.97)
with the vector of misclosures w2for the constraints. The linearized condition equations (3.63) and the
constraints (3.97) represent the linearized functional model of this adjustment problem. Taking into account
the stochastic information of the observed quantities, results in the case of a GHM with constraints. A least
squares solution can be found by minimizing the Lagrange function
K(x,v,k1,k2) = vTPv −2kT
1(Bv +Ax +w1)−2kT
2(Cx +w2)→min,(3.98)
with the vectors of Lagrange multipliers k1and k2and the vector of misclosures w1for the linearized
condition equations 4. From the standard procedure for obtaining a least squares estimate for the unknowns,
4The subscript “1” is introduced here to diferentiate with the vector of misclosures w2for the linearized constraint equations.
3.1. Traditional geodetic solutions 47
the partial derivatives of K(x,v,k1,k2) with respect to all unknown parameters are computed and set equal
to zero:
∂K
∂vT= 2 Pv −BTk1=0
⇒v=QllBTk1,(3.99)
∂K
∂xT=−2k1TA+k2TC=0
⇒ATk1+CTk2=0,(3.100)
∂K
∂k1T=−2 (Bv +Ax +w1) = 0,(3.101)
∂K
∂k2T=−2 (Cx +w2) = 0.(3.102)
Introducing the vector of residuals from equation (3.99) into (3.101) yields
BQllBTk1+Ax +w1=0.(3.103)
Equations (3.100), (3.102) and (3.103) can be expressed as the block matrices
BQllBTA 0
AT0 CT
0 C 0
k1
x
k2
=
−w1
0
−w2
,(3.104)
with the least squares solution for the unknown parameters being obtained by
ˆ
k1
ˆ
x
ˆ
k2
=
BQllBTA 0
AT0 CT
0 C 0
−1
−w1
0
−w2
.(3.105)
For an equivalent solution of the problem, the matrices
R=BQllBT(3.106)
and
M=−ATBQllBT−1A=−ATR−1A(3.107)
48 Chapter 3. Solutions of nonlinear least squares problems
can be introduced. If matrix Ris regular, then the vector of Lagrange multipliers k1is
k1=−R−1(Ax +w1) (3.108)
which can be substituted in equation (3.100), that yields
−ATR−1(Ax +w1) + CTk2=0
⇒Mx −ATR−1w1+CTk2=0.
(3.109)
A least squares solution can be obtained by expressing equations (3.102) and (3.109) as
M CT
C 0
x
k2
=
ATR−1w1
−w2
,(3.110)
resulting in
ˆ
x
ˆ
k2
=
M CT
C 0
−1
ATR−1w1
−w2
.(3.111)
For a regular Mmatrix, the last equation system can be equivalently written as
ˆ
x
ˆ
k2
=
Q11 Q12
Q21 Q22
ATR−1w1
−w2
,(3.112)
with the respective matrices being
Q22 =−hCMTM−1MTCTi−1
,
Q12 =MTM−1MTCTQ22,
Q21 =QT
12,
Q11 =MTM−1MT(Im−CTQ12).
(3.113)
Thus, the vector of corrections can be expressed explicitly by
ˆ
x=Q11(ATR−1w1)−Q12w2(3.114)
and the vector of Lagrange multipliers
ˆ
k2=Q21 ATR−1w1−Q22w2.(3.115)
3.2. Total least squares 49
An iterative procedure is necessary also in this adjustment case. A local minimizer of the Lagrange function
(3.98) is derived, which results in the least squares solution for the unknown parameters ˆ
X, the residuals
ˆ
vand the vector of adjusted observations ˆ
L, similarly to the discussed adjustment cases of the previous
subsections.
3.1.2.4 Error estimation within the GHM with constraints
Starting point is the solution for the vector of corrections within the GHM with constraints
ˆ
x=Q11(ATR−1w1)−Q12w2.
Introducing the same concept as in section 3.1.2.2, the vector of misclosures w1can be related linearly with
the vector of observed quantities lby
w1=B l.
Thus, by substituting w1into (3.114) gives
ˆ
x=Q11(ATR−1Bl)−Q12w2.(3.116)
Applying the propagation law of vaiances and covariances and after some investigation, the cofactor matrix
for ˆ
xcan be computed by
Qˆxˆx =Q11.(3.117)
Taking into account the number of constraint equations for computing the redundancy of the problem, the
estimated variance of the unit weight is
s2
0=vTPv
rd
,with redundancy : rd=r−m+nc.(3.118)
The VC matrix for the corrections is
Σˆxˆx =s2
0Qˆxˆx.(3.119)
Furthermore, the VC and cofactor matrices of the estimated unknown parameters ˆ
X, the adjusted observa-
tions ˆ
Land the residuals ˆ
vcan be computed as in section 3.1.2.2.
3.2 Total least squares
Modern and sophisticated algorithms have been presented by the mathematical community since the 1980s,
for the solution of nonlinear adjustments. These algorithms deal with a class of nonlinear least squares
problems, which can be expressed within an EIV model and solved by TLS. A solution coming from TLS
does not involve any kind of linearization of the functional model but presupposes the use of SVD, as it
was defined in (Golub and Van Loan 1980) or (Van Huffel and Vandewalle 1991, p. 33 ff.). Thus, a TLS
solution is obtained by computing the roots of a polynomial (i.e. by solving the characteristic equation of
50 Chapter 3. Solutions of nonlinear least squares problems
the eigenvalues) and a direct solution can be possible depending on the polynomial’s degree. Such solutions
have been presented in the literature when postulating, in most cases, equally weighted and uncorrelated
measurements.
Various approaches and algorithms have been implemented for the solution of this class of nonlinear least
squares problems when different precision is associated with each measurement. The solutions from these
algorithms are iterative, they do not include a linearization of the functional model and have been published
under the name WTLS. For example, Schaffrin and Wieser (2008) presented a WTLS solution for linear
regression, which inspired Shen et al. (2011), Fang (2011), Amiri-Simkooei and Jazaeri (2012) and Mahboub
(2012) to present modern WTLS algorithms. Despite the name TLS, in all above cases the solution has been
obtained iteratively and does not follow the definition that was established by Golub and Van Loan (1980),
i.e. direct solution using SVD. A clear overview of these type of algorithmic solutions has been presented
in (Snow 2012), covering also the special cases of cofactor matrices being singular. In that work the term
TLS has been used in a more general sense, as it was implied by the following statement“the terms TLS and
TLS solution as used in this dissertation will mean the least squares solution within the EIV model without
linearization”.
Two different perspectives can be distinguished for the terms TLS and WTLS solution from the discussion
above. The first follows the definition of Golub and Van Loan (1980), and the second that of Snow (2012).
In this dissertation the term TLS will refer to the former and the term WTLS to the latter definition. Thus,
the main characteristics of a least squares solution of an adjustment problem within the EIV model will be
distinguished here by
the TLS approach:
- postulating equally weighted and uncorrelated observations,
- treatment of the nonlinear adjustment problem,
- direct solution derived by SVD,
- global optimization,
the WTLS approach:
- postulating individually weighted and correlated/uncorrelated observations,
- reduction of the derived normal equations,
- iterative solution,
- local optimization.
3.2.1 Nonlinear adjustments within the EIV model
In this section, the modelling of nonlinear least squares problems within the EIV model will be introduced.
Therefore, a nonlinear functional model that implicitly relates the observations li, the residuals vi(with
i= 1,2, ..., n) and the unknown parameters xj(with j= 1,2, ..., m) is under consideration. Similarly to
3.2. Total least squares 51
equation (3.54), the discussed functional model can be written as the system of nonlinear condition equations
φ1(l1+v1, ..., ln+vn, x1, ..., xm) = 0,
φ2(l1+v1, ..., ln+vn, x1, ..., xm) = 0,
.
.
.
φr(l1+v1, ..., ln+vn, x1, ..., xm) = 0.
(3.120)
with φidenoting nonlinear differentiable functions of the unknown parameters and the residuals. For a
certain class of such nonlinear adjustment problems5it is possible to formulate this equation system in
matrix notation by the functional model
L+vL= (A+VA)X,
dim(A) = n×m,
rank(A) = m < n,
(3.121)
where Land vLare the vectors of observations and their residuals6, respectively. Matrix Acontains the
coefficients of the functional model with respect to the unknown parameters xj, except the residuals vi
which are stored in the residual matrix VA.Xis the vector containing the unknown parameters. Here, it
is worth mentioning the differences, by definition, of the observation vector Lwith the vector of reduced
observations land the vector of unknown parameters Xwith the vector of corrections x, as it has already
been explained in section 3.1.1. The presented functional model in (3.121), accompanied by the stochastic
model of the measurements, leads to the nonlinear mathematical model known as “Errors In Variables”. A
definition of the EIV model can be found alternatively in (Golub and Van Loan 1980), (Bickel and Ritov
1987), (Van Huffel and Vandewalle 1989) or (Van Huffel and Vandewalle 1991, p. 5).
In contrast to the classical representation of the functional model of an adjustment problem (see for example
the GMM or the GHM), the latter formulation involves a coefficient matrix Athat includes measured
quantities that are under the influence of random errors. Therefore, the necessary residuals are added to the
measurements, symbolized in A, by means of the residual matrix VA. It is of course not the elements (or
the variables) of matrix Athat are subject to errors, but the measurements that are symbolized by these
elements of A. The following simple example illustrates this type of functional modelling:
Example 3.2.1. Assuming that the coordinates of a set of points in 2D have been observed in both
directions (i.e. in xand ydirection). The question is how to fit a straight line to the observed points. The
simplest representation of a straight line in plane is
y=a x +b. (3.122)
5These adjustment problems can be solved using a traditional geodetic approach, which involved a linearization of the
functional model that resulted in the formulation of the GHM and an iterative procedure for obtaining a least squares solution.
6For notation reasons residuals and residual vectors are introduced here and not the notion of errors and error vectors as
it is usual in the TLS literature, see e.g. (Golub and Van Loan 1980, Van Huffel and Vandewalle 1989).
52 Chapter 3. Solutions of nonlinear least squares problems
Adding the necessary residuals to the measurements yields
y1+vy1=a(x1+vx1) + b,
y2+vy2=a(x2+vx2) + b,
.
.
.
yn+vyn=a(xn+vxn) + b.
(3.123)
yand xare the observed coordinates of the 2D points, with their corresponding residuals denoted by vyand
vx.aand bare the unknown line parameters and need to be estimated. This nonlinear equation system can
be formulated equivalently by the EIV model (3.121):
L=
y1
y2
.
.
.
yn
,vL=
vy1
vy2
.
.
.
vyn
,X="a
b#,A=
x11
x21
.
.
.
xn1
,VA=
vx10
vx20
.
.
.
vxn0
.(3.124)
In this adjustment example it can be easily seen that the elements of the “design matrix” Asymbolize some
of the measured quantities of the adjustment problem, that are subject to errors and thus the corresponding
residuals are listed in matrix VA. Therefore, the term/name “Errors in Variables” is misleading. Always,
the measured quantities are subject to errors and not the variables of a matrix.
It is necessary to point out that the stochastic model within the EIV model, involves both the stochastic
properties of the measurements in vector L, as well as in the design matrix A. Thus, an appropriate weight
matrix can be described by
P="PLPLA
PAL PA#,(3.125)
with the cofactor matrix
QLL ="QLQLA
QAL QA#(3.126)
and the variance-covariance matrix
ΣLL ="ΣLΣLA
ΣAL ΣA#.(3.127)
For uncorrelated observations the terms off the diagonal in matrices P,QLL and ΣLL become zero. In case of
normally distributed errors the most probable solution for the undetermined parameters of this adjustment
problem can be obtained by employing the method of least squares. Therefore, two individual solution
strategies are presented in the following sections regarding adjustment problems that can be expressed by
an EIV model. The first is direct and presumes the use of an orthogonal decomposition (TLS), while the
second is iterative but without involving any kind of linearization (WTLS).
3.2. Total least squares 53
3.2.1.1 Least squares parameter estimation using TLS
By the definition of TLS (Golub and Van Loan 1980), (Van Huffel and Vandewalle 1991, p. 33), a solution
of an adjustment problem within the EIV model is based on the minimization of the objective function
|| [VA,vl]||F=|| ˆ
A,ˆ
L]−[A,L]||F→min,(3.128)
with || ||Fbeing the Frobenius norm of a matrix, defined in (Van Huffel and Vandewalle 1991, p. 21) or in
(Felus and Burtch 2009) with
|| [VA,vl]||F=qtrace ([VA,vl]T[VA,vl]).(3.129)
The adjusted matrix ˆ
Aand vector ˆ
Lare
[ˆ
A,ˆ
L]=[A,L]+[VA,vL].(3.130)
A solution for vector Xhas been presented in the TLS literature by decomposing the augmented matrix
[A,L] (i.e. the matrix containing the coefficient matrix Aand the observation vector L) with the help of
SVD, resulting in
U Σ WT= [A,L],(3.131)
with the following matrices described in (Bronshtein et al. 2005, p. 285) :
•Matrix U∈Rn×nis orthogonal and contains the left singular vectors (u) of matrix [A,l]:
U= [u1,u2,··· ,un],with UTU=In(3.132)
and nis the number of observation equations (this is the number of rows of matrix A).
•Matrix W∈R(m+1)×(m+1) is orthogonal and contains the right singular vectors (w) of matrix [A,l]:
W= [w1,w2,··· ,wm+1],with WTW=Im+1 (3.133)
and mis the number of unknown parameters (this is the number of columns of matrix A).
•Matrix Σ∈Rn×(m+1) has the form
Σ="Σ10
0 0 #,(3.134)
with the diagonal matrix Σ1∈R(m+1)×(m+1) carrying the singular values (σ) of matrix [A,l]:
Σ1=
σ10
σ2
...
0σm+1
,(3.135)
54 Chapter 3. Solutions of nonlinear least squares problems
with σ1=. . . =σm+1 =0.
According to the procedure of (Van Huffel and Vandewalle, 1991, p. 35) or (Felus and Schaffrin 2005), the
TLS solution can be derived by scaling appropriately the right singular vector (wmin) of matrix Wthat
corresponds to the minimum singular value (σmin). This is the last column of Wand can be written as
wmin =wm+1 = [w1,m+1,··· , wm,m+1, wm+1,m+1]T.(3.136)
The TLS solution for the vector of unknowns is
ˆ
X=−1
wm+1,m+1
[w1,m+1,··· , wm,m+1]T.(3.137)
It must be mentioned that the last equation is usually presented with a negative sign, as in (Felus and
Schaffrin 2005), which is caused by the form of the functional model. However, the functional model
can always be expressed in such a way that the negative sign is not necessary anymore, see for instance
(Malissiovas et al. 2016).
An equivalent formulation of the objective function
To derive the TLS solution, Schaffrin et al. (2012) and (Snow 2012) minimized the sum of the weighted
squared residuals
Ω(vL,vA) = vT
LPLvL+vT
APAvA→min,
with vA:= vec(VA),
(3.138)
for the case of non-singular cofactor matrices QLand QA. “vec” implies a function that stacks the columns
of the residual matrix VAinto one vector. Postulating uncorrelated observed quantities of equal precision,
the last equation can be formulated equivalently by
Ω(vL,vA) = vLTvL+vATvA→min,(3.139)
which is equal to the objective function (3.128). Thus, it is already visible that least squares is the method
being used for the solution of the adjustment problem. Several contributions have already pointed out that
TLS can be regarded as an approach/solution strategy for a special class of nonlinear least squares problems
and not as a different method than least squares, for example (Neitzel and Petrovic 2008), (Reinking 2008),
(Neitzel 2010) or (Malissiovas et al. 2016).
In this dissertation, the terms TLS and TLS approach will refer to the least squares solution of an adjustment
problem within the EIV model, with equally weighted and uncorrelated observations. The solution is derived
by minimizing the objective function (3.139) through SVD of the augmented matrix [A,L].
3.2.1.2 Least squares parameter estimation using WTLS
The least squares solution of adjustment problems within the EIV model can be derived using WTLS,
especially for cases of individually weighted or correlated observations. A variety of WTLS algorithms exists
3.2. Total least squares 55
in the literature, like for example in (Schaffrin and Wieser 2008), (Fang 2011), (Mahboub 2012), (Snow
2012) or (Schaffrin and Snow 2014).
For obtaining a WTLS solution, the objective function (3.138) is combined with the condition equations in
(3.121) to build the Lagrangian
K (vL,vA,k,X) = vT
LPLvL+vT
APAvA+ 2vT
LPLAvA−2kT(L+vL−(A+VA)X),(3.140)
with kdenoting the vector of Lagrange multipliers. The authors dealing with WTLS, e.g. Schaffrin and
Wieser (2008), introduced the Kronecker-Zehfuss product (symbolized by ⊗) to express the developed La-
grange function equivalently as
K(vL,vA,k,X) = vT
LPLvL+vT
APAvA+ 2vT
LPLAvA−2kTL+vL−AX −XT⊗InvA.(3.141)
The necessary stationary points can be obtained by computing the partial derivatives of K with respect
to all unknowns and setting the solution to zero, which according to (Schaffrin and Snow 2014) yields the
nonlinear normal equation system
∂K
∂vT
L
= 2 (PLvL+PLAvA−k) = 0,(3.142)
∂K
∂vT
A
= 2 (PAvA+PALvL+ (X⊗In)k) = 0,(3.143)
∂K
∂XT= 2 ATk+Im⊗kTvA=0
⇒(A+VA)Tk=0,(3.144)
∂K
∂kT=−2L+vL−AX −(XT⊗In)vA=0.(3.145)
Two individual iterative approaches are developed in the following of this section for solving the system of
normal equations (3.142)-(3.145).
WTLS - Approach 1
A solution for the unknown parameters can be obtained by reducing appropriately the derived normal
equations. According to (Snow 2012), if QLL is regular then equations (3.142) and (3.143) can be rewritten
as
vL= [QL−QLA (X⊗In)] k,(3.146)
vA= [QAL −QA(X⊗In)] k,(3.147)
56 Chapter 3. Solutions of nonlinear least squares problems
using a “bidirectional” substitution. Inserting the explicit expressions of the residual vectors into (3.145)
yields
L+ [QL−QLA (X⊗In)] k−AX +XT⊗In[QAL −QA(X⊗In)] k=0
⇒QL−QLA(X⊗In)−(XT⊗In)QAL +XT⊗InQA(X⊗In)
|{z }
Q1
k= (AX −L).(3.148)
Introducing approximate values for the vector of unknowns X0only on the left hand side of the last equation,
it is possible to express the vector of Lagrange multipliers by
k=Q−1
1(AX −L),(3.149)
with the auxiliary matrix7
Q1=QL−QLA(X0⊗In)−(X0T⊗In)QAL +X0T⊗InQA(X⊗In).(3.150)
Consequently, substituting kin equations (3.146) and (3.147) results in the residual vectors
vL= [QL−QLA (X⊗In)] Q−1
1(AX −L),(3.151)
vA= [QAL −QA(X⊗In)] Q−1
1(AX −L).(3.152)
Substituting vAand kin (3.144) yields
ATk+ (Im⊗k)TvA=0
⇒ATQ−1
1(AX −L) = −(Im⊗k)T[QAL −QA(X⊗In)] Q−1
1
|{z }
R1
(AX −L)
⇒ATQ−1
1(AX −L) = R1(AX −L),
(3.153)
with the auxiliary matrix8
R1=−Im⊗k0TQAL −QAX0⊗InQ−1
1.(3.154)
In this last equation the vector of Lagrange multipliers needs also to be approximated with
k0=Q−1
1AX0−L.(3.155)
7The auxiliary matrix Q1coincides with that in (Snow 2012, p. 23) presented in equation (2.11) and is identical to the
product of matrices BQllBTfrom (Fang 2011, p. 22) from equation (4.16). It is usual in TLS literature that matrix Q1is
built without introducing approximate values for the vector of unknowns (X0). However, this can mislead the readers to think
that they deal with a linear problem.
8It must be pointed out that “Algorithm 1” of section 2.1 in (Snow 2012) contains a typo. This is matrix R1in “Algorithm
1” that differs from the correct definition of R1from equation (2.13c) in that dissertation.
3.2. Total least squares 57
Furthermore, rearranging appropriately equation (3.153) gives
ATQ−1
1−R1AX=ATQ−1
1−R1L,(3.156)
which under a non-singular product of matrices ATQ−1
1−R1A, yields the solution for the vector of
unknown parameters
ˆ
X=ATQ−1
1−R1A−1ATQ−1
1−R1L.(3.157)
Iterative solution for the adjustment results
A least squares solution for the unknown parameters can be obtained iteratively following the WTLS pro-
cedure. The auxiliary matrices Q1and R1are functions of unknown parameters. Thus, an initial approxi-
mation X0for the vector of unknowns, gives
Q1i=QL−QLA(X0
i⊗In)−(X0
i
T⊗In)QAL +X0
i
T⊗InQA(X0
i⊗In),(3.158)
ki= (Q1i)−1(AX0
i−L),(3.159)
and
R1i=−(Im⊗ki)TQAL −QAX0
i⊗InQ1
−1
i,(3.160)
with the superscript iimplying the iteration step. A solution for the unknown parameters is obtained by
ˆ
X=AT(Q1i)−1−R1iA−1AT(Q1i)−1−R1iL(3.161)
and is further introduced as an initial approximation for the unknown parameters in the next iteration step
X0
i+1 =ˆ
Xi.
The final solutions can be obtained after a sufficient stopping criterion is fulfilled. Due to the fact that a
linearization has not been applied in any step of the adjustment, this iterative procedure will be terminated
only after the “computational error” condition has been fulfilled, as it has been defined in subsection 3.1.1.
Therefore, let the vector of corrections being computed from the difference between an estimated vector of
unknown parameters and its approximation in an iteration step, expressed by
∆Xi=ˆ
Xi−X0
i.
The necessary condition for the iteration stop is then given by
max|∆X| ≤ . (3.162)
58 Chapter 3. Solutions of nonlinear least squares problems
The elements of the vector of corrections ∆Xwith the maximum absolute value should become smaller or
at least equal to a predefined threshold . A similar stopping criterion can be found in (Snow 2012). The
developed WTLS procedure is similar to the one presented in (Snow 2012) as “Algorithm 1” and has been
primarily developed and presented by Fang (2011) as “Algorithm 2”.
WTLS - Approach 2
Following (Snow 2012), an alternative WTLS approach exists for obtaining the least squares solution for the
vector of unknown parameters. The first step is to reformulate equation (3.144) to
ATk=−VT
Ak
⇒ATQ−1
1(AX −L) = −VT
AQ−1
1(AX −L)
⇒(A+VA)TQ−1
1A X = (A+VA)TQ−1
1L.
(3.163)
Adding the term (A+VA)TQ−1
1VAXto both sides of the last equation yields
h(A+VA)TQ−1
1(A+VA)iX= (A+VA)TQ−1
1(L+VAX).(3.164)
The vector of unknown parameters reads
ˆ
X=h(A+VA)TQ−1
1(A+VA)i−1(A+VA)TQ−1
1(L+VAX).(3.165)
Iterative solution for the adjustment results
A solution can be achieved, also in this case, iteratively. An initial approximation of the vector of unknowns
X0, as well as the residual matrix V0
A, are necessary and lead to
Q1i=QL−QLA(X0
i⊗In)−(X0
i
T⊗In)QAL +X0
i
T⊗InQA(X0
i⊗In),(3.166)
ˆ
Xi=hA+V0
AiT(Q1i)−1A+V0
Aii−1A+V0
AiT(Q1i)−1L+V0
AiX0
i(3.167)
and
ˆvAi=hQAL −QAˆ
Xi⊗IniQ1
−1
iAˆ
Xi−L,(3.168)
with idenoting the iteration step. The estimates for ˆ
Xand ˆvAcan be used to update the initial values for
the next iteration step
X0
i+1 =ˆ
Xiand V0
Ai+1 = invec(ˆvAi),
with “invec” implying the inverse operator of “vec”, i.e. rearranging a vector back into a matrix. The
iterative procedure should continue until an appropriate “break-off” condition is met. The solution for the
3.3. Discussion and open questions 59
unknown parameters of equation (3.165) has been firstly presented by Fang (2011) within “Algorithm 3”
and is identical to the solution of (Snow 2012) presented in “Algorithm 2”.
3.3 Discussion and open questions
Two existing strategies have been analysed in this chapter for the solution of nonlinear adjustment problems
with the method of least squares. The first has been traditionally used in geodesy and is based on the
Gauss-Newton approach. It involves a linearization of the nonlinear functional model, which allows the
representation of the mathematical model within a GMM or a GHM. This is a local optimization approach,
as the solution from the iterative procedure converges to a local minimum of the objective function. After
the assumption of “good” initial values for the unknown parameters, the estimated least squares solution
would correspond to the global minimum.
The second approach that has been discussed is TLS, as proposed by Golub and Van Loan (1980) for the
direct solution of a class of nonlinear least squares problems using SVD. In that work the solutions of two
individual adjustment problems were presented for fitting a straight line in 2D. The least squares solution was
estimated when only the y-coordinates of the points were regarded as measurements and the x-coordinates
as error free (called ordinary least squares), in contrast to the TLS solution where both coordinates of the
points were measurements. Petrovic (2003) has already pointed out that this comparison caused a confusion
regarding the method of least squares, as many investigations draw the conclusion that TLS is a different
method than least squares, or as stated by Groen (1996) that TLS is a generalization of the least squares
method. For geodesists it has been already clear that the most important steps for the adjustment of
observations is to build a correct mathematical model and minimize the objective function composed of
correct residuals.
The relationship between nonlinear least squares problems and TLS was first placed under scrutiny by
Neitzel and Petrovic (2008) and Neitzel (2010) for two individual nonlinear adjustment problems, this of
fitting a straight line to equally weighted 2D data and for the 2D similarity transformation of coordinates. It
has been shown that the TLS solution is identical to the least squares solution within the GHM, concluding
that TLS can be regarded as a special case of least squares within the GHM. Additionally, Reinking (2008)
showed that the TLS solution can be obtained using the traditional geodetic approaches. From (Neitzel and
Petrovic 2008) and (Neitzel 2010) it can be seen that TLS is not a new method, but a new strategy or an
approach for the solution of a class of nonlinear least squares problems. Therefore, the investigations in the
next chapter try to answer the following arising questions:
- If it is possible to solve an adjustment problem with TLS and SVD, is it also possible to obtain the
same eigenvalue problem from a classical least squares approach and solve the problem directly?
- Are there additional nonlinear least squares problems (besides the generally well-known case of the
straight line fitting to equally weighted 2D data) which can be solved directly?
- Is it possible to classify those nonlinear least squares problems with a direct solution and solve them
by using a systematic approach?
Furthermore, the cases of weighted nonlinear least squares problems, as well as the solution by using WTLS
will be examined in a later chapter.
61
Part II - Methodological contributions
63
4 Direct solutions of nonlinear least squares problems
with equal weights
The current chapter is based on the study of Malissiovas et al. (2016). It is an extended version of this
article and includes the most important facts for the solution of adjustment problems with TLS. A clear
mathematical relationship is presented between TLS and direct least squares solutions. Additionally, a
systematic approach has been developed as a by-product of this investigation, for the direct solution of a
class of nonlinear adjustment problems.
4.1 Basic idea and general methodology
The centre of interest is a class of nonlinear least squares problems that can be transformed into solving a
polynomial equation (or the characteristic equation of an eigenvalue problem) and have a direct solution,
depending on the degree of the resulting polynomial. Solutions for these adjustment problems have been
presented in the TLS literature by using SVD. Therefore, the mathematical relationship is examined between
direct solutions of nonlinear least squares and solutions coming from TLS for the following four adjustment
cases:
1. Fitting of a straight line in 2D;
2. Fitting of a straight line in 3D;
3. Fitting of a plane in 3D;
4. 2D similarity transformation of coordinates.
In all four cases under investigation the coordinates in all directions are regarded as measurements. In TLS
literature these problems are often distinguished as EIV model. Moreover, a regular adjustment model is
always postulated here with the observed quantities being equally weighted and uncorrelated.
The concept of solving nonlinear least squares problems applied here is based directly on (Joviˇci´c et al.
1982), where the adjustment problem of fitting a straight line to a set of points in 3D space was examined.
In that work, the least squares estimate has been obtained by solving an eigenvalue problem, which is one
of the key elements of TLS as well. Following the solution strategy from (Joviˇci´c et al. 1982), a systematic
approach for solving the four investigated adjustment problems has been established by Malissiovas et al.
(2016). The proposed mathematical approach involves a sophisticated parametrization of the problem which
can be always solved by building a Lagrange function that results in a quadratic or cubic algebraic equation.
64 Chapter 4. Direct solutions of nonlinear least squares problems with equal weights
In the following sections the least squares solution of the proposed approach from (Malissiovas et al. 2016)
is derived and compared with the TLS solution for the four problems under investigation. It is shown that
a clearly defined mathematical model of the adjustment problem, leading to an objective function based on
the principle of least squares, is the Occam’s razor1for TLS. A flowchart presenting both ways of solving
directly the discussed nonlinear least squares problems is depicted in Figure 4.1.
Special cases
of nonlinear
least-squares
problems
Least-Squares
approach
Total Least-
Squares approach
Sophisticated
parametrization Augmented matrix
Lagrange function Singular value
decomposition
Characteristic
polynomial
Direct
solution
for the
unknowns
Figure 4.1: Flowchart for two possible direct solutions of a class of nonlinear least squares problems.
4.2 Fitting of a straight line in 2D
One of the first attempts to solve the nonlinear problem of least squares for fitting a straight line to a
set of points in plane (i.e. in the 2D space) non-iteratively was done by Adcock (1878), who provided an
elegant way of finding the direct solution to the problem. Pearson (1901) investigated the same problem by
minimizing the sum of the squared orthogonal distances of every point to the requested line and he extended
his study to fitting a plane to a set of points in the 3D space as well. On the other hand, the work of Golub
and Van Loan (1980) provided an analysis of the TLS solution followed by the contributions of Groen (1996),
Van Huffel (2004), Markovsky and Van Huffel (2007) and Schaffrin (2007). These authors always comprised
the example of the straight line fit as the most appropriate example for illustrating the idea of TLS.
At the beginning an amount of 2D data is observed, e.g. a set of points with coordinates in xand in y
direction. The question is how to fit a straight line to the measured points. The general form of a straight
1The Occam’s razor can be defined as “the principle (attributed to William of Occam) that in explaining a thing no more
assumptions should be made than are necessary. The principle is often invoked to defend reductionism or nominalism.“, see
for exampe the Oxford dictionary.
4.2. Fitting of a straight line in 2D 65
line in 2D2is (Bronshtein et al. 2005, p. 194)
ax +by +c= 0,(4.1)
with the constant line parameters a,b,c. The first two parameters denote the components of a vector
normal to the requested straight line, which intersepts the x-axis at c
aand the y-axis at c
b, as it is depicted
in Figure 4.2.
-4 -3 -2 -1 1 2 3 4
-1
1
2
3
4
Figure 4.2: Representation of a straight line in 2D using equation (4.1).
The Hessian normal form of the straight line can be derived by multiplying equation (4.1) with the normal-
ization parameter 3
f=±1
√a2+b2.(4.2)
Introducing residuals vxfor the coordinates in the xdirection and vyfor the coordinates in the ydirection
results in the nonlinear system of condition equations
a(xi+vxi) + b(yi+vyi) + c= 0,(4.3)
with i= 1, . . . , n,ndenoting the number of observed points. Since the system of equations (4.3) is under-
determined, the least squares criterion can be used for an “optimal” solution by minimizing the sum of the
squared residuals n
X
i=1
v2
xi+v2
yi→min.(4.4)
2The same problem has been investigated in (Malissiovas et al., 2016), where the straight line in 2D has been represented
in coordinate form.
3The sign of the scaling factor fis opposite to the sign of parameter c, as it is explained in (Bronshtein et al. 2005, p. 195).
66 Chapter 4. Direct solutions of nonlinear least squares problems with equal weights
4.2.1 Least squares adjustment with a direct solution
In this section, a direct least squares solution is developed for fitting a straight line in 2D when both
coordinates are measurements and subject to errors. The unknown line parameters can be estimated directly
by constructing and minimizing an appropriate Lagrange function and by solving a system of homogeneous
normal equations. The goal is to show that the proposed approach leads, according to the chosen technique,
to the solution of such algebraic equations that are equivalent to TLS. The solution for fitting a straight line
in 2D by TLS is presented and analysed in the following subsection.
4.2.1.1 Definition of the problem
For solving an adjustment problem, it is important to clarify which quantities are observations and hence
subject to random errors. This is necessary in order to define the objective function of the problem in an
appropriate way. In this investigation both coordinates (in the direction of xand y) are subject to mea-
surement errors. Furthermore, let all measurements be uncorrelated and of the same precision. Therefore,
the aim is to find the shortest distance of each “measured” point to the requested straight line. As noticed
already by Adcock (1878) the same precision of all coordinate measurements corresponds to the normal
distances
D2
i=v2
xi+v2
yi,(4.5)
as measure of deviations, with i= 1, ..., n (nis the number of observed points). This problem is depicted
in Figure 4.3. Moreover, the normal distance of every point to the requested line can be expressed by
(Bronshtein et al. 2005, p. 195)
Di=a xi+b yi+c
√a2+b2.(4.6)
There are infinitely many choices for a condition that connects the three unknown parameters a,band cfor
the general equation of the straight line. It is possible to restrict the problem to the usual a= 1 or b= 1,
but in these cases some lines in plane are excluded4. From all remaining restrictions, the most appropriate
for this study is
a2+b2= 1,(4.7)
as it allows all lines in the plane to be calculated. Geometrically this restriction can be seen as a normalization
of the orthogonal distances from every point to the requested line (i.e. the denominator of the orthogonal
distance of equation (4.6) becomes 1), which results in
Di=a xi+b yi+c. (4.8)
The developed expressions for the orthogonal distances can serve as observation equations and be utilized
as an alternative to the nonlinear condition equations (4.3) for estimating the unknown line parameters.
An important remark is that the point coordinates xiand yiin this transformed functional model can be
treated as fixed parameters. The orthogonal distances Diare serving as random deviations, thus the observed
quantities are zero pseudo-observations denoting the Euclidean distances of the points to the requested line.
4choosing a= 1, then there is no solution for lines parallel to the xdirection and for b= 1 no solution for lines parallel to
the ydirection.
4.2. Fitting of a straight line in 2D 67
012345678910
0
1
2
3
4
5
6
7
8
9
10
Figure 4.3: Example of fitting a straight line to points in 2D with both xand ycoordinates
subject to measurement errors.
This leads to the linear observation equations5for the distances
0i+Di=a xi+b yi+c. (4.9)
Equivalently to the objective function (4.4), the least squares criterion can be applied to obtain the minimum
normal distances from a set of points to the fitted line by minimizing the objective function
Ω(a, b, c) =
n
X
i=1
v2
xi+v2
yi=
n
X
i=1
D2
i=
n
X
i=1
(a xi+b yi+c)2.(4.10)
We seek for a least squares solution for the unknown line parameters a,band cthat minimizes (4.10),
subject to the restriction (4.7). Consequently, the Lagrangian
K(a, b, c, λ) = Ω(a, b, c)−k(a2+b2−1),(4.11)
5Although the observation equations (4.9) are linear, the least squares problem is nonlinear, due to the nonlinear constraint
(4.7).
68 Chapter 4. Direct solutions of nonlinear least squares problems with equal weights
can be introduced, with kdenoting the Lagrange multiplier. Differentiating function K with respect to all
unknown parameters and setting the result to zero, yields the system of normal equations
∂K
∂a = 2 a n
X
i=1
x2
i−k!+b
n
X
i=1
yixi+c
n
X
i=1
xi!= 0,(4.12)
∂K
∂b = 2 a
n
X
i=1
yixi+b n
X
i=1
y2
i−k!+c
n
X
i=1
yi!= 0,(4.13)
∂K
∂c = 2 a
n
X
i=1
xi+b
n
X
i=1
yi+c n!= 0 (4.14)
and
∂K
∂k =−a2+b2−1= 0.(4.15)
Rearranging equation (4.14), yields a solution for parameter
c=−a 1
n
n
X
i=1
xi!−b 1
n
n
X
i=1
yi!,(4.16)
in terms of aand b. Introducing cinto the normal equations (4.12) and (4.13) results in the reduced system
of normal equations
a
n
X
i=1
xi2−1
n n
X
i=1
xi!2
−k
+b"n
X
i=1
xiyi−1
n n
X
i=1
xi
n
X
i=1
yi!#= 0,(4.17)
a"n
X
i=1
xiyi−1
n n
X
i=1
xi
n
X
i=1
yi!#+b
n
X
i=1
yi2−1
n n
X
i=1
yi!2
−k
= 0.(4.18)
If the Lagrange multiplier kwere known, then equations (4.17) and (4.18) would form a homogeneous system
of linear equations in aand b. Thus, the determinant of the equation system is equal to zero for a nontrivial
solution
n
X
i=1
xi2−1
n n
X
i=1
xi!2
−k
"n
X
i=1
xiyi−1
n n
X
i=1
xi
n
X
i=1
yi!#
"n
X
i=1
xiyi−1
n n
X
i=1
xi
n
X
i=1
yi!#
n
X
i=1
yi2−1
n n
X
i=1
yi!2
−k
= 0,(4.19)
which leads to a quadratic characteristic equation with two real and positive solutions for the unknown
parameter k. The minimum solution, denoted by kmin, corresponds to the minimum of the Lagrange func-
tion (4.11). The solution for the unknown line parameters aand bcan be computed by substituting the
Lagrangian factor ˆ
kmin into equations (4.17)-(4.18) subject to the chosen restriction (4.7). An equivalent
solution can be obtained by transforming the equation system (4.17)-(4.18) into an eigenvalue problem.
4.2. Fitting of a straight line in 2D 69
4.2.1.2 Simplification of the problem by substituting one unknown parameter
A simplification of this adjustment problem can be easily achieved by replacing the unknown parameter c
in the functional model (4.3), which leads to more elegant expressions for the orthogonal distances than
equation (4.8). However, such a simplification makes sense only if the requested line will pass through the
center of mass of the measured points, located at
yc=1
n
n
X
i=1
yiand xc=1
n
n
X
i=1
xi.(4.20)
This has been already shown by the developed expression for parameter cin equation (4.16). A similar proof
has been also presented by Joviˇci´c et al. (1982) for the 3D line case and by Adcock (1878) and Malissiovas
et al. (2016) for the 2D line as well. Therefore, introducing parameter cfrom equation (4.16) into (4.1),
yields
a(x−xc) + b(y−yc)=0.(4.21)
The last equation can be further simplified by reducing the coordinates to a coordinate system with its origin
located at the centre of mass of the given points. Geometrically, this is equivalent to shifting the coordinate
system to a point that coincides numerically with the center of mass of the measured set of points, as it is
depicted in Figure 4.4.
Thus, denoting the reduced coordinates of a point by
y0=y−ycand x0=x−xc,(4.22)
leads to a system of simplified condition equations
a(x0
i+vxi) + b(y0
i+vyi) = 0 (4.23)
and to the simplified expression for the normal distances
Di=a x0
i+b y0
i.(4.24)
Consequently, the objective function (4.10) can be rewritten as
Ω(a, b) =
n
X
i=1
D2
i=
n
X
i=1
(a x0
i+b y0
i)2=a2
n
X
i=1
x0
i
2+b2
n
X
i=1
y0
i
2+ 2ab
n
X
i=1
y0
ix0
i.(4.25)
We seek for a least squares solution for the unknown line parameters aand bthat minimizes equation (4.25)
subject to the restriction (4.7). The Lagrangian
K(a, b, k) = Ω(a, b)−k(a2+b2−1),(4.26)
70 Chapter 4. Direct solutions of nonlinear least squares problems with equal weights
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
Figure 4.4: Example of fitting a straight line to points in 2D with coordinates reduced
to the centre of mass of the measured points.
is introduced, where kis the Lagrange multiplier. Differentiating K with respect to all unknowns and setting
the partial derivatives to zero, yields the normal equations
∂K
∂a = 2a n
X
i=1
x0
i
2−k!+ 2b n
X
i=1
y0
ix0
i!= 0,(4.27)
∂K
∂b = 2a n
X
i=1
y0
ix0
i!+ 2b n
X
i=1
y0
i
2−k!= 0 (4.28)
and ∂K
∂k =−a2+b2−1= 0.(4.29)
If the Lagrange multiplier were known, then equations (4.27)-(4.28) would represent a homogeneous system
of equations which is linear in the unknown line parameters aand b. A nontrivial solution can be obtained
4.2. Fitting of a straight line in 2D 71
by setting the determinant of the equation system equal to zero:
n
X
i=1
x0
i
2−k!n
X
i=1
y0
ix0
i
n
X
i=1
y0
ix0
i n
X
i=1
y0
i
2−k!
= 0,(4.30)
which leads to the quadratic characteristic equation
n
X
i=1
x0
i
2−k! n
X
i=1
y0
i
2−k!− n
X
i=1
y0
ix0
i!2
= 0,(4.31)
with one unknown parameter kand two real and positive solutions kmin and kmax. It can be shown that the
smaller of the two solutions for k, denoted by kmin, corresponds to the minimum of the Lagrange function
(4.26). There are two possibilities to determine a solution for the unknown parameters aand b, either by
substituting the Lagrangian factor kmin into equations (4.27)-(4.28) or by solving an eigenvalue problem.
It can be easily seen that both equations (4.19) and (4.30) result in quadratic characteristic equations that
produce identical results for the unknown Lagrange multipliers kand the requested line parameters.
4.2.2 TLS solution with SVD
An alternative solution for finding the line that fits best to a set of points in 2D can be provided by TLS
(Golub and Van Loan 1980, Groen 1996). According to (Golub and Van Loan, 1980) this solution can
be represented geometrically by minimizing the orthogonal distances, as it is depicted in Figure 4.3. It is
noteworthy in that contribution, that the least squares problem of fitting a straight line in 2D was regarded
only when the y-coordinates are observations, in contrast to the definition and solution of the least squares
problem that was presented in the previous section. Thus, the target is to provide an insight into the TLS
approach and show that the TLS solution is equivalent to the one from the developed direct least squares
approach of section 4.2.1.
Equation (4.1) for the straight line can be rewritten6as
y=β x +γ, (4.32)
with
β=−b
a, γ =−c
a.(4.33)
Such a formulation for the straight line implies that the restriction a= 1 is taken into account. Therefore,
it is not possible to describe all straight lines in plane, however, these are limited cases (in this problem
all lines that are parallel to the ydirection). Rearranging the functional model of equation (4.23), which
already contains the reduced coordinates of the measured points to the centre of mass, yields the system of
nonlinear equations
(y0
i+vyi) = β(x0
i+vxi),(4.34)
6Greek letters are chosen to describe the parameters of the straight line in the TLS approach just for readability reasons.
72 Chapter 4. Direct solutions of nonlinear least squares problems with equal weights
leading to the EIV model of equation (3.121) with the respective quantities
L=
y0
1
y0
2
.
.
.
y0
n
,vL=
vy1
vy2
.
.
.
vyn
,X=hβi,A=
x0
1
x0
2
.
.
.
x0
n
,VA=
vx1
vx2
.
.
.
vxn
.(4.35)
Matrix Acontains the coefficients of equation (4.34) with respect to the unknown parameter β, except of
the residuals that are introduced into matrix VA.
4.2.2.1 TLS solution based on the minimum eigenvalue principle
The TLS solution has been presented amongst others by Felus and Schaffrin (2005), as it has been already
explained in subsection 3.2.1.1. The first step is to construct the augmented matrix
[A,L] =
x0
1y0
1
x0
2y0
2
.
.
..
.
.
x0
ny0
n
(4.36)
and decompose it with the help of SVD, which yields
UΣWT= [A,L],(4.37)
where the matrices Uand Wcontain the left and right singular vectors of the augmented matrix respectively
and matrix Σis diagonal carrying the singular values. The TLS solution is in the right singular vector of
matrix Wthat corresponds to the minimum singular value:
wmin =wm+1 = [w1,m+1,··· , wm,m+1, wm+1,m+1]T,(4.38)
with the vector of unknowns computed by
ˆ
X=−1
wm+1,m+1
[w1,m+1 :wm,m+1]T.(4.39)
4.2.2.2 Solution by the eigenvalue/eigenvector decomposition
To understand deeper the operation of SVD and the derivation of the adjusted unknowns of equation (4.39)
it is important to explain SVD as the solution of the eigenproblem of the symmetric non-negative definite
matrices ([A,L]T[A,L]) and ([A,L][A,L]T). According to (Lawson and Hanson 1974, p. 18) matrix W
containing the right singular vectors of [A,L] can be also estimated by the eigenvalue decomposition (EVD)
of the squared matrix ([A,L]T[A,L]) :
WΛWT= [A,L]T[A,L],(4.40)
4.2. Fitting of a straight line in 2D 73
where matrix Λis a diagonal matrix carrying the eigenvalues of [A,L]. A relationship between eigenvalues
and singular values can be found in (Golub and Van Loan 1989, p. 427), expressed as
λi=σi2,(4.41)
with λand σbeing the eigenvalues and singular values, respectively. For an explicit solution of this eigen-
problem, matrix
[A,L]T[A,L] = "x0
1x0
2··· x0
n
y0
1y0
2··· y0
n#
x0
1y0
1
x0
2y0
2
.
.
..
.
.
x0
ny0
n
=G,(4.42)
can be introduced, which can be rewritten in a more compact form as
G=
n
X
i=1
x0
i
2
n
X
i=1
y0
ix0
i
n
X
i=1
y0
ix0
i
n
X
i=1
y0
i
2
.(4.43)
The eigenvalues and eigenvectors of matrix Gcan be computed from the eigenvalue problem
Gy =λy⇒(G−λI)y=0,(4.44)
as explained in (Bronshtein et al. 2005, p. 278). Iis an identity matrix and yan eigenvector of G.
The eigenvalues of Gcan be determined by searching for non-trivial solutions y6= 0, i.e. by solving the
characteristic equation of the eigenvalues
n
X
i=1
x0
i
2−λ
n
X
i=1
y0
ix0
i
n
X
i=1
y0
ix0
i
n
X
i=1
y0
i
2−λ
= 0,(4.45)
or equivalently n
X
i=1
x0
i
2−λ! n
X
i=1
y0
i
2−λ!− n
X
i=1
y0
ix0
i!2
= 0.(4.46)
This quadratic equation has two solutions for the unknown eigenvalues, λmin and λmax. By rearranging the
eigenvalues and eigenvectors appropriately, the TLS solution for the line parameter βcan be found from
equation (4.39). Thus, by normalizing the eigenvector that corresponds to the smallest eigenvalue yields
β=
n
X
i=1
y0
ix0
i
n
X
i=1
x0
i
2−λmin
.(4.47)
74 Chapter 4. Direct solutions of nonlinear least squares problems with equal weights
As expected, the TLS solution is identical with the developed direct least squares solution. This can be seen
by simply comparing the developed characteristic equation of the eigenvalues (4.46), which corresponds to
the quadratic equation (4.31) from the direct least squares solution. The conclusion is that the presented
direct least squares solution for the nonlinear straight line fit in 2D already provides the exact result for
TLS.
4.3 Fitting of a straight line in 3D
The problem of fitting a straight line to points in 3D space has been examined e.g. by Kampmann and Renner
(2004), Kupferer (2004) or Sp¨ath (2004). The last from these authors developed an iterative algorithm for
minimizing the sum of the squared orthogonal distances of the measured points to the fitted line and thus
obtaining a least squares estimate for the unknown line parameters. A similar iterative solution can be
found in the investigations of Snow and Schaffrin (2016), who have solved the problem using several models
and always obtained identical results. Non-iterative adjustment solutions for the straight line fit in space
can be found in the studies of (Joviˇci´c et al. 1982) or (Drixler 1993, p. 46). Both can be transformed into
an eigenvalue problem which gives the motivation for investigating the relationship with the TLS solution.
A representation of a straight line in 3D is given in (Bronshtein et al. 2005, p. 217) , expressed as
y−y0
a=x−x0
b=z−z0
c,(4.48)
for a line that passes through a point with coordinates x0,y0and z0and is parallel to a direction vector
with components a,band c. The target is to minimize the errors in all x,yand zcoordinates, which implies
the nonlinearity of the system of condition equations
a(xi+vxi−x0)−b(yi+vyi−y0)=0,
b(zi+vzi−z0)−c(xi+vxi−x0)=0,
c(yi+vyi−y0)−a(zi+vzi−z0) = 0,
(4.49)
with i= 1, . . . , n,nbeing the number of observed points in 3D space. The best line passing through the 3D
point cloud can be obtained by minimizing the sum of squared residuals from all coordinates
n
X
i=1
v2
xi+v2
yi+v2
zi→min.(4.50)
In order to solve the problem stated above, two additional constraints (or restrictions between the unknown
parameters) have to be taken into account, as it has been already discussed by Snow and Schaffrin (2016).
However, the selection of a proper restriction is avoided at this point. An appropriate parametrization of
the problem is attempted that involves a substitution of some unknown parameters with known, following
the same procedure presented in (Malissiovas et al. 2016).
4.3. Fitting of a straight line in 3D 75
4.3.1 Direct least squares solution for fitting a straight line in 3D
Analogously to the investigated case of the previous section (fitting of a straight line in 2D), the same
precision of all coordinate measurements would correspond to the normal distances
D2
i=v2
xi+v2
yi+v2
zi,(4.51)
as measures of deviations between the observed points and the requested line. As explained in (Bronshtein
et al. 2005, p. 218), the squared normal distance between a point and a line in space is
D2=[a(x−x0)−b(y−y0)]2+ [b(z−z0)−c(x−x0)]2+ [c(y−y0)−a(z−z0)]2
a2+b2+c2.(4.52)
Furthermore, it is possible to reduce the number of the unknowns of the model by replacing the parameters
x0,y0and z0with the coordinates of the centre of mass7
yc=1
n
n
X
i=1
yi, xc=1
n
n
X
i=1
xi, zc=1
n
n
X
i=1
zi,(4.53)
of the n3D points. Therefore, equation (4.48) can be rewritten as
y−yc
a=x−xc
b=z−zc
c.(4.54)
Solution with coordinates reduced to the centre of mass
A reduction of all coordinates to the centre of mass leads to the simplified functional model
y0
a=x0
b=z0
c,(4.55)
and the condition equations
a(x0
i+vxi)−b(y0
i+vyi)=0,
b(z0
i+vzi)−c(x0
i+vxi)=0,
c(y0
i+vyi)−a(z0
i+vzi)=0,
(4.56)
with x0,y0and z0being coordinates reduced to the centre of mass of the 3D point cloud. The squared normal
distances of the reduced points can be formulated as
D2
i=(a x0
i−b y0
i)2+ (b z0
i−c x0
i)2+ (c y0
i−a z0
i)2
a2+b2+c2.(4.57)
In this case the most appropriate restriction between the unknown parameters can be selected as
a2+b2+c2= 1 (4.58)
7The proof that this parameter replacement is allowed can be found in (Joviˇci´c et al. 1982).
76 Chapter 4. Direct solutions of nonlinear least squares problems with equal weights
and makes possible to derive a simplified expression for the squared orthogonal distances
D2
i= (a x0
i−b y0
i)2+ (b z0
i−c x0
i)2+ (c y0
i−a z0
i)2.(4.59)
Therefore, the best line can be estimated by minimizing the objective function
Ω(a, b, c) =
n
X
i=1
v2
xi+v2
yi+v2
zi=
n
X
i=1
D2
i
=
n
X
i=1
[(a x0
i−b y0
i)2+ (b z0
i−c x0
i)2+ (c y0
i−a z0
i)2].
(4.60)
In order to derive the minimum of the objective function under the restriction of equation (4.58), the
Lagrangian
K(a, b, c, k) = Ω(a, b, c)−k(a2+b2+c2−1) (4.61)
can be built. A differentiation of function K with respect to all unknowns and setting the resulting partial
derivatives to zero, leads to the system of normal equations
∂K
∂a = 2a n
X
i=1
x0
i
2+
n
X
i=1
z0
i
2−k!−2b
n
X
i=1
y0
ix0
i−2c
n
X
i=1
y0
iz0
i= 0,(4.62)
∂K
∂b =−2a
n
X
i=1
y0
ix0
i+ 2b n
X
i=1
y0
i
2+
n
X
i=1
z0
i
2−k!−2c
n
X
i=1
x0
iz0
i= 0,(4.63)
∂K
∂c =−2a
n
X
i=1
y0
iz0
i−2b
n
X
i=1
x0
iz0
i+ 2c n
X
i=1
y0
i
2+
n
X
i=1
x0
i
2−k!= 0,(4.64)
and
∂K
∂k =−a2+b2+c2−1= 0.(4.65)
Equations (4.62) to (4.64) can be interpreted as a homogeneous system of equations, with the solution for
parameter kobtained from
(p1−k)q1q2
q1(p2−k)q3
q2q3(p3−k)
= 0,(4.66)
with the respective elements
p1=
n
X
i=1
x0
i
2+
n
X
i=1
z0
i
2, p2=
n
X
i=1
y0
i
2+
n
X
i=1
z0
i
2, p3=
n
X
i=1
y0
i
2+
n
X
i=1
x0
i
2
q1=−
n
X
i=1
y0
ix0
i, q2=−
n
X
i=1
y0
iz0
iand q3=−
n
X
i=1
x0
iz0
i.
(4.67)
Equation (4.66) is a cubic characteristic equation8with the unknown parameter k. The adjusted line
parameters a,band ccan be estimated by substituting kmin into equations (4.62) - (4.64) under the
8see for example (Bronshtein et al. 2005, p. 261) for the calculation of the value of a determinant of third order.
4.3. Fitting of a straight line in 3D 77
specified restriction or by transforming the equation system into an eigenvalue problem.
4.3.2 TLS fitting of a straight line in 3D
A TLS solution for fitting a straight line in 3D using SVD has been presented for the first time in (Malissiovas
et al. 2016). In this section an equivalent solution is presented using a slightly modified functional model.
In order to build the adjustment model of equation (3.121) it is necessary to derive an appropriate functional
model. Thus, rearranging appropriately equation (4.56) yields
α(x0+vxi)−β(y0+vyi)=0,
0α−β(z0+vzi) = x0+vxi,
−α(z0+vzi)+0β=y0+vyi,
(4.68)
with
α=−a
cand β=−b
c.(4.69)
The derived system of nonlinear equations can be expressed within an EIV model, with the respective
quantities being
L=
0
x0
1
y0
1
.
.
.
0
x0
n
y0
n
,vL=
0
vx1
vy2
.
.
.
0
vxn
vyn
,X="α
β#,A=
x0
1−y0
1
0−z0
1
−z0
10
.
.
..
.
.
x0
n−y0
n
0−z0
n
−z0
n0
,VA=
vx1−vy1
0−vz1
−vz10
.
.
..
.
.
vxn−vyn
0−vzn
−vzn0
.(4.70)
The first column of matrix Acontains the coefficients of the functional model (4.68) with respect to the un-
known parameter α, whilst in the second column are the coefficients with respect to the unknown parameter
β.
The augmented matrix is
[A,L] =
x0
1−y0
10
0−z0
1x0
1
−z0
10y0
1
.
.
..
.
..
.
.
x0
n−y0
n0
0−z0
nx0
n
−z0
n0y0
n
.(4.71)
78 Chapter 4. Direct solutions of nonlinear least squares problems with equal weights
The right singular vectors of the augmented matrix can be estimated by the eigenvalue/eigenvector decom-
position. Thus, the squared augmented matrix is
[A,L]T[A,L] =
p1q1q2
q1p2q3
q2q3p3
=G,(4.72)
with the respective elements pand qcorresponding to those of equation (4.67). The eigenvalues and eigen-
vectors of matrix Gcan be found by employing the generalised eigenvalue problem, which results in
(p1−λ)q1q2
q1(p2−λ)q3
q2q3(p3−λ)
= 0.(4.73)
The derived determinant provides the characteristic equation of the eigenvalues. In this case this is a cubic
characteristic equation with three solutions for the unknown eigenvalues λ. The adjusted line parameters ˆα
and ˆ
βcan be found by employing the minimum eigenvalue principle of equation (4.38). The right eigenvector
corresponding to the smallest eigenvalue of matrix Gholds the TLS solution for the 3D line parameters.
Obviously the elements of matrix Gcoincide with those from the direct least squares solution. The de-
terminants (4.66) and (4.73) are equal, leading to identical characteristic equations. Therefore, the TLS
solution for the nonlinear problem of the straight line fit in 3D space is identical with the presented direct
least squares solution.
4.4 Fitting of a plane in 3D
The third case under investigation is the nonlinear problem of fitting a plane to a 3D point cloud with all
coordinates being subject to measurement errors. Also for this case all coordinates of the points are regarded
as uncorrelated observations of equal precision. Several results from various TLS algorithms were presented
for this problem in (Schaffrin et al. 2006), which resulted in a slight deviation from the least squares solution.
Therefore, a mathematical relation between the TLS and least squares solution is built for fitting a plane in
3D, following the same line of thinking as in the previous application cases.
The general equation of a plane in 3D can be found in (Bronshtein et al. 2005, p. 214), which reads
ax +by +cz +d= 0,(4.74)
with x,yand zbeing the 3D coordinates of a point that lies in the plane. a,b,cand dare the plane
parameters. Assuming that the coordinates in all directions are observed quantities, a system of nonlinear
condition equations emerges.
a(xi+vxi) + b(yi+vyi) + c(zi+vzi) + d= 0 (4.75)
4.4. Fitting of a plane in 3D 79
Applying the least squares criterion, the plane that fits best to the observed point cloud can be estimated
by minimizing the sum of squared residuals
n
X
i=1
v2
xi+v2
yi+v2
zi→min.(4.76)
4.4.1 Direct least squares solution for fitting a plane in 3D
Fitting a plane to points in 3D, with all coordinates being subject to measurement errors, is similar to the
case of fitting a straight line in plane. Therefore, the objective function (4.76) is equal to the sum of squared
normal distances of the points to the requested plane
n
X
i=1
v2
xi+v2
yi+v2
zi=
n
X
i=1
D2
i,(4.77)
with the normal distances being expressed by
Di=axi+byi+czi+d
√a2+b2+c2,(4.78)
as it is shown in (Bronshtein et al. 2005, p. 214). A simplification of the problem is also in this case possible
by replacing one unknown parameter. Thus, reducing the coordinates of the point cloud to the centre of
mass (see equation 4.53) results in the substitution of parameter dand the simplified functional model
ax0+by0+cz0= 0,(4.79)
with x0,y0and z0denoting the coordinates of a point reduced to the centre of mass of the 3D point cloud.
Solution with coordinates reduced to the centre of mass
The developed functional model (4.79) leads to the system of nonlinear condition equations
a(x0
i+vxi) + b(y0
i+vyi) + c(z0
i+vzi) = 0 (4.80)
and the orthogonal distances
Di=ax0
i+by0
i+cz0
i
√a2+b2+c2.(4.81)
Since equation (4.81) can be scaled by an arbitrary factor, which means that only two out of the three
parameters a,bor care independent, an appropriate constraint would be
a2+b2+c2= 1.(4.82)
Thus, the expression for the normal distances can be rewritten as
Di=ax0
i+by0
i+cz0
i(4.83)
80 Chapter 4. Direct solutions of nonlinear least squares problems with equal weights
and the objective function under minimization is
Ω(a, b, c) =
n
X
i=1
D2
i=
n
X
i=1
(ax0
i+by0
i+cz0
i)2.(4.84)
A least squares solution for the unknown parameters a,band cis required that minimizes Ω(a, b, c), subject
to the constraint (4.82). The Lagrangian
K(a, b, c, k) = Ω(a, b, c)−k(a2+b2+c2−1),(4.85)
can be built. The differentiation of K with respect to the unknown plane parameters leads, after setting the
partial derivatives to zero, to the system of normal equations
∂K
∂a = 2 "a n
X
i=1
x0
i
2−k!+b n
X
i=1
y0
ix0
i!+c n
X
i=1
x0
iz0
i!#= 0,(4.86)
∂K
∂b = 2 "a n
X
i=1
y0
ix0
i!+b n
X
i=1
y0
i
2−k!+c n
X
i=1
y0
iz0
i!#= 0,(4.87)
∂K
∂c = 2 "a n
X
i=1
x0
iz0
i!+b n
X
i=1
y0
iz0
i!+c n
X
i=1
z0
i
2−k!#= 0 (4.88)
and
∂K
∂k =−a2+b2+c2−1= 0.(4.89)
The solution for the Lagrange multiplier can be derived from
(r1−k)s1s2
s1(r2−k)s3
s2s3(r3−k)
= 0,(4.90)
with the quantities
r1=
n
X
i=1
x0
i
2, r2=
n
X
i=1
y0
i
2, r3=
n
X
i=1
z0
i
2,
s1=
n
X
i=1
y0
ix0
i, s2=
n
X
i=1
x0
iz0
iand s3=
n
X
i=1
y0
iz0
i.(4.91)
This is a cubic characteristic equation and has three solutions for k. The unknown plane parameters a,b
and ccan be estimated, either by inserting ˆ
kmin in equations (4.86) - (4.88), under the restriction (4.82), or
by transforming the equation system into an eigenvalue problem. The presented direct solution for fitting a
plane in 3D coincides with that of Linkwitz (1976).
4.4. Fitting of a plane in 3D 81
4.4.2 TLS fitting of a plane in 3D
The TLS solution for fitting a plane in 3D can be derived analogously to the investigations of Schaffrin
et al. (2006), following however a different functional model. Based on the presented approach for obtaining
a TLS estimate, the functional model of equation (4.79) can be rewritten as
z0=−a
cx0−b
cy0⇒z0
i=αx0+βy0,(4.92)
with
α=−a
cand β=−b
c.(4.93)
Therefore, the system of condition equations (4.80) becomes
z0+vzi=α(x0+vxi) + β(y0+vyi),(4.94)
and can be expressed by an EIV model, after introducing the following matrices:
L=
z0
1
z0
2
.
.
.
z0
n
,vL=
vz1
vz2
.
.
.
vzn
,X="α
β#,A=
x0
1y0
1
x0
2y0
2
.
.
..
.
.
x0
ny0
n
,VA=
vx1vy1
vx2vy2
.
.
..
.
.
vxnvyn
.(4.95)
The first column of the coefficient matrix Acontains the coefficients of the condition equations (4.94) with
respect to the unknown parameter αwhile in the second column are the coefficients with respect to β.
Furthermore, it is possible to build the augmented matrix
[A,L] =
x0
1y0
1z0
1
x0
2y0
2z0
2
.
.
..
.
..
.
.
x0
ny0
nz0
n
(4.96)
and the square matrix
[A,L]T[A,L] =
x0
1x0
2. . . x0
n
y0
1y0
2. . . y0
n
z0
1z0
2. . . z0
n
x0
1y0
1z0
1
x0
2y0
2z0
2
.
.
..
.
..
.
.
x0
ny0
nz0
n
=G.(4.97)
This is equivalent to
G=
r1s1s2
s1r2s3
s2s3r3
,(4.98)
82 Chapter 4. Direct solutions of nonlinear least squares problems with equal weights
with the respective elements being identical to those of equation (4.91). The eigenvalues and eigenvectors
of matrix Gcan be computed from the generalised eigenvalue problem, by solving
(r1−λ)s1s2
s1(r2−λ)s3
s2s3(r3−λ)
= 0.(4.99)
This characteristic cubic equation has three solutions for the eigenvalue λ. The unknown parameters αand
βcan be estimated using the minimum eigenvalue principle. The presented least squares solution for fitting
a plane in 3D coincides perfectly with the TLS solution. Equations (4.90) and (4.99) are identical and only
the name of the unknown (kor λ) is different.
4.5 2D similarity transformation of coordinates
The 2D similarity transformation of coordinates is one of the most frequent geodetic and photogrammetric
applications. A first attempt to estimate the TLS solution of the problem using SVD was that of Felus and
Schaffrin (2005) by presenting a Strucured TLS (STLS) algorithm for solving the problem. Neitzel (2010)
has shown that this algorithm needs to be modified for estimating the correct solution. For this reason,
the same problem has been examined again by Schaffrin et al. (2012). Their modified solution is iterative,
however, they state that a TLS solution using SVD could be possible. Here, a new approach is presented
for a direct solution of the problem (least squares and also TLS solution via SVD).
The well-known equation for the planar coordinate transformation is
"Xi
Yi#="cos φ−sin φ
sin φcos φ#" µ0
0µ#" xi
yi#+"tx
ty#,(4.100)
see for example (Felus and Schaffrin 2005). This can be written equivalently as
Xi= (µcos φ)xi−(µsin φ)yi+tx
Yi= (µsin φ)xi+ (µcos φ)yi+ty,(4.101)
with i= 1, ..., n, where nis the number of observed homologous points in the source xy and target XY
system. The unknown transformation parameters between the two systems are:
-φ= rotation angle
-µ= scale factor
-tx= translation in xdirection
-ty= translation in ydirection
Introducing the parameters
ξ1=µcos φand ξ2=µsin φ, (4.102)
4.5. 2D similarity transformation of coordinates 83
it is possible to obtain the simplified equation system
Xi=ξ1xi−ξ2yi+tx,
Yi=ξ2xi+ξ1yi+ty.(4.103)
If all point coordinates are considered as measured quantities, the necessary residuals are introduced in the
functional model resulting in the nonlinear system of condition equations
Xi+vXi−ξ1(xi+vxi) + ξ2(yi+vyi)−tx= 0,
Yi+vYi−ξ2(xi+vxi)−ξ1(yi+vyi)−ty= 0.(4.104)
The least squares criterion can be employed for an “optimal” solution by minimizing the sum of the squared
residuals in both coordinate systems:
n
X
i=1
v2
xi+v2
yi+v2
Xi+v2
Yi→min.(4.105)
4.5.1 Direct least squares solution for the 2D similarity transformation
For a realistic functional model the translation vector has to be present. However, the substitution of the
translations from the functional model is possible and this can be proven in the same way as for the previous
investigated cases, by showing that
tx=Xc−ξ1xc+ξ2yc,
ty=Yc−ξ2xc−ξ1yc,(4.106)
with xcand ycdenoting the coordinates of the centre of mass of the points in the source system and Xcand
Ycin the target system, computed by
xc=1
n
n
X
i=1
xi, yc=1
n
n
X
i=1
yi, Xc=1
n
n
X
i=1
Xi, Yc=1
n
n
X
i=1
Yi.(4.107)
Therefore, a reduction of all coordinates to their centre of mass leads to a simplified functional model
X0
i+vXi−ξ1(x0
i+vxi) + ξ2(y0
i+vyi)=0,
Y0
i+vYi−ξ2(x0
i+vxi)−ξ1(y0
i+vyi)=0.(4.108)
Appropriate parametrization of the problem
In order to obtain a direct solution in the same manner as in the previous sections, an additional unknown
parameter has to be taken into consideration 9. For this reason the functional model (4.108) can be rewritten
9Here the problem is overparametrized and a meaningful constraint between the unknown parameters is chosen. However,
this is not necessary for obtaining a solution but for being consistent with the solution strategy that was followed in the
adjustment problems of the previous sections. Especially for showing that the TLS solution is identical with the solution from
the proposed direct least squares approach.
84 Chapter 4. Direct solutions of nonlinear least squares problems with equal weights
as
γ(X0
i+vXi) + α(x0
i+vxi)−β(y0
i+vyi)=0,
γ(Y0
i+vYi) + β(x0
i+vxi) + α(y0
i+vyi)=0,(4.109)
with
ξ1=−α
γand ξ2=−β
γ.(4.110)
The enforced additional unknown parameter (γcan be seen as an additional parameter) requires a restriction
between the unknowns. For the purposes of this research, a “meaningful” constraint is chosen as
α2+β2+γ2= 1.(4.111)
Solution with coordinates reduced to the centre of mass
The coordinates of the points in both coordinate systems are subject to measurement errors. By employing
the least squares criterion, the goal is to minimize the errors in all homologous points and in both directions.
This is equivalent to the minimization of the Euclidean distances between the points in the target system
and the transformed homologous points from the source system
n
X
i=1
v2
xi+v2
yi+v2
Xi+v2
Yi=
n
X
i=1
D2
i→min,(4.112)
with the squared distances between two homologous points expressed as
D2
i= (γ X0
i+α x0
i−β y0
i)2+ (γ Y 0
i+β x0
i+α y0
i)2.(4.113)
Therefore, the objective function under minimization becomes
Ω(α, β, γ) =
n
X
i=1
D2
i=
n
X
i=1 h(γ X0
i+α x0
i−β y0
i)2+ (γ Y 0
i+β x0
i+α y0
i)2i.(4.114)
A least squares solution for the unknown transformation parameters α,βand γis desired, that minimizes
Ω under the restriction (4.111). Thus, the Lagrange function can be built as
K(α, β, γ, k) = Ω(α, β, γ)−k(α2+β2+γ2−1).(4.115)
Differentiating the Lagrangian K with respect to all unknown parameters and setting the partial derivatives
to zero, yields the system of normal equations
∂K
∂α = 2 "α n
X
i=1
x0
i
2+
n
X
i=1
y0
i
2−k!+γ n
X
i=1
x0
iX0
i+
n
X
i=1
y0
iY0
i!#= 0,(4.116)
∂K
∂β = 2 "β n
X
i=1
x0
i
2+
n
X
i=1
y0
i
2−k!+γ n
X
i=1
x0
iY0
i−
n
X
i=1
y0
iX0
i!#= 0,(4.117)
4.5. 2D similarity transformation of coordinates 85
∂K
∂γ = 2 "α n
X
i=1
x0
iX0
i+
n
X
i=1
y0
iY0
i!+β n
X
i=1
x0
iY0
i−
n
X
i=1
y0
iX0
i!+γ n
X
i=1
x0
i
2+
n
X
i=1
y0
i
2−k!#= 0 (4.118)
and
∂K
∂k =−α2+β2+γ2−1= 0.(4.119)
Similarly to the previous cases it is possible to estimate kby solving
(v1−k)w1w2
w1(v1−k)w3
w2w3(v2−k)
= 0,(4.120)
which leads to a cubic equation with one unknown parameter. The respective elements are
v1=
n
X
i=1
x0
i
2+
n
X
i=1
y0
i
2, v2=
n
X
i=1
X0
i
2+
n
X
i=1
Y0
i
2,
w1= 0, w2=
n
X
i=1
x0
iX0
i+
n
X
i=1
y0
iY0
iand w3=
n
X
i=1
x0
iY0
i−
n
X
i=1
y0
iX0
i.
(4.121)
The unknown transformation parameters α,βand γcan be estimated either by substituting parameter kmin
into equations (4.116) - (4.118), under the condition (4.119), or by transforming the equation system into
an eigenvalue problem.
4.5.2 TLS 2D similarity transformation
In this section the TLS solution of the 2D similarity transformation is presented. By utilizing the functional
model of equation (4.108) and following the same approach as in the presented TLS solutions (subsections
4.2.2, 4.3.2, 4.4.2), the EIV model is introduced with the relevant matrices
A=
x0
1−y0
1
y0
1x0
1
.
.
..
.
.
x0
n−y0
n
y0
nx0
n
,VA=
vx1−vy1
vy1vx1
.
.
..
.
.
vxn−vyn
vynvxn
,L=
X0
1
Y0
1
.
.
.
X0
n
Y0
n
,vL=
vX1
vY1
.
.
.
vXn
vYn
,ˆ
X="ˆ
ξ1
ˆ
ξ2#.(4.122)
86 Chapter 4. Direct solutions of nonlinear least squares problems with equal weights
The augmented matrix [A,L] can be described in this case by
[A,L] =
x0
1−y0
1X0
1
y0
1x0
1Y0
1
.
.
..
.
..
.
.
x0
n−y0
nX0
n
y0
nx0
nY0
n
.(4.123)
The right eigenvectors of the augmented matrix can be derived by the eigenvalue/eigenvector decomposition
of the squared matrix
[A,L]T[A,L] =
v1w1w2
w1v1w3
w2w3v2
=G,(4.124)
with the respective elements being equal to those of equation (4.121). The solution for the transformation
parameters can be determined from the generalised eigenvalue problem by solving
(v1−λ)w1w2
w1(v1−λ)w3
w2w3(v2−λ)
= 0.(4.125)
As expected, equation (4.125) is the same as equation (4.120) and the resulting cubic polynomial equation
for the solution of kis identical to the characteristic equation of the eigenvalues λ. The translation terms
txand tycan be computed by substituting the estimated parameters into equation (4.106).
4.6 General formulation and classification
The normal equations of the discussed nonlinear least squares problems in this chapter can be transformed
into an eigenvalue problem and be solved directly when the characteristic equation is a polynomial of degree
four or less. Such adjustment cases are the fitting of a straight line in 2D and 3D, the fitting of a plane in 3D
and the 2D similarity transformation of coordinates. The following common features have been identified
for the direct solution of these problems:
1. The measured quantities in all adjustment cases were equally weighted and uncorrelated.
2. In the beginning a nonlinear and over-parametrised functional model was used to express each indi-
vidual problem, see for example equation (4.3) for fitting of a straight line in 2D.
3. Choosing an appropriate restriction between the unknown parameters for the adjustment of each inves-
tigated problem, it was possible to obtain an apparently linear relationship between the observations
and the unknowns.
4. A reduction of the observed coordinates to the centre of mass was in any case proven to be admissible.
This reduction leads everytime to the substitution of some unknown parameters with known ones.
However, it must be mentioned that this parameter substitution is not necessary and is only performed
for simplifying the problem. Thus, an equivalent solution can be obtained from the respective normal
equations, including all unknown parameters.
4.6. General formulation and classification 87
5. The developed objective function for minimising the sum of squared residuals leads to a homogeneous
system of normal equations which is linear with respect to the unknown parameters and has a direct
solution (in case that the derived characteristic equation is a polynomial of degree four or less).
6. The derived direct solutions have been proven to be identical with the TLS solutions obtained by using
SVD.
These features can be used in the future as criteria for identifying easier and quicker those nonlinear least
squares problems that belong to this class. A general formulation of these adjustment problems can be
considered, based on the replacement of the “original” nonlinear functional model with an apparently linear
one, see features (2.) and (3.) above.
General formulation in matrix notation
All discussed adjustments can be solved in their “original” nonlinear form iteratively by linearizing the
condition equations and expressing the problem within a GHM. However, in all cases the problem could be
“transformed” in such a way, so that it could be expressed in matrix notation by the system of observation
equations
L+v=A X,(4.126)
with the nonlinear constraint between the unknown parameters being described by the quadratic function
10
XTX= 1.(4.127)
Vector Lcontains pseudo-observations (zero elements). Vector vincludes the residuals of the pseudo-
observations, which are the orthogonal distances. The design matrix Acontains the coefficients of the linear
observation equations with respect to the unknown parameters in each problem and vector Xholds the
unknown parameters to be estimated.
In order to illustrate this type of functional modeling, the example of fitting of a straight line in 2D can be
considered. For instance, northogonal distances of the measured points to the straight line are describing
the functional model of the problem, with the observation equations
01+D1=a x0
1+b y0
1,
02+D2=a x0
2+b y0
2,
.
.
.
0n+Dn=a x0
n+b y0
n,
(4.128)
10In strict notation the product of matrices will also be a matrix, which in this case will have only one element.
88 Chapter 4. Direct solutions of nonlinear least squares problems with equal weights
under the constraint a2+b2= 1. This system of observation equations can be expressed in matrix notation,
as in equation (4.126), with the respective matrices being defined by
L=
01
02
.
.
.
0n
,v=
D1
D2
.
.
.
Dn
,X="a
b#,A=
x0
1y0
1
x0
2y0
2
.
.
.
x0
ny0
n
.(4.129)
Taking into account the stochastic information of the measured quantities, it can be seen that the problem
can be expressed within a GMM with a quadratic constraint (as in the presented solution of section 3.1).
Avoiding any kind of linearization, a least squares solution can be obtained by minimizing the objective
function
Ω(v,X) = vTv→min,(4.130)
or taking into account the constraint (4.127), by finding the minimum of the Lagrange function
K(v,X,k) = vTv−k(XTX−1) →min,(4.131)
with kdenoting the Lagrange multiplier. Rearranging the observation equations (4.126) and substituting
the solution for the residuals in the Lagrangian yields
K(X,k)=(AX −L)T(AX −L)−k(XTX−1)
⇒K(X,k) = XTATAX −2LTAX +LTL−k(XTX−1).
(4.132)
Taking into consideration that the vector of pseudo-observations Lcontains only zero values, the last function
can be equivalently written as
K(X,k) = XTATAX −k(XTX−1).(4.133)
The minimization of the derived Lagrangian can be found already in (Perovi´c 2005, p. 33) as the solution
of mathematical problems in quadratic forms, with their extrema being derived using an EVD. This can be
proven by taking the partial derivatives of K with respect to the unknowns and setting the solution to zero,
which yields the normal equation system
∂K
∂XT= 2ATAX −2kX=0
⇒ATAX −kX=0,(4.134)
∂K
∂k =XTX−1 = 0.(4.135)
4.7. Discussion and open questions 89
In equation (4.134), parameter kcan be seen as an eigenvalue and Xas an eigenvector of the squared matrix
ATA. The solution can be computed from the generalised eigenvalue problem
ATAX −kX=0⇒(ATA−kI)X=0,(4.136)
with Idenoting an identity matrix. The eigenvalues of matrix ATAcan be determined by searching for
non-trivial solutions X6= 0, i.e. by solving the characteristic equation of the eigenvalues, or equivalently by
det(ATA−kI) = 0.(4.137)
From the latter developments it can be seen that all discussed problems can be expressed within a GMM,
while finding the minimum of the objective function is equivalent to finding the minimum of a quadratic
function by employing EVD.
4.7 Discussion and open questions
In this chapter two individual solution strategies have been examined, the systematic approach for direct least
squares solutions that has been established already in (Malissiovas et al. 2016) and TLS. A mathematical
relationship between the two approaches has been presented by comparing their solutions for four nonlinear
least squares problems, the fitting of a straight line in 2D and 3D, the fitting of a plane in 3D, as well as
the 2D similarity transformation of coordinates. The discussed adjustment problems have been identified
as such, that belong to a certain class of nonlinear least squares and can be transformed into solving a
polynomial equation. Thus, depending on the polynomial’s degree a direct solution can be possible11.
An “optimal” estimate for the unknowns is derived by employing the method of least squares and minimizing
a well-defined Lagrange function. In all discussed cases the normal equations can be solved with various
techniques, for example SVD or EVD and by solving a characteristic equation, which was always identical to
the characteristic equation of the eigenvalues from the corresponding TLS solution. The developed approach
provides a deep understanding of the concept of TLS for the solution of nonlinear least squares problems.
It has been already known from (Neitzel and Petrovic 2008), (Neitzel 2010), as well as (Reinking 2008), that
TLS is not a new method per se but a solution strategy for a class of nonlinear least squares problems. In
addition, the presented direct solutions of this chapter reveal that TLS is an algorithmic approach for the
solution of a class of nonlinear least squares problems using SVD.
Nevertheless, in order to obtain a direct solution for the discussed adjustment problems, either using the
proposed systematic approach or with TLS and SVD, it is always assumed that the observations are un-
correlated and have been obtained with equal precision. When postulating a different precision for each
observation, then different solution strategies can be utilized. A weighted least squares solution can be
obtained in this case using, for example, the Gauss-Newton approach from the traditional geodetic solutions
of section 3.1, or by employing one of the WTLS algorithms from section 3.2.1.2.
11Direct least squares solution have been presented for the investigated adjustment problems, as the roots of the derived
polynomials could be computed directly using known formulas from mathematics, see for more information (Bronshtein et al.
2005, p. 62 ff.)
90 Chapter 4. Direct solutions of nonlinear least squares problems with equal weights
The following questions arise out of the findings from this chapter:
- If it is possible to obtain directly a solution for this class of nonlinear least squares problems by
transforming them into solving a polynomial equation, is it also possible to obtain a similar solution
by transforming the weighted least squares problem?
- Are there specific weighted cases of nonlinear least squares problems (besides the generally well-known
case of equally weighted observations) which can be solved directly?
- Is it possible to detect those weighted nonlinear least squares problems with a direct solution and solve
them by using a systematic approach?
- In cases where a direct weighted least squares solution is not possible, what are the alternative ways?
Is it possible to obtain an iterative solution without making any use of linearization of the problem?
Therefore, possible direct solutions are investigated in the next chapter for the discussed class of adjustment
problems by postulating different weighting cases for the measured quantities. In these scenarios where a
direct solution is not possible, an iterative approach is examined that does not involve a linearization of the
problem at any step of the procedure.
91
5 Direct and iterative solutions of weighted nonlinear
least squares problems
5.1 Basic idea and general methodology
Direct solutions have been presented in the previous chapter for a special class of nonlinear least squares
problems. The established solution strategy involved the transformation of the normal equations into the
solution of a quadratic or cubic algebraic equation (characteristic equation). The mathematical derivations
of those solutions were based on the fact that uncorrelated observations have been obtained with equal
precision. Thus, the postulated weights of the adjustment were in all cases equal.
The investigations in this chapter focus on the aforementioned questions from section 4.7. Three of the
adjustment problems that belong to this class with a direct solution are examined here1, namely:
- Fitting of a straight line in 2D;
- Fitting of a plane in 3D;
- 2D similarity transformation of coordinates,
and for four individual weighting cases for each problem:
1. Same precision for the coordinates in each direction;
2. Individual precision for the coordinates of each point;
3. Individual precision for each coordinate;
4. Individual precision and correlations between the observations (covering also the cases of singular
cofactor matrices).
Direct weighted least squares solutions are proposed in this chapter for the first time for the discussed class
of problems. The general idea of these solutions corresponds to the methodology suggested in (Malissiovas
et al. 2016). It involves a parameterization of the problem and the formation of a Lagrange function that
results in a quadratic or cubic equation. In cases where the problem cannot be transformed to such algebraic
equations, modern iterative algorithms are clearly presented that do not require a linearization of the original
1The case of fitting a straight line in 3D will not discussed further in this chapter in order to avoid repetition, as it has
similarities to the problems of fitting a straight line in 2D and fitting a plane in 3D and therefore could be easily solved following
the same strategy that is presented for the latter problems.
92 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
problem and are based mainly on the approaches from (York 1966) and (Petrovi´c et al. 1983). The iterative
solutions of the developed algorithms can be compared with those from WTLS. Figure 5.1 depicts a flowchart
with the proposed solutions for this class of weighted nonlinear least squares problems.
Class of nonlinear
least-squares problems
(Malissiovas et al 2016)
Constant weights
in each direction
Individual weight
for each point
Individual weight
for each coordinate
Individually
weigthed and cor-
related coordinates
Scaling of the
coordinate system
Sophisticated
parametrization
Lagrange function
Characteristic
polynomial
Sophisticated
parametrization
including weights
Weighted La-
grange function
Characteristic
polynomial
Weigthed La-
grange function
Differentiation
with respect to
all unknowns
Reduced
normal
equations
Direct weigthed
least-squares solution
Iterative weigthed
least-squares solution
without linearization
Figure 5.1: Flowchart for possible direct and iterative solutions of a class of nonlinear weighted least squares
problems.
5.2 Fitting of a straight line in 2D
One of the first attempts to solve the nonlinear least squares problem of fitting a straight line to a set of
points in 2D, that have been observed with different precisions, goes back to (York, 1966). In that article
the problem has been expressed by a pseudo-cubic equation and a solution has been obtained iteratively
and without any kind of linearization. It was further Williamson (1968) who pointed out that the same
adjustment problem can be formulated either as a pseudo-quadratic or even as a pseudo-linear equation. A
year later York (1968) published an iterative least squares solution including this time correlations between
the observations. A thorough investigation and a detailed explanation of the same problem can be found in
(Petrovi´c et al., 1983) as well.
Schaffrin and Wieser (2008) have presented a weighted TLS algorithm for linear regression, that inspired
Shen et al. (2011) and Amiri-Simkooei and Jazaeri (2012) to develope modern TLS algorithms for solving
the same problem. On the other hand, Neitzel and Petrovic (2008) presented a solution within the linearized
GHM, following the traditional geodetic procedure for solving nonlinear least squares problems. This includes
5.2. Fitting of a straight line in 2D 93
a linearization of the condition equations and an iterative process that stops after a predefined threshold,
as it has been discussed in chapter 2.
Point of beginning in this investigation are the coordinates of a set of points in 2D, assuming that they
have been observed with different precisions. The functional model of this problem can be expressed by
the general form of a straight line in 2D, presented already by equation (4.1) in section 4.2. Including the
necessary residuals in the measured quantities results in the nonlinear condition equations
a(xi+vxi) + b(yi+vyi) + c= 0,(5.1)
with i= 1, . . . , n, where nis the number of measured points. The selection of an appropriate restric-
tion between the unknown parameters will be considered at a later point in this section. In the case of
measurements that have been obtained with different precisions, σyifor the y-coordinates and σxifor the
x-coordinates, a least squares solution for the unknown line parameters could be based on the minimization
of the sum of weighted squared residuals
Ω(vxi, vyi) =
n
X
i=1
pxiv2
xi+pyiv2
yi→min,(5.2)
where pxiis a weight for the residual vxiand pyifor vyi. The respective weights have been defined in
(Helmert 1924, p. 81) as
pxi=1
σ2
xi
and pyi=1
σ2
yi
.(5.3)
Therefore, a least squares solution for fitting a straight line in 2D is investigated for four individual weighting
cases that often occur in practice:
1. Same precision σxfor the coordinates in xdirection and σyin ydirection.
2. Individual precision for each point: σxi=σyi∀i.
3. Individual precision for each measured coordinate.
4. Individual precision and correlations between the measured 2D coordinates.
5.2.1 Weighting case 1 - Equally weighted observations in each direction
For the first weighting case the coordinates in xdirection have been observed with the same precision σx,
respectively in ydirection with σy. The weights pxand pycan be computed from equation (5.3) and the
objective function under minimization becomes
Ω(vxi, vyi) =
n
X
i=1
pxv2
xi+pyv2
yi→min,(5.4)
with px
py
=m, m = constant.(5.5)
94 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
This problem will have the geometry that is portrayed in Figure 5.2. Thus, it results in the minimization
of slanted distances from each observed point to the requested line. However, it can be observed that the
ratio between the slanted and the orthogonal distances (i.e. the angles between the orthogonal and the
slanted distances) will be constant for every point. From a geometric perspective, the postulated weights
can be seen as a homogeneous scale of the coordinate system in both xand ydirection. A direct approach
012345678910
0
1
2
3
4
5
6
7
8
9
10
Figure 5.2: Example of fitting a straight line to points in 2D, with observed xand ycoordinates and px,py
individual constant weights for each coordinate axis.
is presented here, for solving the discussed nonlinear weighted least squares problem. The strategy that will
be followed involves the scaling of the observed coordinates beforehand with
xs
i=xi√pxand ys
i=yi√py,(5.6)
with the superscript “s” indicating scale. The scaled coordinates xs
iand ys
ican be utilized to derive the
requested line in a different coordinate system, where the weights of the observations are equal. In this line
of thinking the residuals of the point coordinates will also be scaled accordingly, with
vs
xi=vxi√pxand vs
yi=vyi√py.(5.7)
5.2. Fitting of a straight line in 2D 95
Substituting the scaled coordinates and their residuals from equations (5.6) and (5.7) into the condition
equations (5.1) yields
a1
√pxxs
i+vs
xi+b1
√pyys
i+vs
yi+c= 0.(5.8)
Introducing the auxiliary scaled line parameters
as=a1
√px
,
bs=b1
√py
,
cs=c,
(5.9)
into the condition equations (5.8), yields an alternative functional model to equation (5.1), expressed by
asxs
i+vs
xi+bsys
i+vs
yi+cs= 0.(5.10)
A meaningful constraint2for the solution of this adjustment problem can been chosen here as
as2+bs2= 1.(5.11)
The least squares criterion is employed for a solution of the unknown line parameters by minimizing the
sum of scaled squared residuals n
X
i=1
vs
xi
2+vs
yi
2→min.(5.12)
Thus, it has been shown that the discussed weighted least squares problem can be transformed into a problem
with equal weights in the scaled coordinates, as it is depicted in Figure 5.3.
5.2.1.1 Direct least squares solution in a scaled coordinate system
The sum of squared scaled residuals can be replaced by the sum of squared orthogonal distances
Ω(vs
xi, vs
yi) =
n
X
i=1
vs
xi
2+vs
yi
2=
n
X
i=1
D2
i.(5.13)
The orthogonal distances of the points to the straight line, c.f. equation (4.6), can be expressed for this
problem by
Di=asxs
i+bsys
i+cs
pas2+bs2,(5.14)
which under the constraint (5.11) become
Di=asxs
i+bsys
i+cs.(5.15)
2A “meaningful” constraint is chosen here, in the sense that it will lead to simpler equations for the orthogonal distances
of the 2D points to the requested line, as it has been discussed in subsection 4.2.
96 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
012345678910
0
1
2
3
4
5
6
7
8
9
10
Figure 5.3: Example of fitting a straight line to the scaled points in 2D.
Therefore, using the scaled coordinates it is possible to obtain a direct least squares solution for the unknown
line parameters, following the same procedure as the one presented in section 4.2.1.
Computation of the line parameters in the original coordinate system
The original line parameters can be computed by substituting the estimated “scaled” line parameters ˆas,ˆ
bs
and ˆcsinto equation (5.9):
ˆa= ˆas√px,
ˆ
b=ˆ
bs√py,
ˆc= ˆcs.
(5.16)
However, this solution has been restricted to
as2+bs2=a1
√px2
+b1
√py2
= 1.(5.17)
5.2. Fitting of a straight line in 2D 97
The least squares solution which restricts the line parameters to a2+b2= 1 can be easily derived by
multiplying the original condition equations with the term
1
√a2+b2,(5.18)
which yields
a
√a2+b2(xi+vxi) + b
√a2+b2(yi+vyi) + c
√a2+b2= 0.(5.19)
Using the information of equation (5.16), the line parameters which are restricted to a2+b2= 1 can be
computed by
ˆa=ˆas√px
rˆas√px2+ˆ
bs√py2,
ˆ
b=ˆ
bs√px
rˆas√px2+ˆ
bs√py2,
ˆc=ˆcs
rˆas√px2+ˆ
bs√py2.
(5.20)
5.2.2 Weighting case 2 - Individually weighted points in 2D
In the second weighting case under investigation, each measured point has been obtained with individual
precision
σxi=σyiand pxi=pyi=pi∀i. (5.21)
The ratio between the weights for each point is constant with
pxi
pyi
= 1 ∀i. (5.22)
Taking into consideration (5.21), the objective function (5.2) can be reformulated to
n
X
i=1
pxiv2
xi+pyiv2
yi=
n
X
i=1
pi(v2
xi+v2
yi)→min.(5.23)
5.2.2.1 Direct weighted least squares solution
In case of individually weighted points, the sum of weigthed squared residuals can be expressed equivalently
by the weighted squared orthogonal distances of each point to the requested line:
pxiv2
xi+pyiv2
yi=pi(v2
xi+v2
yi) = pi(D2
i).(5.24)
98 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
The orthogonal distances are
Di=axi+byi+c
√a2+b2,(5.25)
which after taking into account the restriction between the unknown line parameters
a2+b2= 1,(5.26)
can be simplified to
Di=axi+byi+c. (5.27)
Thus, the objective function (5.23) can be equivalently written as
Ω(a, b, c) =
n
X
i=1
pi(v2
xi+v2
yi) =
n
X
i=1
piD2
i=
n
X
i=1
pi(axi+byi+c)2.(5.28)
This adjustment problem is depicted in Figure 5.4.
012345678910
0
1
2
3
4
5
6
7
8
9
10
Figure 5.4: Example of fitting a straight line to points in 2D with xand ymeasured coordinates
and pxi,pyibeing equal weights for each point.
5.2. Fitting of a straight line in 2D 99
We seek for a least squares solution for the unknown line parameters a,band cthat minimizes (5.28),
subject to the restriction (5.26). Consequently, the Lagrangian
K(a, b, c, λ) = Ω(a, b, c)−k(a2+b2−1),(5.29)
can be written, with kdenoting the Lagrange multiplier. By differentiating function K with respect to the
unknown parameters and setting the partial derivatives to zero results in the system of normal equations
∂K
∂a = 2 "an
X
i=1
pixi2−k+bn
X
i=1
piyixi+cn
X
i=1
pixi#= 0,(5.30)
∂K
∂b = 2 "an
X
i=1
piyixi+bn
X
i=1
piyi2−k+cn
X
i=1
piyi#= 0,(5.31)
∂K
∂c = 2 "a
n
X
i=1
pixi+b
n
X
i=1
piyi+c
n
X
i=1
pi#= 0 (5.32)
and ∂K
∂k =−a2+b2−1= 0.(5.33)
Rearranging equation (5.32) leads to
c=−a
n
X
i=1
pixi
n
X
i=1
pi
−b
n
X
i=1
piyi
n
X
i=1
pi
.(5.34)
Introducing the expression for cinto the normal equations (5.30) and (5.31), yields the reduced normal
equations
a
n
X
i=1
pixi2−1
n
X
i=1
pi n
X
i=1
pixi!2
−k
+b
n
X
i=1
piyixi−1
n
X
i=1
pi n
X
i=1
pixi
n
X
i=1
piyi!
= 0 (5.35)
and
a
n
X
i=1
piyixi−1
n
X
i=1
pi n
X
i=1
pixi
n
X
i=1
piyi!
+b
n
X
i=1
piyi2−1
n
X
i=1
pi n
X
i=1
piyi!2
−k
= 0.(5.36)
100 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
Equations (5.35) and (5.36) form a homogeneous system of linear equations with respect to the unknown
line parameters. The determinant of the equation system is equal to zero for a nontrivial solution
n
X
i=1
pixi2−1
n
X
i=1
pi n
X
i=1
pixi!2
−k
n
X
i=1
piyixi−1
n
X
i=1
pi n
X
i=1
pixi
n
X
i=1
piyi!
n
X
i=1
piyixi−1
n
X
i=1
pi n
X
i=1
pixi
n
X
i=1
piyi!
n
X
i=1
piyi2−1
n
X
i=1
pi n
X
i=1
piyi!2
−k
= 0,(5.37)
which leads to a quadratic characteristic equation with two real and positive solutions for k. The minimum
solution, denoted by kmin, corresponds to the minimum of the Lagrange function (5.29). The solution for
the unknown line parameters aand bcan be computed by substituting the Lagrangian factor kmin into
equations (5.35)-(5.36), subject to the chosen restriction, or by transforming the equation system into an
eigenvalue problem.
5.2.3 Weighting case 3 - Individually weighted 2D coordinates
The third weighting case under investigation is more general than the previous two, as the measured coor-
dinates have been observed with individual precisions. As far as known, a direct least squares solution for
this nonlinear problem has not been found. Iterative solutions, however, have been presented by various
authors in the past. Some of those do not make use of any linearization of the original problem, such as for
example York (1966), York (1968) or Petrovi´c et al. (1983). In the next subsection it is shown that a direct
solution for the discussed adjustment problem is not possible. However, an iterative procedure is presented
that is based to a large extent on the derivations of (Petrovi´c et al. 1983).
Minimizing the sum of weighted squared residuals
Starting from the condition equations (5.1) and the objective function (5.2), it is possible to build the
Lagrangian
K(vxi, vyi, a, b, c, ki) = Ω(vxi, vyi)−2
n
X
i=1
ki(a(xi+vxi) + b(yi+vyi) + c),(5.38)
with kidenoting the Lagrange multipliers.The normal equation system of this adjustment problem can be
described by
∂K
∂vxi
= 2pxivxi−2aki= 0,
⇒vxi=aki
pxi
,(5.39)
5.2. Fitting of a straight line in 2D 101
∂K
∂vyi
= 2pyivyi−2bki= 0,
⇒vyi=bki
pyi
,(5.40)
∂K
∂ki
=−2 [a(xi+vxi) + b(yi+vyi) + c]=0,(5.41)
∂K
∂a =−2
n
X
i=1
ki(xi+vxi)=0,(5.42)
∂K
∂b =−2
n
X
i=1
ki(yi+vyi)=0,(5.43)
∂K
∂c =−2
n
X
i=1
ki= 0.(5.44)
Equations (5.39) to (5.44) form a nonlinear system of 3n+ 3 equations. Introducing the residuals from
equations (5.39) and (5.40) into (5.41), yields the expression for the Lagrange multipliers
ki=wi(axi+byi+c),(5.45)
with the auxiliary weighting factors3
wi=−a2
pxi
+b2
pyi−1
.(5.46)
Introducing kiinto equation (5.44), results in the expression for parameter
c=−
a
n
X
i=1
wixi+b
n
X
i=1
wiyi
n
X
i=1
wi
.(5.47)
Additionally, substituting kiinto equations (5.39) and (5.40) yields explicit expressions for the residual
vectors
vxi=−wi
px
a(axi+byi+c) (5.48)
and
vyi=−wi
py
b(axi+byi+c).(5.49)
3It is interesting to note the relationship between the developed weighting factors wi, the coefficients ”Li” and the weighting
factors ”Wi” from (Deming 1964, p. 134,181) and (York 1966).
102 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
Utilizing the developed expressions for vyiand vxi, a minimum of the original objective function Ω can be
found, instead of minimizing the Lagrangian K. This approach gives the possibility to show why a direct
least squares is not possible for this weighted case. Therefore, substituting the developed residuals directly
in the objective function (5.2) yields
Ω(vxi, vyi) =
n
X
i=1
pxiv2
xi+pyiv2
yi
=
n
X
i=1
pxi−wi
pxi
a(axi+byi+c)2
+pyi−wi
pyi
b(axi+byi+c)2
=
n
X
i=1 −a2
pxi
+b2
pyi(axi+byi+c)2
a2
pxi
+b2
pyi2
=
n
X
i=1 −1
a2
pxi
+b2
pyi(axi+byi+c)2
=
n
X
i=1
wi(axi+byi+c)2
=
n
X
i=1
D2
i.
(5.50)
From the last equation it can be seen that the problem of minimizing the sum of weighted squared residuals
can be transformed into the minimization of the slanted distances
Di=−1
sa2
pxi
+b2
pyi
(axi+byi+c) = √wi(axi+byi+c).(5.51)
A direct solution for this weighted case is not possible, as there is no restriction for the unknown parameters
that could lead to a linear formulation of the distances in equation (5.51). From a different perspective
it could be said that a direct least squares solution is possible, when the auxiliary weighting factors wi
in equation (5.46) can be set equal to a constant value, by selecting a meaningful restriction between the
unknown parameters. The geometry of this problem is depicted in Figure (5.5).
5.2.3.1 Iterative least squares solution without linearization
An iterative solution for the discussed adjustment problem can be obtained without performing any kind of
linearization, similar to (York 1966) and (Petrovi´c et al. 1983). The unknown line parameters aand bcan
be estimated by introducing the residuals from equations (5.48) - (5.49) and kifrom (5.45), into equations
5.2. Fitting of a straight line in 2D 103
012345678910
0
1
2
3
4
5
6
7
8
9
10
Figure 5.5: Example of fitting a straight line to points in 2D with observed xiand yicoordinates and pxi,pyi
individual weights for the coordinates.
(5.42) and (5.43). This yields the reduced normal equations
a
n
X
i=1
k2
i
pxi
=−
n
X
i=1
wi(axi+byi+c)xi(5.52)
and
b
n
X
i=1
k2
i
pyi
=−
n
X
i=1
wi(axi+byi+c)yi.(5.53)
Rearranging appropriately equations (5.52) and (5.53) gives
af1=bf2(5.54)
and
bf3=af2,(5.55)
104 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
with the respective quantities being
f1=
n
X
i=1
k2
i
pxi−
n
X
i=1
wix2
i+1
n
X
i=1
wi n
X
i=1
wixi!2
,
f2=
n
X
i=1
(wixiyi)−1
n
X
i=1
wi n
X
i=1
wiyi
n
X
i=1
wixi!,
f3=
n
X
i=1
k2
i
pyi−
n
X
i=1
wiy2
i+1
n
X
i=1
wi n
X
i=1
wiyi!2
.
(5.56)
Furthermore, a simplification of the problem is feasible by setting a restriction between the unknown line
parameters aand b. It is possible to take into account the general restriction a2+b2= 1, however, for
convenience the problem is restricted here to b= 1. Thus, a solution for the remaining unknown is
a=f2
f1
,(5.57)
with f1and f2containing some unknown parameters, according to their definition in (5.56). Thus, equation
(5.57) becomes pseudo-linear after selecting an approximate value a0. The estimated line parameters can be
utilized as new starting values in each iteration step, until a break-off condition is met. As a linearization
has not been applied in any step of the adjustment, this iterative procedure can be terminated after the
condition of the “computational error” is fulfilled, as it was presented in chapter 3. Algorithm 1 has been
developed for estimating the weighted least squares solution for fitting a line to a set of points in 2D, based
on the presented procedure of this subsection.
Algorithm 1 Least squares fitting of a straight line in 2D with general weights
1: Choose approximate value for a0.
2: Define parameter b= 1.
3: Set threshold for the break-off condition of the iteration process.
4: Set parameter da=|ˆa−a0|=∞, for entering the iteration process.
5: while da> do
6: Compute parameters wi,kiand estimate ˆc.
7: Compute the coefficients f1and f2.
8: Estimate parameter ˆa.
9: Compute parameter da=|ˆa−a0|.
10: Update the approximate values with the estimated ones (a0= ˆa).
11: end while
12: return ˆaand ˆc, with b= 1.
5.2. Fitting of a straight line in 2D 105
Iterative procedure of pseudo-quadratic and pseudo-cubic equations
For the sake of a complete overview of the iterative algorithms that can produce the weighted least squares
solution for fitting a straight line in 2D, two alternatives to the developed pseudo-linear equations are
presented here. A thorough analysis of (5.52) and (5.53) can lead to a pseudo-cubic equation instead, as it
has been shown by York (1966), which reads
a3
n
X
i=1
w2
ix0
i
2
pxi−2a2
n
X
i=1
w2
ix0
iy0
i
pxi−an
X
i=1
wix0
i
2−
n
X
i=1
w2
iy0
i
2
pxi+
n
X
i=1
wix0
iy0
i,(5.58)
restricted to b= 1. The coordinates x0and y0are the reduced coordinates to the pseudo-centre of the mass
and can be computed by
x0
i=xi−
n
X
i=1
wixi
n
X
i=1
wi
and y0
i=yi−
n
X
i=1
wiyi
n
X
i=1
wi
.(5.59)
The term ”pseudo-centre” originates from (Deming 1964, p. 134,181) and owes its name to the auxiliary
parameters wi, which will change their values in each iteration step. Taking into account the interesting
comments of Williamson (1968), this pseudo-cubic equation can be alternatively expressed by a pseudo-
quadratic or even as a pseudo-linear one. Such a pseudo-quadratic equation has been presented in (York,
1968) including correlations between the observations, which is equivalent to the development of (Petrovi´c
et al., 1983) when setting the correlations equal to zero and is expressed by
a2
n
X
i=1
w2
i
pxi
(pyix0
iy0
i) + an
X
i=1
w2
i
pxi
y0
i
2−
n
X
i=1
w2
i
pyi
x0
i
2−
n
X
i=1
w2
i
pyi
x0
i
2y0
i
2,(5.60)
under the restriction of b= 1. Iterative algorithms can be easily built for the least squares solution of the
line parameters, by making use either of the pseudo-cubic (5.58) or the pseudo-quadratic equation (5.60).
5.2.4 Weighting case 4 - Individually weighted and correlated 2D coordinates
The developed iterative procedure of the previous subsection can be extended to include correlations between
the observed quantities. Correlations are often considered in geodetic applications, as it is typical that the 2D
coordinates of points are not the original measured quantities, but they have been obtained for example by
polar measurements. In another example, these coordinates are the outcome of some previous adjustment,
for instance of a 2D network. The precision of the adjusted 2D coordinates is obtained in most cases from a
linear error propagation that sometimes results in a cofactor matrix that includes correlations between the
observations, depending on the adjustment problem.
It is very important to point out here that the precisions of the coordinates coming from a linearized
error propagation is just an approximation, as it has been discussed in section 2.3. Therefore, when the
original measurements are polar coordinates, then these should be utilized for obtaining a rigorous least
squares solution. However, the adjusted 2D Cartesian coordinates of points and their approximated stochastic
model from the error propagation are often used in practice. Therefore, this subsection is dedicated to
those practical least squares solutions for fitting straight lines in 2D, taking into account the approximated
variances and covariances of the 2D point coordinates.
106 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
It would be beneficial at this point to introduce vector/matrix notation, in order to derive simpler equations
for the solution of the adjustment problem. Firstly, the cofactor matrix QLL is given or obtained from a
previous adjustment and can be written as
QLL ="Qxx Qxy
Qyx Qyy #,with Qxy =QT
yx.(5.61)
Qxx and Qyy are the cofactor matrices of the measured 2D coordinates and matrices Qxy,Qyx hold the
correlations between the coordinates. The respective weight matrices
P=Q−1
LL ="Pxx Pxy
Pyx Pyy #,with Pxy =PT
yx.(5.62)
The nonlinear condition equations (5.1) can be expressed equivalently in vector notation by
a(xc+vx) + b(yc+vy) + ce=0,(5.63)
with vectors xcand yclisting the coordinates of the measured points
xc=
x1
x2
.
.
.
xn
,yc=
y1
y2
.
.
.
yn
(5.64)
and vectors vxand vycarrying the corresponding residuals
vx=
vx1
vx2
.
.
.
vxn
,vy=
vy1
vy2
.
.
.
vyn
.(5.65)
Vector eis a vector of ones
e=
1
.
.
.
1
,(5.66)
with dimension being equal to the number of nmeasured points. A least squares solution of the problem
can be derived by minimizing an objective function expressed in matrix notation by
Ω(vx,vy) = vT
xPxxvx+vT
yPyyvy+ 2 vT
xPxyvy.(5.67)
5.2. Fitting of a straight line in 2D 107
5.2.4.1 Iterative least squares solution without linearization
Combining the developed condition equations (5.63) and the objective function (5.67), leads to the Lagrange
function
K(a, b, c, vx,vy,k) = Ω(vx,vy)−2kT[a(xc+vx) + b(yc+vy) + ce],(5.68)
with kdenoting the vector of Lagrange multipliers. Following the same procedure as in the previous sections,
a minimum for K is obtained by differentiation with respect to all unknown parameters and setting the partial
derivatives to zero, resulting in the system of nonlinear normal equations
∂K
∂vT
x
= 2 (Pxxvx+Pxyvy−ak) = 0,(5.69)
∂K
∂vT
y
= 2 (Pyyvy+Pyxvx−bk) = 0,(5.70)
∂K
∂kT=−2 [a(xc+vx) + b(yc+vy) + ce] = 0,(5.71)
∂K
∂a =−2kT(xc+vx)=0,(5.72)
∂K
∂b =−2kT(yc+vy) = 0,(5.73)
∂K
∂c =−2kTe= 0.(5.74)
A linearisation or approximation of the original problem is avoided here. Explicit expressions for the residuals
can be obtained by expressing equations (5.69) and (5.70) using block matrices:
"Pxx Pxy
Pyx Pyy #" vx
vy#="ak
bk#.(5.75)
The residual vectors can be computed by
"vx
vy#="Pxx Pxy
Pyx Pyy #−1"ak
bk#="Qxx Qxy
Qyx Qyy #" ak
bk#,(5.76)
which yields
vx= (aQxx +bQxy)k(5.77)
and
vy= (aQyx +bQyy)k.(5.78)
108 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
Equivalently, the residual vectors can be computed by using the properties of inverting block matrices, as it
has been discussed in (Snow 2012, p. 22). Moreover, introducing the expressions for vxand vyinto equation
(5.71) yields
a2Qxx +b2Qyy +abQxy +abQyxk=−(axc+byc+ce).(5.79)
Introducing appropriate approximate values a0and b0in the left hand side of the last equation, allows us
to build the auxiliary matrix
W=a02Qxx +b02Qyy +a0b0Qxy +a0b0Qyx (5.80)
and write equation (5.79) as
Wk =−(axc+byc+ce).(5.81)
In case of regular cofactor matrices, matrix Wis also regular and invertible (the case of singular cofactor
matrices in Wis discussed in the next subsection). This leads to the vector of Lagrange multipliers
k=−W−1(axc+byc+ce).(5.82)
Furthermore, substituting vector kinto the normal equation (5.74) results in
eTk= 0
⇒ −eTW−1(axc+byc+ce)=0
⇒c=−aheTW−1e−1eTW−1xci−bheTW−1e−1eTW−1yci.
(5.83)
The solution for the unknown line parameters aand bcan be obtained by analyzing further equations (5.72)
and (5.73). Taking into account the solution for the vector of Lagrange multipliers from equation (5.82) and
for the residual vectors from (5.77) and (5.78), yields the reduced system of equations
−xT
cW−1(axc+byc+ce) + kT(aQxxk+bQxyk) = 0 (5.84)
and
−yT
cW−1(axc+byc+ce) + kT(aQxyk+bQyyk)=0.(5.85)
Substituting also parameter cfrom (5.83), the last two equations can be equivalently written as
a f1+b f2= 0 (5.86)
5.2. Fitting of a straight line in 2D 109
and
b f3+a f2= 0,(5.87)
with the respective quantities being
f1=kTQxxk−xT
cW−1xc+xT
cW−1eeTW−1e−1eTW−1xc,
f2=kTQxyk−xT
cW−1yc+yT
cW−1eeTW−1e−1eTW−1xc,
f3=kTQyyk−yT
cW−1yc+yT
cW−1eeTW−1e−1eTW−1yc.
(5.88)
Equations (5.86) and (5.87) form a system of pseudo-linear equations with two unknown parameters (i.e.
the line parameters aand b). The term “pseudo-linear” comes from the fact that parameters f1,f2and f3
contain matrix W, which has been built using the approximations a0and b0. Furthermore, a restriction of
the problem to b= 1 leads to the solution for the remaining unknown parameter
a=−f2
f1
.(5.89)
An iterative procedure for estimating the unknown line parameters can be found in Algorithm 2. A similar
iterative solution for this weighting case has been presented in (York 1968).
Algorithm 2 Least squares fitting of a straight line in 2D with general weights and correlations
1: Choose approximate value for a0.
2: Define parameter b= 1.
3: Set threshold for the break-off condition of the iteration process.
4: Set parameter da=|ˆa−a0|=∞, for entering the iteration process.
5: while da> do
6: Compute the auxiliary matrix W, and the vector of Lagrange multipliers k.
7: Estimate parameter ˆc.
8: Compute the coefficients f1and f2.
9: Estimate parameter ˆa.
10: Compute parameter da=|ˆa−a0|.
11: Update the approximated parameter with the estimated (a0= ˆa).
12: end while
13: return ˆaand ˆc, with b= 1.
5.2.4.2 Solution for singular cofactor matrices
Postulating regular cofactor matrices in equation (5.81) permitted the inversion of matrix W. This led
to the reduction of the normal equations and the solution for the unknown line parameters. However, if
the given cofactor matrices are singular, then a solution for the vector of Lagrange multipliers cannot be
obtained using equation (5.82), as long as matrix Wis not invertible anymore. However, a solution with
iterations is possible also for the case of singular cofactor matrices following a rather different procedure.
110 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
Taking into account equation (5.81), together with the normal equations (5.72)-(5.74), the following system
of equations emerges:
Wk =−(axc+byc+ce),
kT(xc+vx)=0,
kT(yc+vy)=0,
kTe= 0.
(5.90)
If the chosen restriction between the unknown parameters is b= 1, then the developed normal equations
can be written as4
Wk =−(axc+yc+ce),
kT(xc+vx)=0,
kTe= 0.
(5.91)
This equation system can be expressed in a block matrix form, after introducing approximate values for the
residual vector v0
x
W xce
xc+v0
xT0 0
eT0 0
k
a
c
=
−yc
0
0
.(5.92)
This can be equivalently written as
N"k
X#=n,(5.93)
with matrices
N=
W xce
xc+v0
xT0 0
eT0 0
,n=
−yc
0
0
(5.94)
and the vector of unknown parameters
X="a
c#.(5.95)
4The third equation, kT(yc+vy) = 0, is not taken into account as we set bas a fixed parameter.
5.2. Fitting of a straight line in 2D 111
A least squares solution for this adjustment problem can be computed by
"ˆ
k
ˆ
X#=N−1n,(5.96)
without applying any linearization to the original problem.
Solution with a symmetric normal matrix N
The developed normal matrix Nfrom equation (5.94) is nonsymmetric. An equivalent solution of the
problem using, however, a symmetric matrix Ncan be obtained by adding the term avxto both sides of
equation (5.81)5. This leads to the equation system
Wk +a(xc+vx) + ce=−yc+avx,
kT(xc+vx)=0,
kTe= 0
(5.97)
or written in block matrix form
"W A
AT0#" k
X#="w
0#,(5.98)
with matrix
A=xc+v0
x,e(5.99)
and vector
w=−yc+a0v0
x.(5.100)
It is worth noticing the similarity of matrix Aand vector wwith those from the GHM, as explained in
section 3.1.2, without however applying a linearization to the functional model. Therefore, a solution of the
adjustment problem can be obtained by equation (5.96), after introducing
N="W A
AT0#,n="w
0#,(5.101)
where Nis in this case symmetric.
The inversion of matrix Ndepends on the rank deficiency of matrix W, which is not invertible as long as the
cofactor matrices of the adjustment problem are singular. An elegant way to ensure that a unique solution
exists, even when singular cofactor matrices must be employed, is the fullfilment of the Neitzel-Schaffrin
5A similar procedure for deriving a symetric normal matrix Nis already known in the literature dealing with WTLS
algorithms. An example can be found in (Snow 2012, pp. 25-26).
112 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
(NS) criterion that was proposed by Neitzel and Schaffrin (2016). As a linearization of the problem has been
avoided in the presented solution strategy, a similar criterion is presented in the following that will ensure a
unique solution for the unknown parameters. Starting with the equation system from equation (5.98), with
- rank of W≤n, with n= number of condition equations;
- rank of A=m, with m= number of unknown parameters;
- redundancy : rd=n−m;
while the rank of matrix Wwill be smaller than nin cases of singular cofactor matrices. Similar to the NS
criterion for the GHM, a unique solution will exist if the rank of the augmented matrix [W|A] is equal to
the number of condition equations nof the problem (which is equal to the rank of matrix Bin the GHM).
A criterion that would ensure a unique solution of the problem can be described in this case by
rank ([W|A]) = n. (5.102)
An iterative procedure is presented in Algorithm 3, for the special cases of fitting straight lines in 2D with
singular cofactor matrices. It must be pointed out that this is a general procedure and can be used to derive
a solution also for the previous examined weighting cases for fitting a straight line in 2D. However, this
iteration process involves more complicated normal equations and thus must be preferred when a singular
cofactor matrix is given.
Algorithm 3 Least squares fitting of a straight line in 2D with singular cofactor matrices
1: Choose approximate values for a0,v0
xand v0
y.
2: Define parameter b= 1.
3: Set threshold for the break-off condition of the iteration process.
4: Set parameter da=|ˆa−a0|=∞, for entering the iteration process.
5: while da> do
6: Compute matrices W,Aand vector w.
7: Build matrix Nand vector n.
8: Estimate the vector of unknowns "ˆ
k
ˆ
X#.
9: Compute the residual vectors vxand vy.
10: Compute parameter da=|ˆa−a0|.
11: Update the approximate values with the estimated ones, with a0= ˆa,v0
x=vxand v0
y=vy.
12: end while
13: return ˆaand ˆc, with b= 1.
5.3. Fitting of a plane in 3D 113
5.3 Fitting of a plane in 3D
The problem of fitting a plane to an observed point cloud in 3D is investigated here, with the observed
coordinates being obtained with different precisions. The general form of a plane in 3D has been already
presented in subsection 4.4:
ax +by +cz +d= 0,
with x,yand zdenoting the 3D coordinates of a point that lies in the plane. a,b,cand dare the unknown
plane parameters. Random errors will influence the observed quantities, thus individual residuals can be
introduced in the functional model, leading to the condition equations
a(xi+vxi) + b(yi+vyi) + c(zi+vzi) + d= 0.(5.103)
An “optimal” solution for the unknown plane parameters is possible by minimizing the sum of weighted
squared residuals.The least squares solution of the best plane is investigated, for four different weighting
cases:
1. Same precision σxfor the coordinates in xdirection, σyin ydirection and σzin zdirection.
2. Individual precision for each point: σxi=σyi=σzi∀i.
3. Individual precision for each coordinate.
4. Individual precision and correlations between the measured 3D coordinates.
5.3.1 Weighting case 1 - Equally weighted observations in each direction
The fitting a plane to a 3D point cloud is examined in this subsection, with the measured point coordinates
being observed with the same precision in each direction. The estimation of the unknown plane parameters
is based on the minimization of the sum of weighted squared residuals
n
X
i=1
pxv2
xi+pyv2
yi+pzv2
zi→min,(5.104)
with the constant weights px,pyand pzbeing computed by
px=1
σ2
x
, py=1
σ2
y
and pz=1
σ2
z
.(5.105)
Following the same solution strategy as in subsection 5.2.1, the observed 3D coordinates are multiplied
beforehand with the respective weights leading to the scaled coordinates
xs
i=xi√px, ys
i=yi√pyand zs
i=zi√pz(5.106)
and the respective residuals
vs
xi=vxi√px, vs
yi=vyi√pyand vs
zi=vzi√pz.(5.107)
114 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
Substituting the scaled coordinates and their residuals into the condition equations (5.103) yields
a1
√pxxs
i+vs
xi+b1
√pyys
i+vs
yi+c1
√pzzs
i+vs
zi+d= 0.(5.108)
Furthermore, introducing the auxiliary scaled plane parameters
as=a1
√px
,
bs=b1
√py
,
cs=c1
√pz
,
ds=d,
(5.109)
in equation (5.108), results in the functional model
asxs
i+vs
xi+bsys
i+vs
yi+cszs
i+vs
zi+ds= 0.(5.110)
Similar to the procedure of subsection 4.4.1, a “meaningful” restriction between the unknown plane param-
eters is
as2+bs2+cs2= 1.(5.111)
The least squares solution for the unknown plane parameters can be derived by minimizing the sum of
squared scaled residuals n
X
i=1
vs
xi
2+vs
yi
2+vs
zi
2→min.(5.112)
Direct least squares solution in a scaled coordinate system
Utilizing the scaled 3D coordinates, the original problem is transformed to a problem with equal weights.
Thus, the sum of squared residuals is equal to the sum of squared orthogonal distances
Ω(vs
xi, vs
yi) =
n
X
i=1
vs
xi
2+vs
yi
2+vs
zi
2=
n
X
i=1
D2
i,(5.113)
with the distances of the measured points to the requested plane
Di=asxs
i+bsys
i+cszs
i+ds
pas2+bs2+cs2.(5.114)
Taking into account the selected restriction from equation (5.111), the expressions for the orthogonal dis-
tances become
Di=asxs
i+bsys
i+cszs
i+ds.(5.115)
A least squares solution for the unknown plane parameters can be derived in this case directly, following the
same procedure as the one presented in subsection 4.4.
5.3. Fitting of a plane in 3D 115
Computation of the plane parameters in the original coordinate system
Introducing the estimated plane parameters ˆas,ˆ
bs, ˆcsand ˆ
dsinto equation (5.109), yields the plane param-
eters in the original coordinate system
ˆa= ˆas√px,
ˆ
b=ˆ
bs√py,
ˆc= ˆcs√pz,
ˆ
d=ds,
(5.116)
however, being restricted to
as2+bs2+cs2=a
√px2
+b
√py2
+c
√pz2
= 1.(5.117)
Therefore, a least squares solution that is restricted to a2+b2+c2= 1 can be computed by
ˆa=ˆas√px
rˆas√px2+ˆ
bs√py2+ˆcs√pz2
,
ˆ
b=ˆ
bs√px
rˆas√px2+ˆ
bs√py2+ˆcs√pz2
,
ˆc=ˆcs√pz
rˆas√px2+ˆ
bs√py2+ˆcs√pz2
,
ˆ
d=ˆ
ds
rˆas√px2+ˆ
bs√py2+ˆcs√pz2
.
(5.118)
5.3.2 Weighting case 2 - Individually weighted points in 3D
In the second weighting scenario for fitting a plane in 3D, every point has been observed with individual
precision:
σxi=σyi=σzi∀i
⇒pxi=pyi=pzi=pi∀i.
(5.119)
Also in this case the ratio between the weights of each point is constant:
pxi
pyi
= constant ,pyi
pzi
= constant and pxi
pzi
= constant.(5.120)
116 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
A least squares solution for the unknown plane parameters can be found by minimizing
Ω(vxi, vyi, vzi) =
n
X
i=1
pxiv2
xi+pyiv2
yi+pziv2
zi,(5.121)
which after taking into account the stochastic model of equation (5.119), becomes
Ω(vxi, vyi, vzi) =
n
X
i=1
pi(v2
xi+v2
yi+v2
zi)→min.(5.122)
Direct weighted least squares solution
The orthogonal distances of the measured points to the requested plane are
Di=axi+byi+czi+d
√a2+b2+c2(5.123)
and after selecting the restriction
a2+b2+c2= 1,(5.124)
can be equivalently written as
Di=axi+byi+czi+d. (5.125)
The sum of weighted squared residuals of this adjustment problem is equal to the sum of weighted squared
orthogonal distances
piD2
i=piv2
xi+v2
yi+v2
zi.(5.126)
Thus, the objective function is
Ω(a, b, c, d) =
n
X
i=1
pi(v2
xi+v2
yi+v2
zi) = piD2
i=pi(axi+byi+czi+d)2.(5.127)
To obtain a solution for the unknown plane parameters which minimizes equation (5.127) under the chosen
restriction, the Lagrangian
K(a, b, c, k) =
n
X
i=1
pi(axi+byi+czi+d)2−k(a2+b2+c2−1),(5.128)
can be built. The normal equation system for this problem is
∂K
∂a = 2 "a n
X
i=1
pixi2−k!+b n
X
i=1
piyixi!+c n
X
i=1
pixizi!+d n
X
i=1
pixi!#= 0,(5.129)
∂K
∂b = 2 "a n
X
i=1
piyixi!+b n
X
i=1
piyi2−k!+c n
X
i=1
piyizi!+d n
X
i=1
piyi!#= 0,(5.130)
5.3. Fitting of a plane in 3D 117
∂K
∂c = 2 "a n
X
i=1
pixizi!+b n
X
i=1
piyizi!+c n
X
i=1
pizi2−k!+d n
X
i=1
pizi!#= 0,(5.131)
∂K
∂d = 2 "a
n
X
i=1
pixi+b
n
X
i=1
piyi+c
n
X
i=1
pizi+d
n
X
i=1
pi#= 0,(5.132)
∂K
∂k =−a2+b2+c2−1= 0.(5.133)
Equation (5.132) can be rearranged to
d=−1
n
X
i=1
pi a
n
X
i=1
xi+b
n
X
i=1
yi+c
n
X
i=1
zi!.(5.134)
The expression for dcan be subsequently introduced into equations (5.129) - (5.131). This yields the reduced
normal equations
a
n
X
i=1
pixi2−1
n
X
i=1
pi n
X
i=1
pixi!2
−k
+b
n
X
i=1
pixiyi−1
n
X
i=1
pi n
X
i=1
pixi
n
X
i=1
piyi!
+
c
n
X
i=1
pixizi−1
n
X
i=1
pi n
X
i=1
pixi
n
X
i=1
pizi!
= 0,
(5.135)
a
n
X
i=1
pixiyi−1
n
X
i=1
pi n
X
i=1
pixi
n
X
i=1
piyi!
+b
n
X
i=1
piyi2−1
n
X
i=1
pi n
X
i=1
piyi!2
−k
+
c
n
X
i=1
piyizi−1
n
X
i=1
pi n
X
i=1
piyi
n
X
i=1
pizi!
= 0
(5.136)
and
118 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
a
n
X
i=1
pixizi−1
n
X
i=1
pi n
X
i=1
pixi
n
X
i=1
pizi!
+b
n
X
i=1
piyizi−1
n
X
i=1
pi n
X
i=1
piyi
n
X
i=1
pizi!
+
c
n
X
i=1
pizi2−1
n
X
i=1
pi n
X
i=1
pizi!2
−k
= 0,
(5.137)
which represent a homogeneous system of equations. A nontrivial solution is possible by setting the deter-
minant equal to zero:
(f1−k)g1g2
g1(f2−k)g3
g2g3(f3−k)
= 0,(5.138)
with the respective quantities being
f1=
n
X
i=1
pixi2−1
n
X
i=1
pi n
X
i=1
pixi!2
, f2=
n
X
i=1
piyi2−1
n
X
i=1
pi n
X
i=1
piyi!2
,
f3=
n
X
i=1
pizi2−1
n
X
i=1
pi n
X
i=1
pizi!2
,
g1=
n
X
i=1
pixiyi−1
n
X
i=1
pi n
X
i=1
pixi
n
X
i=1
piyi!, g2=
n
X
i=1
pixizi−1
n
X
i=1
pi n
X
i=1
pixi
n
X
i=1
pizi!,
g3=
n
X
i=1
piyizi−1
n
X
i=1
pi n
X
i=1
piyi
n
X
i=1
pizi!.
(5.139)
Equation (5.138) is a cubic characteristic equation with three solutions for k. The unknown plane parameters
can be estimated either by substituting kmin into equations (5.135)-(5.137) or by solving an eigenvalue
problem.
5.3.3 Weighting case 3 - Individually weighted 3D coordinates
For the third weighting case of fitting a plane to a 3D point cloud, the observed 3D coordinates of each point
have been obtained with individual precisions. The objective function under minimization can be expressed
in this case as
Ω(vxi, vyi, vzi) =
n
X
i=1
pxiv2
xi+pyiv2
yi+pziv2
zi.(5.140)
5.3. Fitting of a plane in 3D 119
Iterative least squares solution without linearization
A least squares solution can be computed for this nonlinear adjustment by minimizing the Lagrange function
K(a, b, c, vxi, vyi, vzi, ki) = Ω(vxi, vyi, vzi)−2
n
X
i=1
ki[a(xi+vxi) + b(yi+vyi) + c(zi+vzi) + d].(5.141)
Avoiding any kind of linearization of the problem, a differentiation of function K with respect to all unknown
parameters and setting the partial derivatives equal to zero yields the system of normal equations
∂K
∂vxi
= 2pxivxi−2aki= 0
⇒vxi=aki
pxi
,(5.142)
∂K
∂vyi
= 2pyivyi−2bki= 0
⇒vyi=bki
pyi
,(5.143)
∂K
∂vzi
= 2pzivzi−2cki= 0
⇒vzi=cki
pzi
,(5.144)
∂K
∂k =−2 [a(xi+vxi) + b(yi+vyi) + c(zi+vzi) + d]=0,(5.145)
∂K
∂a =−2
n
X
i=1
ki(xi+vxi)=0,(5.146)
∂K
∂b =−2
n
X
i=1
ki(yi+vyi)=0,(5.147)
∂K
∂c =−2
n
X
i=1
ki(zi+vzi)=0,(5.148)
∂K
∂d =−2
n
X
i=1
ki= 0.(5.149)
Equations (5.142)-(5.149) represent a nonlinear system of 4n+4 equations. The residuals from (5.142)-(5.144)
are introduced into equation (5.145), that yields the expression for the Lagrange multipliers
ki=wi(axi+byi+czi+d),(5.150)
120 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
with the auxiliary weighting factors
wi=−a2
pxi
+b2
pyi
+c2
pzi −1
.(5.151)
Introducing kiinto (5.149) returns
d=−
a
n
X
i=1
wixi+b
n
X
i=1
wiyi+c
n
X
i=1
wizi
n
X
i=1
wi
.(5.152)
Substituting the residuals vxi,vyi,vziand the Lagrange multipliers kiinto equations (5.146), (5.147) and
(5.148), results in the system of reduced equations
a
n
X
i=1
k2
i
pxi
=−
n
X
i=1
wi(axi+byi+czi+d)xi,(5.153)
b
n
X
i=1
k2
i
pyi
=−
n
X
i=1
wi(axi+byi+czi+d)yi(5.154)
and
c
n
X
i=1
k2
i
pzi
=−
n
X
i=1
wi(axi+byi+czi+d)zi.(5.155)
Taking into account parameter dyields
af1=bf2+cf3,(5.156)
bf4=af2+cf5(5.157)
and
cf6=af3+bf5,(5.158)
5.3. Fitting of a plane in 3D 121
with the quantities
f1=
n
X
i=1
k2
i
pxi−
n
X
i=1
wix2
i+1
n
X
i=1
wi n
X
i=1
wixi!,
f2=
n
X
i=1
(wixiyi)−1
n
X
i=1
wi n
X
i=1
wixi
n
X
i=1
wiyi!,
f3=
n
X
i=1
(wixizi)−1
n
X
i=1
wi n
X
i=1
wixi
n
X
i=1
wizi!,
f4=
n
X
i=1
k2
i
pyi−
n
X
i=1
wiy2
i+1
n
X
i=1
wi n
X
i=1
wiyi!,
f5=
n
X
i=1
(wiyizi)−1
n
X
i=1
wi n
X
i=1
wiyi
n
X
i=1
wizi!,
f6=
n
X
i=1
k2
i
pzi−
n
X
i=1
wiz2
i+1
n
X
i=1
wi n
X
i=1
wizi!.
(5.159)
At this stage a meaningful restriction between the unknown parameters can be selected. For the sake of
convenience the restriction c= 1 is chosen. Therefore, solving equation (5.157) for band introducing it in
(5.156) gives
ˆa=f1−f2
2
f4−1f2f5
f4
+f3(5.160)
and
ˆ
b= ˆaf2
f4+f5
f4
.(5.161)
Equations (5.160) and (5.161) become pseudo-linear after approximating functions f1, f2, . . . , f6. Therefore,
initial values for a0and b0are necessary for computing the auxiliary weighting factors wi, the Lagrange
multipliers ki, as well as functions f1, f2, . . . , f6. The estimated plane parameters can be utilized as new
starting values in each iteration step, until a break-off condition is met. Based on the presented procedure,
Algorithm 4 has been developed for estimating the weighted least squares solution of fitting a plane to a 3D
point-cloud.
122 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
Algorithm 4 Least squares fitting of a plane to points in 3D with general weights
1: Choose approximate values for a0,b0.
2: Define parameter c= 1.
3: Set threshold for the break-off condition of the iteration process.
4: Set parameter da=|ˆa−a0|=∞and db=|ˆ
b−b0|=∞, for entering the iteration process.
5: while da> or db> do
6: Compute parameters ki,piand estimate ˆ
d.
7: Compute the coefficients f1, f2, . . . , f5.
8: Estimate parameters ˆaand ˆ
b.
9: Compute parameter da=|ˆa−a0|and db=|ˆ
b−b0|.
10: Update the approximate values with the estimated ones (a0= ˆaand b0=ˆ
b).
11: end while
12: return ˆa,ˆ
band ˆ
d, with c= 1.
5.3.4 Weighting case 4 - Individually weighted and correlated 3D coordinates
For the fourth weighted adjustment problem under investigation, correlations are introduced between the
measurements. Therefore, the cofactor matrix of the 3D point coordinates is given or obtained from a
previous adjustment and is expressed by
QLL =
Qxx Qxy Qxz
Qyx Qyy Qyz
Qzx Qzy Qzz
,with Qxy =QT
yx ,Qxz =QT
zx and Qyz =QT
zy.(5.162)
Qxx,Qyy and Qzz represent the cofactor matrices for the x,yand zcoordinates, respectively. Matrices Qxy,
Qxz,Qyz,Qyx,Qzx and Qzy are holding the correlations between the 3D point coordinates. The respective
weight matrices are
P=Q−1
LL =
Pxx Pxy Pxz
Pyx Pyy Pyz
Pzx Pzy Pzz
.(5.163)
The nonlinear condition equations (5.103) can be expressed equivalently in vector notation by
a(xc+vx) + b(yc+vy) + c(zc+vz) + de=0.(5.164)
Vectors xc,ycand zcinclude the 3D point coordinates
xc=
x1
x2
.
.
.
xn
,yc=
y1
y2
.
.
.
yn
and zc=
z1
z2
.
.
.
zn
(5.165)
5.3. Fitting of a plane in 3D 123
and the residual vectors vx,vyand vzlist the corresponding residuals
vx=
vx1
vx2
.
.
.
vxn
,vy=
vy1
vy2
.
.
.
vyn
and vz=
vz1
vz2
.
.
.
vzn
.(5.166)
eis a vector of ones with length being equal to the number of measured points. A solution for this weighted
least squares problem can be derived by minimizing the objective function
Ω(vx,vy,vz) = vT
xPxxvx+vT
yPyyvy+vT
zPzzvz+ 2 vT
xPxyvy+ 2 vT
xPxzvz+ 2 vT
yPyzvz.(5.167)
Iterative least squares solution without linearization
The objective function (5.167) can be combined with the nonlinear condition equations (5.164) to build the
Lagrangian
K(a, b, c, d, vx,vy,vz,k) = Ω(vx,vy,vz)−2kT[a(xc+vx) + b(yc+vy) + c(zc+vz) + de].(5.168)
kis the vector of Lagrange multipliers. The resulting normal equations are
∂K
∂vT
x
= 2 (Pxxvx+Pxyvy+Pxzvz−ak) = 0,(5.169)
∂K
∂vT
y
= 2 (Pyyvy+Pyxvx+Pyzvz−bk) = 0,(5.170)
∂K
∂vT
z
= 2 (Pyyvy+Pzxvx+Pzyvy−ck) = 0,(5.171)
∂K
∂kT=−2 [a(xc+vx) + b(yc+vy) + c(zc+vz) + de] = 0,(5.172)
∂K
∂a =−2kT(xc+vx)=0,(5.173)
∂K
∂b =−2kT(yc+vy) = 0,(5.174)
∂K
∂c =−2kT(zc+vz)=0,(5.175)
∂K
∂d =−2kTe= 0.(5.176)
124 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
A solution for the unknown plane parameters can be obtained by analyzing the derived normal equations.
Expressing equations (5.169)-(5.171) with block matrices
Pxx Pxy Pxz
Pyx Pyy Pyz
Pzx Pzy Pzz
vx
vy
vz
=
ak
bk
ck
,(5.177)
it is possible to derive the residual vectors
vx
vy
vz
=
Pxx Pxy Pxz
Pyx Pyy Pyz
Pzx Pzy Pzz
−1
ak
bk
ck
=
Qxx Qxy Qxz
Qyx Qyy Qyz
Qzx Qzy Qzz
ak
bk
ck
,(5.178)
or equivalently
vx= (aQxx +bQxy +cQxz)k,(5.179)
vy= (aQyx +bQyy +cQyz)k,(5.180)
and
vz= (aQzx +bQzy +cQzz)k.(5.181)
Substituting further vx,vyand vzinto equation (5.172) gives
Wk =−(axc+byc+czc+de).(5.182)
The auxiliary matrix
W=a02Qxx +b02Qyy +c02Qzz +a0b0(Qxy +Qyx) + a0c0(Qxz +Qzx) + b0c0(Qyz +Qzy) (5.183)
can be constructed after introducing approximate values for parameters a0,b0and c0. In case of regular
cofactor matrices, matrix Wis also regular and invertible. Thus, a solution for the vector of Lagrange
multipliers can be obtained by
k=−W−1(axc+byc+czc+de).(5.184)
5.3. Fitting of a plane in 3D 125
Introducing vector kinto the normal equation (5.176) yields the expression
eTk= 0
⇒ −eTW−1(axc+byc+czc+de)=0
⇒d=−aheTW−1e−1eTW−1xci−bheTW−1e−1eTW−1yci−cheTW−1e−1eTW−1zci.
(5.185)
The solution for the unknown plane parameters a,band ccan be computed by analyzing further the normal
equations (5.173)-(5.175). Taking into account vector k, as well as the residual vectors vx,vyand vz, yields
the reduced system of equations
−xT
cW−1(axc+byc+cyc+de) + kT(aQxx +bQxy +cQxz)k= 0,(5.186)
−yT
cW−1(axc+byc+cyc+de) + kT(aQxx +bQxy +cQxz)k= 0,(5.187)
and
−zT
cW−1(axc+byc+cyc+de) + kT(aQzx +bQzy +cQzz)k= 0.(5.188)
Introducing parameter dfrom (5.185) into these three equations results in
af1+bf2+cf3= 0,(5.189)
bf4+af2+cf5= 0 (5.190)
cf6+af3+bf5= 0,(5.191)
with the defined parameters
f1=kTQxxk−xT
cW−1xc+xT
cW−1eeTW−1e−1eTW−1xc,
f2=kTQxyk−xT
cW−1yc+yT
cW−1eeTW−1e−1eTW−1xc,
f3=kTQxzk−xT
cW−1zc+zT
cW−1eeTW−1e−1eTW−1xc,
f4=kTQyyk−yT
cW−1yc+yT
cW−1eeTW−1e−1eTW−1yc,
f5=kTQyzk−yT
cW−1zc+zT
cW−1eeTW−1e−1eTW−1yc,
f6=kTQzzk−zT
cW−1zc+yT
cW−1eeTW−1e−1eTW−1zc.
(5.192)
126 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
Equations (5.189) - (5.191) form a system of pseudo-linear equations with three unknown parameters. A
restriction between the plane parameters can be chosen at this point. Selecting the restriction of c= 1, the
remaining unknown plane parameters are
ˆa=f2f5−f3f4
f1f4−f2
2
(5.193)
and
ˆ
b=−ˆaf2
f4−f5
f4
.(5.194)
An iterative procedure for estimating the unknown plane parameters, based on the presented approach, can
be found in Algorithm 5.
Algorithm 5 Least squares fitting of a plane in 3D with general weights and correlations
1: Choose approximate values for a0and b0.
2: Define parameter c= 1.
3: Set threshold for the break-off condition of the iteration process.
4: Set parameters da=|ˆa−a0|=∞and db=|ˆ
b−b0|=∞, for entering the iteration process.
5: while da> or db> do
6: Compute the auxiliary matrix W, and the vector of Lagrange multipliers k.
7: Estimate parameter ˆ
d.
8: Compute the coefficients f1, f2, . . . , f5.
9: Estimate parameters ˆaand ˆ
b.
10: Compute parameters da=|ˆa−a0|and db=|ˆ
b−b0|.
11: Update the approximate values with the estimated ones, a0= ˆaand b0=ˆ
b.
12: end while
13: return ˆa,ˆ
band ˆ
d, with c= 1.
Solution for singular cofactor matrices
An iterative solution for the case of singular cofactor matrices is possible also for this adjustment problem,
following the same procedure that has been presented in subsection 5.2.4.2. Therefore, putting together
equation (5.182) with the normal equations (5.173)-(5.176), results in the system of nonlinear equations
Wk =−(axc+byc+czc+de),
kT(xc+vx)=0,
kT(yc+vy)=0,
kT(zc+vz)=0,
kTe= 0.
(5.195)
5.3. Fitting of a plane in 3D 127
Furthermore, selecting c= 1 as a meaningful restriction between the unknown parameters and introducing
approximate values for the residual vectors v0
xand v0
y, the last equation system can be expressed by6
W xcyce
xc+v0
xT0 0 0
yc+v0
yT0 0 0
eT0 0 0
k
a
b
d
=
−zc
0
0
0
.(5.196)
This can be equivalently written as
N"k
X#=n,(5.197)
after introducing matrices
N=
W xcyce
xc+v0
xT0 0 0
yc+v0
yT0 0 0
eT0 0 0
,n=
−zc
0
0
0
(5.198)
and the vector of unknown parameters
X=
a
b
d
.(5.199)
A solution of this adjustment problem can be derived by
"ˆ
k
ˆ
X#=N−1n.(5.200)
Solution with a symmetric normal matrix N
Similarly to the procedure of subsection 5.2.4.2, an equivalent solution of the problem using a symmetric
matrix Ncan be obtained by adding the term avx+bvyto both sides of (5.182). Therefore, the equation
system (5.196) can be expressed as
"W A
AT0#" k
X#="w
0#,(5.201)
with matrix
A=xc+v0
x,yc+v0
y,e(5.202)
6The fourth equation, kT(zc+vz) = 0, is not taken into account as parameter cis treated as known.
128 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
and vector
w=−zc+a0v0
x+b0v0
y.(5.203)
A solution of the adjustment problem can be obtained by equation (5.200), after introducing
N="W A
AT0#,n="w
0#,(5.204)
with Nbeing symmetric.
The inversion of matrix Ndepends on the rank deficiency of matrix W, that will be faced when employ-
ing a singular cofactor matrix. Similar to the adjustment problem from subsection 5.2.4.2, the presented
criterion (5.102) will ensure the existence of a unique solution.
An iterative procedure is presented in Algorithm 6 for estimating the plane parameters when singular cofactor
matrices are given.
Algorithm 6 Least squares fitting of a plane in 3D with singular cofactor matrices.
1: Choose approximate values for a0,b0,v0
x,v0
yand v0
z.
2: Define parameter c= 1.
3: Set threshold for the break-off condition of the iteration process.
4: Define parameters da=|ˆa−a0|=∞and db=|ˆ
b−b0|=∞, for entering the iteration process.
5: while da> or db> do
6: Compute matrices W,Aand vector w.
7: Build matrix Nand vector n.
8: Estimate the unknown vector "ˆ
k
ˆ
X#.
9: Compute the residual vectors vx,vyand vz.
10: Compute parameters da=|ˆa−a0|and db=|ˆ
b−b0|.
11: Update the approximate values with the estimated ones, with a0= ˆa,b0=ˆ
b,v0
x=vx,v0
y=vyand
v0
z=vz.
12: end while
13: return ˆa,ˆ
band ˆ
d, with c= 1.
5.4 2D similarity transformation of coordinates
This subsection deals with the least squares solution of the 2D similarity transformation of coordinates,
where homologous points in two coordinate systems have been measured with different precisions. This
problem has been treated partly in (Teunissen, 1985, p. 148 ff.) as “symmetric Helmert transformation”. A
direct solution was obtained in terms of an eigenvalue problem, when cofactor matrices with special block
diagonal strucure were applied to the coordinates of the points in the two coordinate systems. Additionaly,
Marx (2017) derived direct solutions for “point-wise” and proportional weight matrices in the two coordinate
systems.
5.4. 2D similarity transformation of coordinates 129
The functional model of this problem has been already presented in subsection 4.5, by the system of equations
Xi=ξ1xi−ξ2yi+tx,
Yi=ξ2xi+ξ1yi+ty.
Assuming that all point coordinates are measured quantities the necessary residuals are introduced in the
functional model, resulting in the system of nonlinear condition equations
Xi+vXi−ξ1(xi+vxi) + ξ2(yi+vyi)−tx= 0,
Yi+vYi−ξ2(xi+vxi)−ξ1(yi+vyi)−ty= 0,(5.205)
with i= 1, . . . , n indicating the number of observed homologous points. This functional model can be
expressed equivalently by the overparameterized system
γ(Xi+vXi) + α(xi+vxi)−β(yi+vyi) + tx= 0,
γ(Yi+vYi) + β(xi+vxi) + α(yi+vyi) + ty= 0,(5.206)
with
ξ1=−α
γand ξ2=−β
γ.(5.207)
Following the procedure of subsection 4.5.1, a constraint can been chosen as
α2+β2+γ2= 1.(5.208)
The least squares criterion is utilized for an “optimal” estimate of the unknown transformation parameters,
by minimizing the sum of weighted squared residuals
n
X
i=1 pXivXi
2+pYivYi
2+pxivxi
2+pyivyi
2→min,(5.209)
where pXi,pYiare the weights influencing the residuals of the coordinates in the target system, respectively
pxi,pyithe weights in the source system. This adjustment problem is investigated in the next sections for
four different weighting cases:
1. Same precision σX=σY, for the coordinates in Xand Ydirection of the points in the target coordinate
system. Same precision σx=σy, for the coordinates in the directions of xand yof the points in the
source coordinate system.
2. Individual precision for each pair of homologous points in the two systems: σxi=σyi=σXi=σYi∀i.
3. Individual precision for each coordinate.
4. Individual precisions and correlations between the measured 2D coordinates in each system.
130 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
5.4.1 Weighting case 1 - Equally weighted observations in each coordinate sys-
tem
For the first weighting case the measured points in the target system have been observed with the same
precision σX=σYand the points in the source system with σx=σy. Thus, the objective function under
minimization becomes
Ω(vXi, vYi, vxi, vyi) =
n
X
i=1
pXY vXi
2+vYi
2+pxy vxi
2+vyi
2→min.(5.210)
with 1
σX2
i
=1
σY2
i
=pXY =constant ∀i,
1
σx2
i
=1
σy2
i
=pxy =constant ∀i.
(5.211)
From a geometric perspective the postulated weights can be seen as a homogeneous scale in each coordinate
system, according to the respective weight. For obtaining a direct solution, the coordinates are transformed
linearly by multiplying them with the respective weights
Xs
i=Xi√pXY , Y s
i=Yi√pXY and xs
i=xi√pxy , ys
i=yi√pxy.(5.212)
The scaled coordinates Xs
i,Ys
ifrom the target coordinate system and xs
i,ys
ifrom the source coordinate
system can be used to derive the requested transformation parameters with equally weighted observations.
In this line of thinking the residuals are
vs
Xi=vXi√pXY , vs
Yi=vYi√pXY and vs
xi=vxi√pxy , vs
yi=vyi√pxy.(5.213)
Substituting the scaled coordinates and their residuals from equations (5.212) and (5.213) into the condition
equations (5.206) yields
γ1
√pXY
(Xs
i+vs
Xi) + α1
√pxy
(xs
i+vs
xi)−β1
√pxy
(ys
i+vs
yi) + tx= 0,
γ1
√pXY
(Ys
i+vs
Yi) + β1
√pxy
(xs
i+vs
xi) + α1
√pxy
(ys
i+vs
yi) + ty= 0.
(5.214)
Introducing the scaled transformation parameters
αs=α1
√pxy
,
βs=β1
√pxy
,
γs=γ1
√pXY
,
ts
x=tx,
ts
y=ty,
(5.215)
5.4. 2D similarity transformation of coordinates 131
into equation (5.214), results in
γsXs
i+vs
Xi+αsxs
i+vs
xi−βsys
i+vs
yi+ts
x= 0,
γsYs
i+vs
Yi+βsxs
i+vs
xi+αsys
i+vs
yi+ts
y= 0.(5.216)
A meaningful constraint between the transformation parameters is
αs2+βs2+γs2= 1 (5.217)
and an “optimal” solution is possible by minimizing the sum of scaled squared residuals
n
X
i=1
vs
Xi
2+vs
Yi
2+vs
xi
2+vs
yi
2→min.(5.218)
Direct least squares solution
The scaling of the measured 3D coordinates leads to the transformation of the original problem into a
problem with equal weights. This means that the sum of squared residuals is equal to the sum of squared
Euclidean distances between the points in the target system and the transformed points from the source
system: n
X
i=1
vs
Xi
2+vs
Yi
2+vs
xi
2+vs
yi
2=
n
X
i=1
D2
i→min.(5.219)
The squared distances between the homologous points are
D2
i= (γsXs
i+αsxs
i−βsys
i+ts
x)2+ (γsYs
i+βsxs
i+αsys
i+ts
y)2.(5.220)
A least squares solution for the transformation parameters can be obtained directly, following the procedure
of subsection 4.5.
Computation of the transformation parameters in the original coordinate system
The original trasnformation parameters can be computed by substituting the estimated scaled tranformation
parameters into equation (5.215):
ˆα= ˆαs√pxy,
ˆ
β=ˆ
βs√pxy,
ˆγ= ˆγs√pXY ,
ˆ
tx=ˆ
ts
x,
ˆ
ty=ˆ
ts
y.
(5.221)
However, the developed solution is restricted to
αs2+βs2+γs2=α1
√pxy 2
+β1
√pxy 2
+γ1
√pXY 2
= 1.(5.222)
132 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
The least squares solution which restricts the transformation parameters to α2+β2+γ2= 1 can be found
by
ˆα=ˆαs√pxy
rˆαs√pxy2+ˆ
βs√pxy2+ˆγs√pXY 2
,
ˆ
β=
ˆ
βs√pxy
rˆαs√pxy2+ˆ
βs√pxy2+ˆγs√pXY 2
,
ˆγ=ˆγs√pXY
rˆαs√pxy2+ˆ
βs√pxy2+ˆγs√pXY 2
,
ˆ
tx=ˆ
ts
x
rˆαs√pxy2+ˆ
βs√pxy2+ˆγs√pXY 2
,
ˆ
ty=ˆ
ts
y
rˆαs√pxy2+ˆ
βs√pxy2+ˆγs√pXY 2
.
(5.223)
5.4.2 Weighting case 2 - Individual weight for each pair of homologous points
in both systems
In the second weighting case for the 2D similarity transformation the homologous points in the source and
target system have been measured with individual precisions
σXi=σYi=σxi=σyi∀i, (5.224)
with the respective weights
pXi=pYi=pxi=pyi=pi.(5.225)
A least squares solution invlolves the minimization of the objective function (5.209), which taking into
account the postulated weights becomes
n
X
i=1
pXivXi
2+pYivYi
2+pxivxi
2+pyivyi
2=
n
X
i=1
pivXi
2+vYi
2+vxi
2+vyi
2→min.(5.226)
Direct weighted least squares solution
It has been already shown in section 4.5 that the sum of squared residuals of this problem is equal to the
sum of squared Euclidean distances
n
X
i=1
D2
i=
n
X
i=1
(γXi+αxi−βyi+tx)2+ (γYi+βxi+αyi+ty)2.(5.227)
5.4. 2D similarity transformation of coordinates 133
For equally weighted homologous points, it is also true that
n
X
i=1
pivXi
2+vYi
2+vxi
2+vyi
2=
n
X
i=1
piD2
i.(5.228)
Therefore, the objective function under minimization can be written as
Ω(α, β, γ, tx, ty) =
n
X
i=1
piD2
i=
n
X
i=1
pi(γXi+αxi−βyi+tx)2+ (γYi+βxi+αyi+ty)2.(5.229)
A meaningful restriction for the unknown parameters is chosen also for this weighting case as
α2+β2+γ2= 1.(5.230)
We attempt to obtain the least squares solution for the unknown transformation parameters, that minimizes
equation (5.229) under the chosen restriction. Thus, the Lagrangian can be written as
K(α, β, γ, tx, ty, k) = Ω(α, β, γ, tx, ty)−k(α2+β2+γ2−1),(5.231)
with the normal equations
∂K
∂α = 2 "α n
X
i=1
pixi2+
n
X
i=1
piyi2−k!+γ n
X
i=1
pixiXi+
n
X
i=1
piyiYi!
+tx
n
X
i=1
pixi+ty
n
X
i=1
piyi#= 0,
(5.232)
∂K
∂β = 2 "β n
X
i=1
pixi2+
n
X
i=1
piyi2−k!+γ n
X
i=1
pixiYi−
n
X
i=1
piyiXi!
−tx
n
X
i=1
piyi+ty
n
X
i=1
pixi#= 0,
(5.233)
∂K
∂γ = 2 "α n
X
i=1
pixiXi+
n
X
i=1
piyiYi!+β n
X
i=1
pixiYi−
n
X
i=1
piyiXi!+γ n
X
i=1
pixi2+
n
X
i=1
piyi2−k!
+tx
n
X
i=1
piXi+ty
n
X
i=1
piYi#= 0,
(5.234)
∂K
∂tx
= 2 tx
n
X
i=1
pi+α
n
X
i=1
pixi−β
n
X
i=1
piyi+γ
n
X
i=1
piXi!= 0,(5.235)
∂K
∂ty
= 2 ty
n
X
i=1
pi+α
n
X
i=1
piyi+β
n
X
i=1
pixi+γ
n
X
i=1
piYi!= 0 (5.236)
134 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
and
∂K
∂k =−α2+β2+γ2−1= 0.(5.237)
A solution for the trasnlation parameters can be derived by rearanging equations (5.235) and (5.236), which
yields
tx=−α
n
X
i=1
pixi
n
X
i=1
pi
+β
n
X
i=1
piyi
n
X
i=1
pi
−γ
n
X
i=1
piXi
n
X
i=1
pi
(5.238)
and
ty=−α
n
X
i=1
piyi
n
X
i=1
pi
−β
n
X
i=1
pixi
n
X
i=1
pi
−γ
n
X
i=1
piYi
n
X
i=1
pi
.(5.239)
Substituting txand tyinto equations (5.232) - (5.234), results in the system of reduced normal equations
α
n
X
i=1
pixi2+
n
X
i=1
piyi2− n
X
i=1
pixi!2
n
X
i=1
pi
− n
X
i=1
piyi!2
n
X
i=1
pi
−k
+
γ
n
X
i=1
pixiXi+
n
X
i=1
piyiYi−
n
X
i=1
pixi
n
X
i=1
piXi
n
X
i=1
pi
−
n
X
i=1
piyi
n
X
i=1
piYi
n
X
i=1
pi
= 0,
(5.240)
β
n
X
i=1
pixi2+
n
X
i=1
piyi2− n
X
i=1
pixi!2
n
X
i=1
pi
− n
X
i=1
piyi!2
n
X
i=1
pi
−k
+
γ
n
X
i=1
pixiYi−
n
X
i=1
piyiXi+
n
X
i=1
piyi
n
X
i=1
piXi
n
X
i=1
pi
−
n
X
i=1
pixi
n
X
i=1
piYi
n
X
i=1
pi
= 0,
(5.241)
5.4. 2D similarity transformation of coordinates 135
and
α
n
X
i=1
pixiXi+
n
X
i=1
piyiYi−
n
X
i=1
pixi
n
X
i=1
piXi
n
X
i=1
pi
−
n
X
i=1
piyi
n
X
i=1
piYi
n
X
i=1
pi
+
β
n
X
i=1
pixiYi−
n
X
i=1
piyiXi+
n
X
i=1
piyi
n
X
i=1
piXi
n
X
i=1
pi
−
n
X
i=1
pixi
n
X
i=1
piYi
n
X
i=1
pi
+
γ
n
X
i=1
piXi2+
n
X
i=1
piYi2− n
X
i=1
piXi!2
n
X
i=1
pi
− n
X
i=1
piYi!2
n
X
i=1
pi
−k
= 0.
(5.242)
The Lagrange multiplier kcan be calculated by solving
(v1−k)w1w2
w1(v1−k)w3
w2w3(v2−k)
= 0,(5.243)
which is a cubic characteristic equation with the unknown parameter k.
The respective elements are
v1=
n
X
i=1
pixi2+
n
X
i=1
piyi2− n
X
i=1
pixi!2
n
X
i=1
pi
− n
X
i=1
piyi!2
n
X
i=1
pi
,
v2=
n
X
i=1
piXi2+
n
X
i=1
piYi2− n
X
i=1
piXi!2
n
X
i=1
pi
− n
X
i=1
piYi!2
n
X
i=1
pi
,
w1= 0,
w2=
n
X
i=1
pixiXi+
n
X
i=1
piyiYi−
n
X
i=1
pixi
n
X
i=1
piXi
n
X
i=1
pi
−
n
X
i=1
piyi
n
X
i=1
piYi
n
X
i=1
pi
,
w3=
n
X
i=1
pixiYi−
n
X
i=1
piyiXi+
n
X
i=1
piyi
n
X
i=1
piXi
n
X
i=1
pi
−
n
X
i=1
pixi
n
X
i=1
piYi
n
X
i=1
pi
.
(5.244)
136 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
The solution for the transformation parameters α,βand γcan be estimated either by substituting parameter
kmin into the reduced normal equations (5.240) - (5.242) or by transforming them and solve an eigenvalue
problem.
5.4.3 Weighting case 3 - Individually weighted coordinates
In the third investigated weighting case for the 2D similarity transformation the coordinates of the points
have been measured with different precisions, leading to individual weights for the residuals of each coordi-
nate. The variances of the observed coordinates in the source system can be stored in the variance-covariance
matrix
ΣLL1="Σxx 0
0 Σyy #=QLL1,for σ2
0= 1,(5.245)
with the sub-matrices
Σxx =
σ2
x10
σ2
x2
...
0σ2
xn
and Σyy =
σ2
y10
σ2
y2
...
0σ2
yn
.(5.246)
Selecting the variance of the unit weight being equal to one, the cofactor of the coordinates in the source
system is equal to the variance-covariance matrix (QLL1=ΣLL1) and the respective weight matrix is
P1=QLL
−1
1="Pxx 0
0 Pyy #,(5.247)
with the weight sub-matrices being expressed by
Pxx =
px10
px2
...
0pxn
and Pyy =
py10
py2
...
0pyn
.(5.248)
Analogously, the variance-covariance matrix of the observed coordinates in the target system can be written
as
ΣLL2="ΣXX 0
0 ΣYY #=QLL2,for σ2
0= 1,(5.249)
with
ΣXX =
σ2
X10
σ2
X2
...
0σ2
Xn
and ΣYY =
σ2
Y10
σ2
Y2
...
0σ2
Yn
.(5.250)
5.4. 2D similarity transformation of coordinates 137
Thus, the weight matrix for the coordinates in the target system is
P2=QLL
−1
2="PXX 0
0 PYY #,(5.251)
with the weight sub-matrices
PXX =
pX10
pX2
...
0pXn
and PYY =
pY10
pY2
...
0pYn
.(5.252)
The nonlinear condition equations (5.205) can be equivalently written in vector notation as
Xc+vX−ξ1(xc+vx) + ξ2(yc+vy)−txe=0,
Yc+vY−ξ2(xc+vx)−ξ1(yc+vy)−tye=0,(5.253)
with vectors Xc,Yclisting the coordinates of the points in the target system and xc,ycthe coordinates in
the source system:
Xc=
X1
X2
.
.
.
Xn
,Yc=
Y1
Y2
.
.
.
Yn
,xc=
x1
x2
.
.
.
xn
,yc=
y1
y2
.
.
.
yn
,(5.254)
while vectors vX,vY,vxand vycontain the residuals of the corresponding coordinates
vX=
vX1
vX2
.
.
.
vXn
,vY=
vY1
vY2
.
.
.
vYn
,vx=
vx1
vx2
.
.
.
vxn
,vy=
vy1
vy2
.
.
.
vyn
.(5.255)
eis a vector of ones, with length being equal to the number of homologous points. A solution based on
the least squares principle can be obtained by minimizing the objective function (5.209), written in matrix
notation as
Ω(vX,vY,vx,vy) = vT
XPXXvX+vT
YPYYvY+vT
xPxxvx+vT
yPyyvy.(5.256)
138 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
Iterative least squares solution without linearization
Utilizing the nonlinear functional model of equation (5.253) and the objective function (5.256), a least
squares solution can be derived by minimizing the Lagrange function
K(ξ1, ξ2, tx, ty,vX,vY,vx,vy,k1,k2) = Ω(vX,vY,vx,vy)
−2kT
1[−(Xc+vX) + ξ1(xc+vx)−ξ2(yc+vy) + txe]
−2kT
2[−(Yc+vY) + ξ2(xc+vx) + ξ1(yc+vy) + tye],
(5.257)
with k1and k2denoting the vectors of Lagrange multipliers. A linearization of the problem is avoided also
here. Differentiating the Lagrangian with respect to all unknowns and setting the partial derivatives to zero
yields
∂K
∂vT
X
= 2 (PXXvX+k1) = 0,(5.258)
∂K
∂vT
Y
= 2 (PYYvY+k2) = 0,(5.259)
∂K
∂vT
x
= 2 (Pxxvx−ξ1k1−ξ2k2) = 0,(5.260)
∂K
∂vT
y
= 2 (Pyyvy+ξ2k1−ξ1k2) = 0,(5.261)
∂K
∂kT
1
=−2 [−(Xc+vX) + ξ1(xc+vx)−ξ2(yc+vy) + txe] = 0,(5.262)
∂K
∂kT
2
=−2 [−(Yc+vY) + ξ2(xc+vx) + ξ1(yc+vy) + tye] = 0,(5.263)
∂K
∂ξ1
=−2kT
1(xc+vx) + kT
2(yc+vy)= 0,(5.264)
∂K
∂ξ2
=−2−kT
1(yc+vy) + kT
2(xc+vx)= 0,(5.265)
∂K
∂tx
=−2kT
1e= 0 (5.266)
5.4. 2D similarity transformation of coordinates 139
and
∂K
∂ty
=−2kT
2e= 0.(5.267)
Equations (5.258)-(5.267) represent a nonlinear system of 6n+ 4 equations. Substituting the residuals from
(5.258)-(5.261) into equations (5.262) and (5.263) yields
ξ2
1Qxx +ξ2
2Qyy +QXXk1+ (ξ1ξ2Qxx −ξ1ξ2Qyy)k2=−(ξ1xc−ξ2yc+txe−Xc)(5.268)
and
(ξ1ξ2Qxx −ξ1ξ2Qyy)k1+ξ2
2Qxx +ξ2
1Qyy +QYYk2=−(ξ2xc+ξ1yc+tye−Yc).(5.269)
Introducing approximate values for the unknown transformation parameters only in the left-hand side of
the last two equations, it is possible to write
W1k1+W2k2=−(ξ1xc−ξ2yc+txe−Xc) (5.270)
and
W2k1+W3k2=−(ξ2xc+ξ1yc+tye−Yc),(5.271)
with the auxiliary matrices defined as
W1=ξ0
1
2Qxx +ξ0
2
2Qyy +QXX,
W2=ξ0
1ξ0
2Qxx −ξ0
1ξ0
2Qyy,
W3=ξ0
2
2Qxx +ξ0
1
2Qyy +QYY.
(5.272)
Since the introduced cofactor matrices from equations (5.245) and (5.249) are diagonal and therefore regular,
then matrices W1,W2and W3are also regular and invertible. In this case a solution for the vectors of
Lagrange multipliers can be found by
k1=W5(ξ2xc+ξ1yc+tye−Yc)−W4(ξ1xc−ξ2yc+txe−Xc) (5.273)
and
k2=W5(ξ1xc−ξ2yc+txe−Xc)−W6(ξ2xc+ξ1yc+tye−Yc).(5.274)
The respective matrices are
W4=W1−W2W−1
3W2−1,
W5=W1−W2W−1
3W2−1W2W−1
3,W6=W3−W2W−1
3W2−1
.(5.275)
140 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
Inserting the expressions for k1and k2into equations (5.266) and (5.267) gives the solution for the translation
parameters
txeTW4e=tyeTW5e+eTW5(ξ2xc+ξ1yc−Yc)−eTW4(ξ1xc−ξ2yc−X),(5.276)
tyeTW6e=txeTW5e+eTW5(ξ1xc−ξ2yc−Xc)−eTW6(ξ2xc+ξ1yc−Yc).(5.277)
Estimates for the unknown transformation parameters ξ1and ξ2can be computed by substituting the derived
residual vectors from equations (5.258) - (5.261), the vectors of Lagrange multipliers k1,k2from (5.273)
and (5.274), as well as the translation parameters (5.276) and (5.277) into the normal equations (5.264) and
(5.265). This results in the reduced system of equations
ξ1f1+ξ2f2+f3= 0 (5.278)
and
ξ2f4+ξ1f2+f5= 0,(5.279)
with the respective quantities
f1=kT
1Qxxk1+kT
2Qyyk2+xT
cW5yc−xT
cW4xc+yT
cW5xc−yT
cW6yc,
f2=kT
1Qxxk2−kT
2Qyyk1+xT
cW5xc+xT
cW4yc−yT
cW5yc−yT
cW6xc,
f3=yT
cW5−xT
cW4(txe−Xc) + xT
cW5−yT
cW6(tye−Yc),
f4=kT
2Qxxk2+kT
1Qyyk1−xT
cW5yc−xT
cW6xc−yT
cW4yc−yT
cW5xc,
f5=xT
cW5+yT
cW4(txe−Xc)−xT
cW6+yT
cW5(tye−Yc).
(5.280)
Last but not least, solving (5.279) for ξ2and introducing it in (5.278) leads to the least squares estimate for
ˆ
ξ1=f1f4−f2
2−1(f2f5−f3f4) (5.281)
and
ˆ
ξ2=−ξ1
f2
f4−f5
f4
.(5.282)
An iterative solution is possible by choosing meaningful approximate values for the unknown transforma-
tion parameters. Thus, the last two equations become pseudo-linear with the functions f1, f2, . . . , f5being
approximated. An iterative procedure can be found in Algorithm 7 for the weighted least squares solution
of the 2D similarity transformation of coordinates.
5.4. 2D similarity transformation of coordinates 141
Algorithm 7 Least squares 2D similarity transformation of coordinates with general weights
1: Choose approximate values for ξ0
1and ξ0
2.
2: Set threshold for the break-off condition of the iteration process.
3: Set parameters dξ1=|ˆ
ξ1−ξ0
1|=∞and dξ2=|ˆ
ξ2−ξ0
2|=∞, for entering the iteration process.
4: while dξ1=|ˆ
ξ1−ξ0
1|> or dξ2=|ˆ
ξ2−ξ0
2|> do
5: Compute the auxiliary matrices W1,W2,...,W6.
6: Estimate the translation parameters ˆ
txand ˆ
ty.
7: Compute the vectors of Lagrange multipliers k1and k2.
8: Compute the coefficients f1, f2, . . . , f5.
9: Estimate parameters ˆ
ξ1and ˆ
ξ2.
10: Compute parameter dξ1=|ˆ
ξ1−ξ0
1|and dξ2=|ˆ
ξ2−ξ0
2|.
11: Update the approximate values with the estimated ones (ξ0
1=ˆ
ξ1,ξ0
2=ˆ
ξ2) .
12: end while
13: return ˆ
ξ1,ˆ
ξ1,ˆ
txand ˆ
ty.
5.4.4 Weighting case 4 - Individually weighted and correlated coordinates in
each coordinate system
In the last investigated weighting case for the 2D similarity transformation, correlations are introduced
between the measured coordinates of the points in each coordinate system. Therefore, two cofactor matrices
are given. The first is related to the coordinates in the source system
QLL1="Qxx Qxy
Qyx Qyy #,with Qxy =QT
yx,(5.283)
while the second concerns the coordinates in the target system
QLL2="QXX QXY
QYX QYY #,with QXY =QT
YX.(5.284)
The respective weights of this problem can be computed by
P1=QLL
−1
1="Pxx Pxy
Pyx Pyy #(5.285)
and
P2=QLL
−1
2="PXX PXY
PYX PYY #.(5.286)
142 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
Taking into account the stochastic model described above, the objective function (5.256) can be extended
to
Ω(vX,vY,vx,vy) = vT
XPXXvX+vT
YPYYvY+vT
xPxxvx+vT
yPyyvy+ 2 vT
XPXYvY+ 2 vT
xPxyvy.(5.287)
Iterative least squares solution without linearization
The developed objective function can be combined with the nonlinear condition equations (5.253) to form
the Lagrange function
K(ξ1, ξ2, tx, ty,vX,vY,vx,vy,k1,k2) = Ω(vX,vY,vx,vy)
−2kT
1[−(Xc+vX) + ξ1(xc+vx)−ξ2(yc+vy) + txe]
−2kT
2[−(Yc+vY) + ξ2(xc+vx) + ξ1(yc+vy) + tye],
(5.288)
with k1and k2denoting vectors of Lagrange multipliers. Differentiating the Lagrangian with respect to all
unknowns and setting the result to zero, yields the system of normal equations
∂K
∂vT
X
= 2 (PXXvX+PXYvY+k1) = 0,(5.289)
∂K
∂vT
Y
= 2 (PYYvY+PYXvX+k2) = 0,(5.290)
∂K
∂vT
x
= 2 (Pxxvx+Pxyvy−ξ1k1−ξ2k2) = 0,(5.291)
∂K
∂vT
y
= 2 (Pyyvy+Pyxvx+ξ2k1−ξ1k2) = 0,(5.292)
∂K
∂kT
1
=−2 [−(Xc+vX) + ξ1(xc+vx)−ξ2(yc+vy) + txe] = 0,(5.293)
∂K
∂kT
2
=−2 [−(Yc+vY) + ξ2(xc+vx) + ξ1(yc+vy) + tye] = 0,(5.294)
∂K
∂ξ1
=−2kT
1(xc+vx) + kT
2(yc+vy)= 0,(5.295)
∂K
∂ξ2
=−2−kT
1(yc+vy) + kT
2(xc+vx)= 0,(5.296)
5.4. 2D similarity transformation of coordinates 143
∂K
∂tx
=−2kT
1e= 0 (5.297)
and
∂K
∂ty
=−2kT
2e= 0.(5.298)
The first two of the developed normal equations can be expressed using block matrices:
"PXX PXY
PYX PYY #" vX
vY#=−"k1
k2#.(5.299)
Thus, a solution for the residual vectors in the target system can be computed by
"vX
vY#=−"PXX PXY
PYX PYY #−1"k1
k2#=−"QXX QXY
QYX QYY #" k1
k2#,(5.300)
or equivalently by
vX=−QXXk1−QXYk2(5.301)
and
vY=−QYXk1−QYYk2.(5.302)
In the same manner, explicit expressions for the residual vectors of the coordinates in the source system can
be obtained by utilizing equations (5.291) and (5.292), which yields
vx=ξ1(Qxxk1+Qxyk2) + ξ2(−Qxyk1+Qxxk2) (5.303)
and
vy=ξ1(Qyxk1+Qyyk2) + ξ2(−Qyyk1+Qyxk2).(5.304)
For a solution of k1and k2, all residual vectors are introduced into equations (5.293) and (5.294), resulting
in
ξ2
1Qxx −ξ1ξ2Qxy −ξ1ξ2Qyx +ξ2
2Qyy +QXXk1+ξ1ξ2Qxx +ξ2
1Qxy −ξ2
2Qyx −ξ1ξ2Qyy +QXYk2=
−(ξ1xc−ξ2yc+txe−Xc)
(5.305)
144 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
and
ξ1ξ2Qxx +ξ2
1Qyx −ξ2
2Qxy −ξ1ξ2Qyy +QYXk1+ξ2
2Qxx +ξ1ξ2Qxy +ξ1ξ2Qyx +ξ2
1Qyy +QYYk2=
−(ξ2xc+ξ1yc+tye−Yc).
(5.306)
Introducing approximate values for the unknown transformation parameters only in the left-hand side of
the last two equations, it is possible to rewrite them as
W1k1+W2k2=−(ξ1xc−ξ2yc+txe−Xc) (5.307)
and
W3k2+W4k1=−(ξ2xc+ξ1yc+tye−Yc),(5.308)
with the auxiliary matrices
W1=ξ0
1
2Qxx −ξ0
1ξ0
2Qxy −ξ0
1ξ0
2Qyx +ξ0
2
2Qyy +QXX,
W2=ξ0
1ξ0
2Qxx +ξ0
1
2Qxy −ξ0
2
2Qyx −ξ0
1ξ0
2Qyy +QXY,
W3=ξ0
2
2Qxx +ξ0
1ξ0
2Qxy +ξ0
1ξ0
2Qyx +ξ0
1
2Qyy +QYY,
W4=ξ0
1ξ0
2Qxx +ξ0
1
2Qyx −ξ0
2
2Qxy −ξ0
1ξ0
2Qyy +QYX.
(5.309)
If the cofactor matrices are regular, then matrices W1,W2,W3and W4are also regular and invertible.
Consequently, a solution for the vectors of Lagrange multipliers can be found by
k1=W5(ξ2xc+ξ1yc+tye−Yc)−W6(ξ1xc−ξ2yc+txe−Xc) (5.310)
and
k2=W7(ξ1xc−ξ2yc+txe−Xc)−W8(ξ2xc+ξ1yc+tye−Yc),(5.311)
after introducing the matrices
W5=W1−W2W−1
3W4−1W2W−1
3,
W6=W1−W2W−1
3W4−1,
W7=W3−W4W−1
1W2−1W4W−1
1,
W8=W3−W4W−1
1W2−1.
(5.312)
5.4. 2D similarity transformation of coordinates 145
Substituting the derived vectors of Lagrange multipliers k1and k2into the normal equations (5.297) and
(5.298) yields the translation vectors
txeTW6e=tyeTW5e+eTW5(ξ2x + ξ1y−Y) −eTW6(ξ1x−ξ2y−X)(5.313)
and
tyeTW8e=txeTW7e+eTW7(ξ1x−ξ2y−X) −eTW8(ξ2x + ξ1y−Y).(5.314)
Furthermore, estimates for the transformation parameters ξ1and ξ2can be computed by substituting the
residual vectors, the vectors of Lagrange multipliers, as well as the translation parameters into equations
(5.295) and (5.296). This results in the reduced normal equations
ξ1f1+ξ2f2+f3= 0 (5.315)
and
ξ2f5+ξ1f4+f6= 0,(5.316)
with the respective quantities
f1=kT
1Qxxk1+kT
1Qxyk2+kT
2Qyxk1+kT
2Qyyk2+xT
cW5yc−xT
cW6xc+yT
cW7xc−yT
cW8yc,
f2=kT
1Qxxk2−kT
1Qxyk1+kT
2Qyxk2−kT
2Qyyk1+xT
cW5xc+xT
cW6yc−yT
cW7yc−yT
cW8xc,
f3=yT
cW7−xT
cW6(txe−Xc) + xT
cW5−yT
cW8(tye−Yc),
f4=kT
1Qyxk1+kT
1Qyyk2−kT
2Qxxk1−kT
2Qxyk2+yT
cW5yc−yT
cW6xc−xT
cW7xc+xT
cW8yc,
f5=kT
1Qyxk2+kT
1Qyyk2−kT
2Qxxk2+kT
2Qxyk1+yT
cW5xc+yT
cW6yc+xT
cW7yc+xT
cW8xc,
f6=−yT
cW6−xT
cW7(txe−Xc) + yT
cW5+xT
cW8(tye−Yc).
(5.317)
Finally, solving equation (5.316) for ξ2and introducing it in (5.315) leads to the least squares estimate for
ˆ
ξ1= (f1f5−f2f4)−1(f2f6−f3f5) (5.318)
and
ˆ
ξ2=−ˆ
ξ1
f4
f5−f6
f5
.(5.319)
146 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
Choosing appropriate approximate values for the unknown transformation parameters, an iterative process
could give the solution to this adjustment problem. Equations (5.318) and (5.319) are pseudo-linear with
the auxiliary functions f1, f2, . . . , f6being approximated in each iteration step. Algorithm 8 includes the
presented iterative least squares solution for the 2D similarity transformation of coordinates with correlated
observations.
Algorithm 8 Least squares 2D similarity transformation of coordinates with correlated observations
1: Choose approximate values for ξ0
1and ξ0
2.
2: Set threshold for the break-off condition of the iteration process.
3: Set parameters dξ1=|ˆ
ξ1−ξ0
1|=∞and dξ2=|ˆ
ξ2−ξ0
2|=∞, for entering the iteration process.
4: while dξ1=|ˆ
ξ1−ξ0
1|> or dξ2=|ˆ
ξ2−ξ0
2|> do
5: Compute the auxiliary matrices W1,W2,...,W8.
6: Estimate the translation parameters ˆ
txand ˆ
ty.
7: Compute the vectors of Lagrange multipliers k1and k2.
8: Compute the coefficients f1, f2, . . . , f6.
9: Estimate parameters ˆ
ξ1and ˆ
ξ2.
10: Compute parameter dξ1=|ˆ
ξ1−ξ0
1|and dξ2=|ˆ
ξ2−ξ0
2|.
11: Update the approximate values with the estimated ones (ξ0
1=ˆ
ξ1,ξ0
2=ˆ
ξ2) .
12: end while
13: return ˆ
ξ1,ˆ
ξ1,ˆ
txand ˆ
ty.
Solution for singular cofactor matrices
An iterative least squares solution is possible also for the case of singular cofactor matrices, following the
same procedure as in the previous application cases. Grouping equations (5.307) and (5.308), together with
the normal equations (5.295)-(5.298), results in the system of equations
W1k1+W2k2=−(ξ1xc−ξ2yc+txe−Xc),
W3k2+W4k1=−(ξ2xc+ξ1yc+tye−Yc),
kT
1(xc+vx) + kT
2(yc+vy)=0,
kT
1(yc+vy)−kT
2(xc+vx)=0,
kT
1e= 0,
kT
2e= 0.
(5.320)
5.4. 2D similarity transformation of coordinates 147
Introducing approximate values for the residual vectors v0
x,v0
y,v0
Xand v0
X, the developed equation system
can be expressed by
W1W2xc−yce0
W4W3ycxc0e
xc+v0
xTyc+v0
yT0 0 0 0
yc+v0
yT−xc+v0
xT0 0 0 0
eT0 0 0 0 0
0eT0 0 0 0
k1
k2
ξ1
ξ2
tx
ty
=
Xc
Yc
0
0
0
0
,(5.321)
which can be equivalently formulated as
N"k
X#=n.(5.322)
The introduced matrices are
N=
W1W2xc−yce0
W4W3ycxc0e
xc+v0
xTyc+v0
yT0 0 0 0
yc+v0
yT−xc+v0
xT0 0 0 0
eT0 0 0 0 0
0eT0 0 0 0
,n=
Xc
Yc
0
0
0
0
(5.323)
with the vector of unknown transformation parameters
X=
ξ1
ξ2
tx
ty
.(5.324)
The least squares solution of this adjustment problem is
"ˆ
k
ˆ
X#=N−1n.(5.325)
The estimated parameters for the 2D similarity transformation can be utilized as new approximations and
the procedure is repeated until the necessary predefined condition is met.
Solution with a symmetric normal matrix N
Similarly to the cases of subsections 5.2.4.2 and 5.3.4, a symmetric matrix Ncan be obtained by adding the
terms ξ1vx−ξ2vyand ξ2vx+ξ1vyto both sides of equations (5.307) and (5.308) respectively. In this way,
148 Chapter 5. Direct and iterative solutions of weighted nonlinear least squares problems
the equation system (5.321) becomes
"W A
AT0#" k
X#="w
0#,(5.326)
with the relevant matrices being expressed for this problem as
W="W1W2
W4W3#,(5.327)
A="xc+v0
x−yc+v0
ye0
yc+v0
yxc+v0
x0e#(5.328)
and
w=−zc+a0v0
x+b0v0
y.(5.329)
A solution of this adjustment problem can be computed by equation (5.325), after introducing
N="W A
AT0#,n="w
0#,(5.330)
with Nbeing symmetric.
The rank deficiency of matrix W, respectively of the matrices W1,W2,W3and W4, depends on the cofactor
matrices of the problem and is important for the inversion of matrix N. Similar to the adjustment cases from
subsections 5.2.4.2 and 5.3.4, the presented criterion (5.102) will ensure the existence of a unique solution.
The developed iterative procedure is presented in Algorithm 9 for obtaining a weighted least squares solution
for the 2D transformation parameters, when singular cofactor matrices are given.
Algorithm 9 Least squares 2D similarity transformation of coordinates with singular cofactor matrices
1: Choose approximate values for ξ0
1,ξ0
2,v0
x,v0
yand v0
X,v0
Y.
2: Set threshold for the break-off condition of the iteration process.
3: Define parameters dξ1=|ˆ
ξ1−ξ0
1|=∞and dξ2=|ˆ
ξ2−ξ0
2|=∞, for entering the iteration process.
4: while dξ1=|ˆ
ξ1−ξ0
1|> or dξ2=|ˆ
ξ2−ξ0
2|> do
5: Compute matrices W,Aand vector w.
6: Build matrix Nand vector n.
7: Estimate the vector of unknown parameters "ˆ
k
ˆ
X#.
8: Compute the residual vectors vx,vyand vX,vY.
9: Compute parameter dξ1=|ˆ
ξ1−ξ0
1|and dξ2=|ˆ
ξ2−ξ0
2|.
10: Update the approximate values with the estimated ones, with ξ0
1=ˆ
ξ1,ξ0
2=ˆ
ξ2,v0
x=vx,v0
y=vy,
v0
X=vXand v0
Y=vY.
11: end while
12: return ˆ
ξ1,ˆ
ξ1,ˆ
txand ˆ
ty.
5.5. Discussion of weighted nonlinear least squares solutions 149
5.5 Discussion of weighted nonlinear least squares solutions
In this chapter, three adjustment cases that belong to a class of nonlinear least squares problems with a
direct solution have been examined: the fitting of straight line in 2D, the fitting of a plane in 3D and the
2D similarity transformation of coordinates. Further, four individual weighting scenarios were presumed in
each adjustment problem that often occur in practice: constant weights for the coordinates in each direction,
individual weights for the coordinates of each point, individual weight for each coordinate and individually
weighted and correlated coordinates. A thorough analysis of these weighted least squares problems has
shown that in certain cases a direct solution is still possible. Otherwise, an iterative solution could be always
developed without performing any kind of linearization of the problem.
A direct solution has been proven to be always possible for the first two weighting cases, regarding the
discussed class of least squares problems. The estimated unknown parameters have been obtained by
parametrizing appropriately the mathematical model and minimizing a clearly defined Lagrange function.
This led to a system of normal equations that can have a nontrivial solution if the determinant is equal
to zero or equivalently by solving an eigenvalue problem. Therefore, two novel systematic approaches have
been established for the direct solution of the first two weighting cases, based on the same solution strategy
of (Malissiovas et al. 2016).
For the last two weighting cases, it has been demonstrated why a direct least squares solution is not possible.
Subsequently, iterative algorithms have been presented that are applicable in all cases without making use
of linearization of the original problem. The general idea of the established iterative systematic approach is
based on the minimization of a Lagrange function and the solution of a reduced system of normal equations,
following (Petrovi´c et al. 1983). In addition, a simple extension to this systematic approach has been
demonstrated for the solution of these adjustment problems when singular cofactor matrices are given. The
developed iterative algorithms produce the WTLS solution and can be compared to those presented by Fang
(2011) and Snow (2012).
151
6 Numerical Investigations
The developed methodologies and implemented algorithms of the previous chapters are tested here for
two application examples: this of fitting a straight line to measured points in 2D and for estimating the
2D similarity transformation parameters between two measured groups of homologous points. Different
weighting cases are examined for each adjustment problem. Moreover, the presented solutions for the
examples in this chapter have been tested and found to be numerically equal to the least squares solution
within the GHM.
6.1 Fitting of a straight line in 2D
This section illustrates the least squares solution for fitting a straight line in 2D. The dataset of the measured
point coordinates is listed in Table 6.1. It originates from the work of Pearson (1901) and since then it has
been utilized by many authors. (Snow, 2012, p. 67) noticed that York (1966) introduced unusual weights
for the observed coordinates and solved the problem iteratively. Different algorithms for estimating the
least squares solution of the same dataset using York’s stochastic model has been presented at least by Neri
et al. (1989), Schaffrin and Wieser (2008), Shen et al. (2011) or Amiri-Simkooei and Jazaeri (2012). Snow
(2012) introduced also correlations between the measured coordinates and solved the problem for regular
and singular cofactor matrices.
Table 6.1: Example dataset of measured points in 2D.
Point No. x-coord. [m] y-coord. [m]
1 0 5.9
2 0.9 5.4
3 1.8 4.4
4 2.6 4.6
5 3.3 3.5
6 4.4 3.7
7 5.2 2.8
8 6.1 2.8
9 6.5 2.4
10 7.4 1.5
152 Chapter 6. Numerical Investigations
The measured coordinates are in both directions of xand yunder the influence of random errors. Least
squares solutions for fitting a straight line to the points are presented below for four different weighting
cases.
Equally weighted coordinates
For the first weighting case the measured coordinates are uncorrelated and have been obtained with equal
precision, i.e. equal weights:
pxi=pyi= 1.(6.1)
For solving the nonlinear problem of fitting a straight line to the ten points of the presented dataset, Pearson
(1901) proposed a direct approach using a functional model equivalent to equation (4.32). The least squares
solution for the slope of the requested line has been found in that article to be ˆ
β=−0.546. A solution has
been also obtained utilizing the algorithm of Neitzel and Petrovic (2008). The results from the GHM are
listed in Table 6.2.
Table 6.2: Solution within the GHM using the algorithm of Neitzel and Petrovic (2008).
Estimated parameter GHM solution
ˆ
β(slope ˆa) -0.545561197521
ˆγ(y-intercept ˆ
b) 5.7840437745301
Further, a direct least squares solution was obtained for the unknown line parameters a,band c, following the
developed methodology of section 4.2.1. The Lagrange function of equation (4.11) leads to a homogeneous
system of equations with one unknown parameter k, that can be estimated by solving
(56.396 −k)−30.43
−30.43 (17.22 −k)
= 0.(6.2)
This yields a quadratic equation with the solutions for the unknown parameters kmin = 0.618572759437049
and kmax = 72.997427240563. The results for the line parameters can be found in Table 6.3. Parameters ˆa,
ˆ
band ˆcwere utilized further to compute parameters ˆ
βand ˆγ.
Table 6.3: Direct least squares solution (section 4.2.1).
Estimated parameter least squares solution
ˆa-0.4789242860482
ˆ
b-0.8778562115935
ˆc5.0775587555999
Computed parameter
ˆ
β=−ˆa
ˆ
b-0.545561197521
ˆγ=−ˆc
ˆ
b5.7840437745301
6.1. Fitting of a straight line in 2D 153
An estimate for the unknown line parameters was derived using the TLS approach. The determinant of the
generalized eigenvalue problem was built following the procedure for the eigenvalue/eigenvector decomposi-
tion of matrix G(equation 4.43). This results in the characteristic equation of the eigenvalues (quadratic
equation) with the solutions λmin = 0.618572759437049 and λmax = 72.997427240563. The TLS solution
for the unknown line parameters is presented in Table 6.4.
Table 6.4: TLS solution (section 4.2.2).
Estimated parameter TLS solution
ˆ
β-0.545561197521
ˆγ5.7840437745301
The developed direct least squares solution is, as expected, identical to the TLS. Both solutions are numeri-
cally consistent with the result of the linearized GHM. The requested straight line is depicted in Figure 6.1,
together with the measured points and their estimated residuals.
-1012345678
0
1
2
3
4
5
6
7
8
Figure 6.1: Fitting a straight line to points in 2D, with observed xand ycoordinates of equal precision.
154 Chapter 6. Numerical Investigations
Equally weighted coordinates in each direction
In this weighting case the measured coordinates are uncorrelated and have been obtained with the same
precision in each direction. The postulated weights are
pxi= 0.5 and pyi= 1.5.(6.3)
A direct solution can be derived in this case from the developed methodology of section 5.2.1. The results
are presented in Table 6.5.
Table 6.5: Direct least squares solution (section 5.2.1)
Estimated parameter Direct least squares solution
ˆa-0.4832580303705
ˆ
b-0.8754779700726
ˆc5.0853141652839
Computed parameter
ˆ
β=−ˆa
ˆ
b-0.5519933646422
ˆγ=−ˆc
ˆ
b5.8086146529331
For comparison reasons the least squares solution has been computed again with the algorithm of (Neitzel
and Petrovic 2008), taking into account the current stochastic model. Both the developed direct solution
and the one within the GHM are numerically identical. The estimated straight line is depicted in Figure 6.2.
6.1. Fitting of a straight line in 2D 155
-1012345678
0
1
2
3
4
5
6
7
8
Figure 6.2: Fitting a straight line to points in 2D, with observed xand ycoordinates and px,pyindividual
constant weights for each coordinate axis.
Individually weighted points
For this weighting example the measured coordinates are uncorrelated and have been obtained with indi-
vidual precision for each measured point. The postulated weights are listed in Table 6.6.
Table 6.6: Individual weights for each point.
Point No. px=1
σx2py=1
σy2
1 1 1
2 1.2 1.2
3 0.8 0.8
4 1.1 1.1
5 0.9 0.9
6 1.15 1.15
7 1 1
8 0.93 0.93
9 1.25 1.25
10 1.13 1.13
156 Chapter 6. Numerical Investigations
A direct weighted least squares solution has been obtained following the procedure of section 5.2.2. The
results for the estimated line parameters can be found in Table 6.7.
Table 6.7: Direct least squares solution (section 5.2.2).
Estimated parameter Direct least squares solution
ˆa-0.4824660036697
ˆ
b-0.875914696362
ˆc5.1014648040614
Computed parameter
ˆ
β=−ˆa
ˆ
b-0.5508139156399
ˆγ=−ˆc
ˆ
b5.8241571071355
The computed straight line is shown in Figure 6.3.
-1012345678
0
1
2
3
4
5
6
7
8
Figure 6.3: Fitting a straight line to points in 2D, with individual weight for the coordinates of each point.
6.1. Fitting of a straight line in 2D 157
Individually weighted 2D coordinates
The stochastic model of York (1966) has been adopted for this adjustment example. The measured coordi-
nates are uncorrelated and have been obtained with individual precision. The postulated weights are listed
in Table 6.8.
Table 6.8: Individual weights for each coordinate.
Point No. px=1
σx2py=1
σy2
1 1000 1
2 1000 1.8
3 500 4
4 800 8
5 200 20
6 80 20
7 60 70
8 20 70
9 1.8 100
10 1 500
An iterative weighted least squares solution has been derived from the developed approach of section 5.2.3,
by employing Algorithm 1. The results are listed in Table 6.9.
Table 6.9: Iterative least squares solution using Algorithm 1 (section 5.2.3).
Estimated parameter Iterative least squares solution
ˆa0.4805334074462
ˆ
b1
ˆc-5.4799102240329
Computed parameter
ˆ
β=−ˆa
ˆ
b-0.4805334074462
ˆγ=−ˆc
ˆ
b5.4799102240329
The presented solution has been found to be numerically identical with the iterative least squares solution
of York (1966), Neri et al. (1989), the pseudoquadratic algorithm from (Petrovi´c et al. 1983) and the WTLS
algrithm of Schaffrin and Wieser (2008). The estimated line is depicted in Figure 6.4.
158 Chapter 6. Numerical Investigations
-1012345678
0
1
2
3
4
5
6
7
8
Figure 6.4: Fitting a straight line to points in 2D, with individual weight for each measured coordinate.
Individually weighted and correlated 2D coordinates
In addition to the stochastic model of Table 6.8, correlations between the measured coordinates of each point
are postulated in this example. This is for instance the case when polar coordinates of points have been
originally measured, while their Cartesian coordinates are utilized in the adjustment, together with their
stochastic properties from a linear error propagation. For comparison reasons, the necessary correlations
between the point coordinates have been taken directly from the numerical investigations of (Snow 2012,
pp. 68-70) (i.e. the case of a regular cofactor matrix) and are listed in Table 6.10.
6.1. Fitting of a straight line in 2D 159
Table 6.10: Individual weights for each coordinate and correlations for each point.
Point No. px=1
σx2py=1
σy2ρxy
1 1000 1 -0.165956
2 1000 1.8 0.440649
3 500 4 -0.999771
4 800 8 -0.395335
5 200 20 -0.706488
6 80 20 -0.815323
7 60 70 -0.627480
8 20 70 -0.308879
9 1.8 100 -0.206465
10 1 500 0.077633
The variance-covariance matrix for this adjustment problem can be computed by
ΣLL ="Σxx Σxy
Σyx Σyy #,(6.4)
with
Σxx =
σ2
x100000000 0
0σ2
x20000000 0
0 0 σ2
x3000000 0
000σ2
x40 0 0 0 0 0
0000σ2
x50000 0
00000σ2
x60 0 0 0
000000σ2
x70 0 0
0000000σ2
x80 0
00000000σ2
x90
000000000σ2
x10
,(6.5)
Σyy =
σ2
y100000000 0
0σ2
y20 0 0 0 0 0 0 0
0 0 σ2
y3000000 0
0 0 0 σ2
y40 0 0 0 0 0
0 0 0 0 σ2
y50 0 0 0 0
00000σ2
y60 0 0 0
000000σ2
y70 0 0
0000000σ2
y80 0
00000000σ2
y90
000000000σ2
y10
(6.6)
and
160 Chapter 6. Numerical Investigations
Σxy =Σyx =
σx1y100000000 0
0σx2y20000000 0
0 0 σx3y3000000 0
000σx4y400000 0
0000σx5y50000 0
00000σx6y60 0 0 0
000000σx7y70 0 0
0000000σx8y80 0
00000000σx9y90
000000000σx10y10
.(6.7)
The individual covariances between the coordinates of each point are
σxy =ρxy σxσy.(6.8)
Setting the variance of the universal weight equal to one (σ2
0= 1) the cofactor matrix of the observations is
QLL ="Qxx Qxy
Qyx Qyy #=σ2
0ΣLL =ΣLL.(6.9)
For a regular cofactor matrix it is possible to compute the weight matrix
P=Q−1
LL.(6.10)
A least squares solution for the unknown line parameters can be estimated iteratively, utilizing Algorithm 2
from the developed approach of section 5.2.4.1. The results are presented in Table 6.11.
Table 6.11: Iterative least squares solution using Algorithm 2 (section 5.2.4.1).
Estimated parameter Iterative least squares solution
ˆa0.4592286797279
ˆ
b1
ˆc-5.357272562041
Computed parameter
ˆ
β=−ˆa
ˆ
b-0.4592286797279
ˆγ=−ˆc
ˆ
b5.357272562041
The results from the proposed approach have been compared and found to be numerically identical with the
solution from the WTLS algorithm presented in (Snow 2012, p. 72). The requested straight line is depicted
in Figure 6.5.
6.1. Fitting of a straight line in 2D 161
-1012345678
0
1
2
3
4
5
6
7
8
Figure 6.5: Fitting a straight line to points in 2D, with individually weighted and correlated coordinates for
each point.
Solution with a singular cofactor matrix
A singular variance-covariance matrix is postulated in this example for the measured point coordinates
including correlations between the observations. This is the case when, for example, the 2D Cartesian
coordinates of the points have been obtained by a least squares adjustment of a free network and their
stochastic properties by a linearized error propagation. The same stochastic model as in (Snow 2012, pp.
71-72) is utilized here, for the case of a singular cofactor matrix which satisfies the NS criterion, in order
to compare the results of the proposed approach. The necessary variances and covariances can be found
in Appendix A.1 and lead to a singular matrix Wwith rank deficiency equal to 2. However, applying the
developed criterion from equation (5.102) results in
rank ([W|A]) = 10 = n, (6.11)
which ensures that a unque solution still exists, with
- rank of W= 8 < n, with n= number of condition equations;
- rank of A=2=m, with m= number of unknown parameters;
- redundancy : rd=n−m= 10 −2 = 8;
162 Chapter 6. Numerical Investigations
Here it can be seen that the rank of matrix Wis smaller than the number of condition equations n, caused
by the rank deficiency of the introduced cofactor matrices.
A least squares solution for the unknown line parameters has been estimated iteratively, using Algorithm 3
from section 5.2.4.2. The results are shown in Table 6.12.
Table 6.12: Iterative least squares solution using Algorithm 3 (section 5.2.4.2).
Estimated parameter Iterative least squares solution
ˆa0.4931726182468
ˆ
b1
ˆc-5.54275204298
Computed parameter
ˆ
β=−ˆa
ˆ
b-0.4931726182468
ˆγ=−ˆc
ˆ
b5.54275204298
The solution for the line parameters is numerically equal to the WTLS solution from (Snow 2012, p. 72)
and the solution within the GHM for the case of a singular cofactor matrix. The computed straight line is
depicted in Figure 6.6.
-1012345678
0
1
2
3
4
5
6
7
8
Figure 6.6: Fitting a straight line to points in 2D, with a singular cofactor matrix.
6.2. 2D similarity transformation of coordinates 163
6.2 2D similarity transformation of coordinates
The least squares solution of the 2D similarity transformation is presented in this subsection following the
developed methodologies and algorithms of chapters 4 and 5. The coordinates of four homologous points
have been measured in two coordinate systems, the target XY and the source xy system and are listed in
Table 6.13.
Table 6.13: Example dataset for the 2D similarity transformation
Point No. Target S. Source S.
i Xi[m]Yi[m]xi[m]yi[m]
1 -117.478 0 17.856 144.794
2 117.472 0 252.637 154.448
3 0.015 -117.41 140.089 32.326
4 -0.014 117.451 130.40 267.027
This dataset originates from (Mikhail et al. 2001, pp. 397-402) and has been utilized in the past at least by
Felus and Schaffrin (2005), Neitzel (2010) and Malissiovas et al. (2016).
Equally weighted coordinates
In the first weighting case of this numerical example the coordinates of the homologous points are equally
weighted and uncorrelated, i.e. equal weights:
pXi=pYi=pxi=pyi= 1.(6.12)
A GHM has been employed by Neitzel (2010) for deriving the least squares solution for the unknown
transformation parameters between the coordinate systems. The results from the GHM are presented in
Table 6.14.
Table 6.14: Results from Neitzel (2010).
Estimated parameter GHM solution
Parameter ˆ
ξ10.99900748077781
Parameter ˆ
ξ2-0.04109806319405
Scale factor ˆµ0.99985248784424
Rotational angle ˆ
φ-2◦21020.7200
Translation parameter ˆ
tx-141.2628 mm
Translation parameter ˆ
ty-143.9316 mm
The developed direct least squares solution for the 2D similarity transformation of section 4.5.1 is presented
in Table 6.15. For comparison reasons, the estimates for α,βand γcan be substituted in equation (4.110)
164 Chapter 6. Numerical Investigations
to derive the parameters ξ1and ξ2. Furthermore, the rotational angle φas well as the scale factor µcan be
computed by substituting ˆ
ξ1and ˆ
ξ2into equation (4.102). Thus, the rotational angle is
ˆ
φ= arctan ˆ
ξ2
ˆ
ξ1!(6.13)
and the scale factor
ˆµ=ˆ
ξ1
cos ˆ
φ.(6.14)
Similarly, the translation parameters txand tycan be derived from equation (4.106).
Table 6.15: Direct least squares solution for the 2D similarity transformation
Estimated parameter Direct least squares solution
ˆα-0.4832580303705
ˆ
β-0.8754779700726
ˆγ5.0853141652839
Computed parameter
ˆ
ξ1=−ˆα
ˆγ0.99900748077781
ˆ
ξ2=−ˆ
β
ˆγ-0.04109806319405
ˆµ0.99985248784424
ˆ
φ-2◦21020.72394355800
ˆ
tx-141.2627900259449
ˆ
ty-143.9316426333377
In addition, the TLS solution of section 4.5.2 has been computed and found to be identical to the solution
from the proposed direct least squares approach. Both coincide numerically with the results from the GHM,
as it can be seen from Tables 6.14 and 6.15.
Equally weighted coordinates in each coordinate system
Equal weights between the coordinates of each coordinate system are assumed for the second weighting case,
with
pXY i= 1.1 and pxyi= 0.9.(6.15)
A direct weighted least squares solution has been derived here, following the developed methodology of
section 5.4.1. The results are presented in Table 6.16.
6.2. 2D similarity transformation of coordinates 165
Table 6.16: Direct least squares solution (section 5.4.1)
Estimated parameter Direct least squares solution
ˆα-0.7064570685103
ˆ
β0.0290628626954
ˆγ0.7071589357166
Computed parameter
ˆ
ξ1=−ˆα
ˆγ0.9990074830836
ˆ
ξ2=−ˆ
β
ˆγ-0.0410980632889
ˆµ0.99985249015199
ˆ
φ-2◦21020.72394355600
ˆ
tx-141.2627903519875
ˆ
ty-143.9316429655667
Equally weighted homologous points in both systems
For this weighting example the coordinates of the homologous points have been measured with individual
precisions. The weights of this stochastic model are listed in Table 6.17.
Table 6.17: Individual weights for homologous points in both systems.
Point No. Target S. Source S.
i pXi=1
σX2
i
pYi=1
σY2
i
pxi=1
σx2
i
pyi=1
σy2
i
1 0.9 0.9 0.9 0.9
2 1.05 1.05 1.05 1.05
3 0.85 0.85 0.85 0.85
4 1.3 1.3 1.3 1.3
A direct weighted least squares solution has been estimated for this example following the solution strategy
of section 5.4.2. Table 6.18 contains the results.
166 Chapter 6. Numerical Investigations
Table 6.18: Direct least squares solution (section 5.4.2)
Estimated parameter Direct least squares solution
ˆα-0.7064686408794
ˆ
β0.0290681956735
ˆγ0.7071471554453
Computed parameter
ˆ
ξ1=−ˆα
ˆγ0.9990404902845
ˆ
ξ2=−ˆ
β
ˆγ-0.0411062894755
ˆµ0.99988580761122
ˆ
φ-2◦21022.13959790500
ˆ
tx-141.2687384001714
ˆ
ty-143.9337541051444
Individually weighted coordinates
Individual precision for each measured coordinate has been postulated in this case. The introduced weights
of this stochastic model are listed in Table 6.19.
Table 6.19: Individual weight for each coordinate.
Point No. Target S. Source S.
i pXi=1
σX2
i
pYi=1
σY2
i
pxi=1
σx2
i
pyi=1
σy2
i
1 0.95 1.2 1 0.8
2 1 0.75 1.15 0.9
3 0.95 1.3 0.7 0.85
4 1 0.85 1.2 1
An iterative weighted least squares solution for the unknown transformation parameters has been computed
by using Algorithm 7 from section 5.4.3. The results are presented in Table 6.20.
6.2. 2D similarity transformation of coordinates 167
Table 6.20: Iterative least squares solution (section 5.4.3)
Estimated parameter Iterative least squares solution
ˆ
ξ10.99899700973504
ˆ
ξ2-0.041113617539
ˆ
tx-141.2636321979249
ˆ
ty-143.9334627584676
Computed parameter
ˆµ0.99984266512622
ˆ
φ-2◦21024.01884146500
Individually weighted and correlated coordinates in each coordinate system
In this adjustment example correlations are present only between the coordinates of each point and in both
coordinate systems. The correlation coefficients have been computed randomly, like in (Snow, 2012) for
the example of fitting a straight line in 2D. Here the randn function has been used from the programming
language GNU Octave (version 4.0.0). The introduced weights, together with the correlation coefficients
are listed in Table 6.21 for the coordinates in the target system and in Table 6.22 for the coordinates in the
source system.
Table 6.21: Weights and correlations for the coordinates of the points in the target system.
Point No. Target S.
i pXi=1
σX2
i
pYi=1
σY2
i
ρXY i
1 0.95 1.2 0.2
2 1 0.75 -0.3
3 0.95 1.3 -0.5
4 1 0.85 0.2
Table 6.22: Weights and correlations for the coordinates of the points in the source system.
Point No. Source S.
i pxi=1
σx2
i
pyi=1
σy2
i
ρxyi
1 1 0.8 0.2
2 1.15 0.9 -0.4
3 0.7 0.85 -0.3
4 1.2 1 0.2
An iterative weighted least squares solution was derived for this adjustment problem utilizing Algorithm 8
from section 5.4.4. The results are presented in Table 6.23.
168 Chapter 6. Numerical Investigations
Table 6.23: Iterative least squares solution (section 5.4.4)
Estimated parameter Iterative least squares solution
ˆ
ξ10.9990276226893
ˆ
ξ2-0.0411159727
ˆ
tx-141.26679907
ˆ
ty-143.93227343
Computed parameter
ˆµ0.99987334903342
ˆ
φ-2◦21024.24459835800
Solution with a singular cofactor matrix
A singular cofactor matrix is postulated in the last weighting example of the 2D similarity transformation.
For the sake of comparison, the dataset from the numerical example in (Snow 2012, pp. 77-82) and (Neitzel
and Schaffrin 2017) is adopted. The observed coordinates are listed in Table 6.24.
Table 6.24: Example dataset from Neitzel and Schaffrin (2016).
Point No. Target S. Source S.
i Xi[m]Yi[m]xi[m]yi[m]
1 400.0040 100.0072 453.8001 137.6099
2 500.0019 299.9994 521.2865 350.7972
3 399.9925 399.9933 406.8728 433.9247
4 100.0059 400.0022 110.5545 386.9880
5 99.9956 99.9978 157.4861 90.6802
The coordinates of the homologous points in the two coordinate systems are the outcome of a free network
adjustment. An approximate solution for the variances and covariances of these coordinates has been also
computed using a linearized error propagation. As it is explained by Neitzel and Schaffrin (2017), the
resulting cofactor matrices are fully populated (i.e. the off-diagonal elements are not zero) and rank deficient,
but still fulfilling the NS criterion. The introduced singular cofactor matrices for this numerical example
can be found in Appendix A.2 and lead to a singular matrix Wwith rank deficiency equal to 2. Also for
this numerical example the developed criterion from equation (5.102) leads to
rank ([W|A]) = 10 = n, (6.16)
which ensures that a unque solution for the unknown parameters exists, with
- rank of W= 8 < n, with n= number of condition equations;
- rank of A= 4 = m, with m= number of unknown parameters;
6.2. 2D similarity transformation of coordinates 169
- redundancy : rd=n−m= 10 −4 = 6;
An iterative least squares solution has been computed for this adjustment case using Algorithm 9 from
section 5.4.4. Table 6.25 contains the resulting estimates for the transformation parameters.
Table 6.25: Iterative least squares solution (section 5.4.4)
Estimated parameter Iterative least squares solution
ˆ
ξ10.9876550155542
ˆ
ξ2-0.1564292113176
ˆ
tx-69.726354301821
ˆ
ty35.0782153796499
Computed parameter
ˆµ0.99996626338233
ˆ
φ-9◦000.00480300600
The developed iterative procedure provides the exact numerical result as the least squares solution within
the GHM presented by Neitzel and Schaffrin (2017) and the WTLS solution from (Snow 2012, p. 80).
171
7 Conclusion and outlook
7.1 Conclusion
The fundamental principles of adjustment calculus and the method of least squares have been briefly dis-
cussed in chapter 2, setting the basis for the methodological developments of this thesis. A review of related
works in adjustment calculus has shown that the mathematical modelling of the measurement results em-
bodies the most fundamental parts of every adjustment problem (i.e. the functional and stochastic model).
Only a correct mathematical model could lead to meaningful estimates for the unknown parameters and
the residuals of each problem. For the purposes of this research an unambiguous definition has been given
in chapter 2 concerning linear and nonlinear least squares problems. Additionally, the evaluation of the ad-
justment results has been discussed in terms of precision and reliability. Various approaches were presented
that are common in geodetic literature for the computation and correct interpretation of the stochastic
parameters of the estimated unknowns.
The main subject of this thesis is the least squares solution of a class of nonlinear adjustment problems.
For this reason, two solution strategies were highlighted in chapter 3. The first is related to the traditional
approaches that are commonly used in geodesy and two famous adjustment models, namely the GMM and
the GHM. Based on the Gauss-Newton approach, a least squares solution can be derived iteratively and by
linearizing the nonlinear functional model. The second strategy includes the most modern algorithms that
have been developed in the last decades by the mathematical/statistical community. In the TLS literature
the discussed problems are often expressed within the EIV model and various algorithmic approaches have
been presented for a solution. Depending on the stochastic model, the presented TLS solutions are direct
and cover the cases of equally weighted and uncorrelated measurements, while the WTLS solutions are
iterative and can deal usually with more general weighting cases. Chapter 3 comes to the conclusion that
TLS and WTLS provide the (weighted) least squares solution of the discussed class of problems. Hence, it
is in agreement with the views of Petrovic (2003) and Neitzel (2010), that TLS is not a new method but a
special case of the least squares method.
In the fourth chapter of this thesis, the solution of four individual nonlinear adjustment problems is discussed:
the fitting of a straight line in 2D and 3D, the fitting of a plane in 3D and the 2D similarity transformation of
coordinates. A mathematical relationship between direct least squares and TLS solutions has been presented
that was based on the publication of Malissiovas et al. (2016). This shows that TLS produces the least
squares solution of a problem using SVD, while the exact solution can also be achieved by following the
standard procedure for the least squares solution of a problem, i.e. by minimizing the sum of squared
residuals. Additionally, a new solution strategy has been established for the direct least squares solution of
the investigated class of problems. The developments of this chapter give an overview of these adjustment
problems that can be transformed into an eigenvalue problem and thus, it clarifies in which cases a solution
172 Chapter 7. Conclusion and outlook
from TLS is possible. This chapter demonstrates that TLS is an algorithmic approach for obtaining the
least squares solution and not a method.
The findings in chapter 5 provide an overview of possible weighted least squares solutions for the discussed
class of nonlinear adjustment problems, i.e. the WTLS solution for the mathematical/statistical community.
Different weighting scenarios and correlations between the observations were postulated for each problem.
It has been shown that for certain weighting cases a direct solution still exists. For these cases two novel
direct approaches have been proposed. Further, general weight matrices have been examined including also
cofactor matrices with correlations between the measurements. New algorithms have been developed and
presented for the iterative weighted least squares solution of this class of problems without linearizing the
original problem in any step of the procedure. In addition, singular cofactor matrices that still fulfill the
criterion of Neitzel and Schaffrin (2016) have been taken into account. The presented algorithms can handle
the latter stochastic model without the need of any special treatment of the adjustment problem.
The implemented algorithms are based on the established solution strategies for obtaining a (weighted) least
squares solution for the investigated class of nonlinear problems. The direct approaches can be employed
in engineering tasks for which efficiency is important, i.e. no need for starting values for the unknown
parameters or iterations to obtain a minimum solution for the objective function. For instance, straight
lines or planes can be fitted directly to measured 3D point clouds from laser scanners, or the similarity
transformation parameters between several sets of homologous points can be estimated, if such a stochastic
model is postulated that can lead to a direct solution. In case of more general weight matrices or taking
correlations between the measurements into account in the stochastic model, a weighted least squares solution
is provided by the presented iterative algorithms.
7.2 Outlook
Complex adjustment problems can be further tackled in future research using the knowledge that was
acquired so far. Thus, all the developed algorithms can be possibly extended in order to provide elegant
solutions for other adjustment cases such as:
•Variance Component Estimation (VCE) of individual groups of observations,
•Tykhonov regularization of ill-posed problems.
Moreover, the discussed (weighted) least squares solutions can be considered as “optimal” only if the random
errors that influence the measurements are normally distributed. In the presence of outliers (or blunders),
however, other adjustment methods may be preferred that are more robust in the sense that the solution
is not falsified by a small amount of outliers. Thus, the developed algorithms and solution strategies can
be extended to become more robust against outliers. Future developments can be based on the following
objectives:
- Robust estimators have been studied since decades by geodesists, mathematicians and statisticians.
These estimators can be divided into two main groups, the M-estimators (based on the maximum
likelihood method) with a breakdown point of 5 to 10% and estimators with higher one, like for
example the L1-norm which can reach a breakdown point of 50% of blunders. It is worth to mention
7.2. Outlook 173
the rigorous solution via linear programming (Dantzig 1963, Dantzig and Thapa 2006), for example
using the simplex algorithm presented in (Dantzig 1949) or (Barrodale and Roberts 1974). However,
the minimization of L1-norm with linear programming has been, interestingly, neglected by the geodetic
literature with some exceptions, for example (Fuchs 1980) or (Fuchs 1982). It is still not known if such
algorithms can be developed for a robust solution of adjustment problems within the EIV model.
- An approximate solution by minimizing a Lp-norm could be obtained by an iterative procedure of
reweighted least squares, which is almost exclusively used by geodesists. Some first examples can be
found in the algorithm of Schlossmacher (1973) and the contributions of Krarup et al. (1980) and
Somogyi and Z´avoti (1993). In fact, reweighted least squares can possibly fail to provide a correct
solution in some cases, as has already been pointed out by Neitzel (2004) for the case of L1. A thorough
analysis of the robustness of various Lp-norms has been presented by Marx (2013), who noticed that Lp
for 1.2<p<1.5 may be less resistant to outliers than L1and proposed L1.05 as an alternative solution.
New robust algorithms can be implemented for the detection of blunders, based on the combination
of a reweighted approach and the direct solutions of weighted nonlinear least squares from this thesis.
- Most of the mentioned procedures fail to provide a resistant solution when leverage points1exist in
the dataset. Rigorous methods for the identification of erroneous data can be global optimization
methods based on systematic or stochastic search, as demonstrated by the study of Marx (2015) for
the detection of blunders by means of a Monte Carlo simulation. Combinatorial approaches can also
be employed in leverage point’s cases. Examples of combinatorial approaches include the maximum
subsample (MSS) method developed theoretically by (Neitzel, 2004, p. 109) and employed successfully
by Neitzel and Marx (2007), Wujanz et al. (2016) and Wujanz (2016) for the detection of deformations
using laser scanning data. The employment of such methods for the implementation of modern and
robust algorithms is of great interest.
1See e.g. (Everitt and Skrondal 2010) for a definition.
175
Appendices
177
A Stochastic models for the numerical investigations
A.1 Singular cofactor matrix for fitting a straight line in 2D
A singular cofactor matrix is presented in this appendix for the application example of fitting a straight
line in 2D. It was computed by utilizing the correlation coefficients from (Snow 2012, pp. 94-95), together
with the necessary precisions of the measured coordinates from Table 6.8. The variance-covariance matrix
for this adjustment problem is
QLL ="Qxx Qxy
Qyx Qyy #,(A.1)
with the sub-matrices:
Qxx =
0.001 −0.00010729581873004 −0.000184367806311119 −0.000162605765335955 −0.000396422204114424 ···
−0.00010729581873004 0.001 −0.000209334092987561 −0.00018594935307068 −0.000458502562871592 ···
−0.000184367806311119 −0.000209334092987561 0.002 −0.000326944096945427 −0.000819451184602405 ···
−0.000162605765335955 −0.00018594935307068 −0.000326944096945427 0.00125 −0.000762997332323675 ···
−0.000396422204114424 −0.000458502562871592 −0.000819451184602405 −0.000762997332323675 0.005 ···
−0.0002625284166187 −0.000318111572725898 −0.00060532385276285 −0.000608691217742683 −0.00177606225648547 ···
−1.30268774106986e−06 −6.44923418774174e−05 −0.000275369953484654 −0.000452568996672073 −0.00192208514323818 ···
0.00055638919876439 0.000525619226802249 0.0006397035186675 0.000228426867755593 −0.000754067056197251 ···
0.00130709043198385 0.00133828286919678 0.00195079375824933 0.00127603026024317 0.00137539801300953 ···
0.00375954407416456 0.00404574061392281 0.00646160268527631 0.00507413806293484 0.00993112604965441 ···
−0.0002625284166187 −1.30268774106986e−06 0.00055638919876439 0.00130709043198385 0.00375954407416456
−0.000318111572725898 −6.44923418774174e−05 0.000525619226802249 0.00133828286919678 0.00404574061392281
−0.00060532385276285 −0.000275369953484654 0.0006397035186675 0.00195079375824933 0.00646160268527631
−0.000608691217742684 −0.000452568996672073 0.000228426867755593 0.00127603026024317 0.00507413806293484
−0.00177606225648547 −0.00192208514323818 −0.000754067056197251 0.00137539801300953 0.00993112604965441
0.0125 −0.00356447023778097 −0.0041685755760135 −0.00394653521323417 −0.000250424830498049
−0.00356447023778097 0.0166666666666667 −0.0159730654839819 −0.0211377773455841 −0.0296837502900246
−0.0041685755760135 −0.0159730654839819 0.05 −0.0415061974365333 −0.069564033429064
−0.00394653521323417 −0.0211377773455841 −0.0415061974365333 0.555555555555556 −0.114601731317248
−0.000250424830498049 −0.0296837502900246 −0.069564033429064 −0.114601731317248 1
,(A.2)
Qyy =
1−0.0799735814606215 −0.06518386303754 −0.0514204579136473 −0.0396422204114424 ···
−0.0799735814606215 0.555555555555556 −0.0551643771471813 −0.0438286828378432 −0.0341747632812917 ···
−0.06518386303754 −0.0551643771471813 0.25 −0.0365534612806129 −0.0289719744741855 ···
−0.0514204579136473 −0.0438286828378432 −0.0365534612806129 0.125 −0.0241280941877523 ···
−0.0396422204114424 −0.0341747632812917 −0.0289719744741855 −0.0241280941877523 0.05 ···
−0.0166037549406539 −0.0149959233498933 −0.0135354528317981 −0.0121738243548537 −0.0112328039935045 ···
−3.8138791846177e−05 −0.00140733827808684 −0.00285034981466199 −0.00418997473652902 −0.00562728909449856 ···
0.00940469397241429 0.00662217980206098 0.00382295973501637 0.0012209929672505 −0.00127460596179143 ···
0.005545515048479 0.00423202202024702 0.002926190637374 0.00171197424195029 0.000583531957097723 ···
0.00531679821802292 0.00426458505408133 0.00323080134263815 0.00226922352718828 0.0014044733149058 ···
178 Appendix A. Stochastic models for the numerical investigations
−0.0166037549406539 −3.8138791846177e−05 0.00940469397241429 0.005545515048479 0.00531679821802292
−0.0149959233498933 −0.00140733827808684 0.00662217980206098 0.00423202202024702 0.00426458505408133
−0.0135354528317981 −0.00285034981466199 0.00382295973501637 0.002926190637374 0.00323080134263815
−0.0121738243548537 −0.00418997473652902 0.0012209929672505 0.00171197424195029 0.00226922352718828
−0.0112328039935045 −0.00562728909449856 −0.00127460596179143 0.000583531957097723 0.0014044733149058
0.05 −0.00660011638235733 −0.00445639474180467 −0.00105896652148659 −2.23986777699e−05
−0.00660011638235733 0.0142857142857143 −0.00790461742025186 −0.00262556138650223 −0.00122902402483588
−0.00445639474180467 −0.00790461742025186 0.0142857142857143 −0.00297656367844959 −0.00166289845870223
−0.00105896652148659 −0.00262556138650223 −0.00297656367844959 0.01 −0.000687610387903489
−2.23986777699e−05 −0.00122902402483588 −0.00166289845870223 −0.000687610387903489 0.002
,(A.3)
and
Qxy =
0.0316227766016838 −0.0025289867005658 −0.00206129473887088 −0.00162605765335955 −0.00125359708006575 ···
−0.00339299170599481 0.0235702260395516 −0.00234042630964225 −0.0018594935307068 −0.00144991241169878 ···
−0.00583022195151901 −0.00493405188950133 0.0223606797749979 −0.00326944096945427 −0.0025913321746667 ···
−0.00514204579136473 −0.00438286828378432 −0.00365534612806129 0.0125 −0.00241280941877523 ···
−0.0125359708006575 −0.0108070090465971 −0.00916174276506853 −0.00762997332323675 0.0158113883008419 ···
−0.00830187747032693 −0.00749796167494667 −0.00676772641589903 −0.00608691217742684 −0.00561640199675225 ···
−4.11946034176041e−05 −0.00152009907587077 −0.00307872967476321 −0.00452568996672073 −0.00607816690940393 ···
0.0175945713361161 0.0123889639864633 0.00715210276593168 0.00228426867755593 −0.0023845694060815 ···
0.0413338287288236 0.0315436297318278 0.021810537267639 0.0127603026024317 0.00434939041038 ···
0.11888722238149 0.0953590207675548 0.072242914239365 0.0507413806293484 0.0314049780471384 ···
−0.0005250568332374 −1.20605449440977e−06 0.000297402536496859 0.000175364583519327 0.000168131922284769
−0.000636223145451796 −5.97083063915161e−05 0.000280955294656434 0.000179549488118848 0.000180931020641263
−0.0012106477055257 −0.000254943037809526 0.000341935913709648 0.000261726447211668 0.00028897165695746
−0.00121738243548537 −0.000418997473652902 0.00012209929672505 0.000171197424195029 0.000226922352718828
−0.00355212451297094 −0.0017795050590842 −0.000403065795849046 0.000184529007192446 0.000444133458802924
0.025 −0.00330005819117866 −0.00222819737090234 0.000529483260743293 −1.119933888495e−05
−0.00712894047556193 0.0154303349962092 −0.00853796263679499 −0.00283593042227886 −0.00132749766951248
−0.008337151152027 −0.0147881850800537 0.0267261241912424 −0.00556864073733696 −0.0031109981507291
−0.00789307042646833 −0.0195697791310586 −0.0221859957479004 0.074535599249993 −0.00512514523129067
−0.000500849660996098 −0.0274818126551341 −0.0371835399333781 −0.015375435693872 0.0447213595499958
,(A.4)
with Qxy =QT
yx.
A.2 Singular cofactor matrix for the 2D similarity transformation
A stochastic model is postulated in the last investigated weighting case for the 2D similarity transformation,
that involves a singular cofactor matrix. This matrix has been firstly introduced in (Snow 2012, pp. 96-97)
and (Neitzel and Schaffrin 2017). However, it is reordered in such a way, so that it fits to the needs and the
formulation of the stochastic model in this work (see section 5.4.4). Therefore, two cofactor matrices are
given. The first matrix is related to the coordinates in the source system
QLL1="Qxx Qxy
Qyx Qyy #,(A.5)
with the relevant sub-matrices:
A.2. Singular cofactor matrix for the 2D similarity transformation 179
Qxx =
36.370281457026799 −10.717856095227498 −8.652980417908310 −3.008722473688215 −13.990722470202799
−10.717856095227498 31.714850184591498 −9.754412805230141 −7.170133519153175 −4.072447764980719
−8.652980417908310 −9.754412805230141 29.490003562291800 −9.264714951570749 −1.817895387582575
−3.008722473688215 −7.170133519153175 −9.264714951570749 26.031496206858098 −6.587925262445999
−13.990722470202799 −4.072447764980719 −1.817895387582575 −6.587925262445999 26.468990885212101
10−6[m2],(A.6)
Qyy =
29.082186568548597 −12.471413471703498 −6.363232761238930 −2.835610062891940 −7.411930272714205
−12.471413471703498 30.958038019578300 −17.720699070083000 −0.556828023714815 −0.209097454076965
−6.363232761238930 −17.720699070083000 38.734832754164593 −10.531447467909800 −4.119453454932779
−2.835610062891940 −0.556828023714815 −10.531447467909800 32.063142073587599 −18.139256519071097
−7.411930272714205 −0.209097454076965 −4.119453454932779 −18.139256519071097 29.879737700795101
10−6[m2],(A.7)
Qxy =QT
yx =
−5.470847531049374 −4.151968395749720 −6.309575380973525 5.061541661693445 10.870849646079201
5.848221379655184 3.083310192383325 −3.437514270957704 −5.891999249139355 0.397981948058545
4.362573210487724 3.061227368299260 6.203628967652264 −10.232648328119298 −3.394781218319955
−0.456888271105760 5.116298153814835 2.979330360651054 −1.237264484444125 −6.401475758915990
−4.283058787987756 −7.108867318747670 0.564130323627930 12.300370400009299 −1.472574616901790
10−6[m2].
(A.8)
The second cofactor matrix concerns the coordinates in the target system
QLL2="QXX QXY
QYX QYY #,(A.9)
with
QXX =
6.794487571800590 −2.067492652515560 −1.752371987000400 −0.451547525907080 −2.523075406377545
−2.067492652515560 6.429318039548140 −1.970512300406600 −1.403742194865920 −0.987570891760060
−1.752371987000400 −1.970512300406600 6.229399700497529 −2.051290948290685 −0.455224464799860
−0.451547525907080 −1.403742194865920 −2.051290948290685 5.079972517721780 −1.173391848658100
−2.523075406377545 −0.987570891760060 −0.455224464799860 −1.173391848658100 5.139262611595570
10−6[m2],
(A.10)
QYY =
6.094989272208480 −2.499242822649745 −1.204799645049460 −0.699300070526570 −1.691646733982720
−2.499242822649745 5.912760395723919 −3.440142089389230 −0.117912645153740 0.144537161468785
−1.204799645049460 −3.440142089389230 7.206462860853270 −1.847407593285470 −0.714113533129125
−0.699300070526570 −0.117912645153740 −1.847407593285470 6.360602565647050 −3.695982256681265
−1.691646733982720 0.144537161468785 −0.714113533129125 −3.695982256681265 5.957205362324320
10−6[m2],
(A.11)
QXY =QT
YX =
−1.246373797420340 −0.879050061336250 −1.163579541268595 0.979701730455350 2.309301669569825
1.090128341653400 0.554510615436640 −0.917558590359990 −0.955341882018735 0.228261515288690
0.938081379301460 0.362129743332930 1.443374807628205 −2.018715860691035 −0.724870069571550
−0.106812374044770 1.212589988419975 0.582948103761090 −0.048153307431655 −1.640572410704635
−0.675023549489755 −1.250180285853295 0.054815220239305 2.042509319686065 −0.172120704582330
10−6[m2]
(A.12)
181
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