Determining Rotational Elemen ts of Planetary
Bo dies
Metho d and Implemen tation of an
Inertial F rame Bundle Blo c k A djustmen t
v orgelegt v on
Dipl.-Math.
Steffi Burmeister
geb. in Sc h w erin
v on der F akultät VI – Planen Bauen Um w elt
der T ec hnisc hen Univ ersität Berlin
zur Erlangung des ak ademisc hen Grades
Doktorin der Naturwissensc haften
– Dr.-rer.-nat. –
genehmigte Dissertation
Promotionsaussc h uss:
V orsitzender: Prof. Dr. F rank Neitzel
Gutac h ter: Prof. Dr. Jürgen Ob erst
Gutac h ter: Prof. Dr. Jürgen Kusc he
T ag der wissensc haftlic hen Aussprac he: 27. Jan uar 2017
Berlin 2017
A c kno wledgemen t
I’d lik e to thank Professor J. Ob erst for the opp ortunit y to write a PhD thesis
and do researc h on a fascinating and inspiring topic, for the gained insigh t in
the field of planetary geo desy and the kno wledge that I w as able to gather
during the past four y ears of w ork under his sup ervision.
I am also grateful to m y colleagues Dr. K. Willner, A. P asew aldt and to F.
Preusk er for v arious discussions and esp ecially for pro viding the data sets of
Phob os and V esta used for this w ork.
Finally I thank Simon F ear for his clear opinion ab out tables, and m y family
and friends for their curiosit y , encouragemen ts and kindness.
Abstract
The rotational elemen ts of a planetary b o dy state its orien tation in space with
resp ect to the In ternational Celestial Reference F rame (ICRF). They are used
for computing con trol p oin t net w orks (CPNs) and for creating e.g. maps or
terrain mo dels of the b o dy . This w ork describ es the inertial fr ame bund le
blo ck adjustment . It is a metho d to determine rotational elemen ts of planetary
b o dies in a direct and analytical w a y from image p oin t measuremen ts. Sim ul-
taneously with the rotational parameters, a corresp onding CPN is computed
and the external orien tation data of the camera are impro v ed. In con trary to
the classical photogrammetric bundle blo c k adjustmen t, here the in terdep en-
dence of the b o dy’s rotational mo del and the camera’s external orien tation is
resolv ed. The metho d w as tested on sim ulated data and applied to t w o real
science cases. The amplitude of forced libration, whic h con tributes to the ori-
en tation of the prime meridian, w as determined for the Martian mo on Phob os.
Using com bined images of the missions Viking and Mars Express, an ampli-
tude of 1 . 13 ◦ ± 0 . 13 w as computed together with a CPN of 680 p oin ts and an
uncertain t y of ∅ 55 m. The amplitude is in agreemen t with previous results,
deriv ed b y indirect metho ds (e.g. Willner et al., 2010; Ob erst et al., 2014).
Secondly , the p ole axis orien tation of V esta w as confirmed using images of the
curren t Da wn mission. The computed CPN of 82 806 p oin ts is compared with
the b o dy-fixed solution of Preusk er et al. (2012), sho wing that b oth net w orks
are plausible within the a v erage in tersection error of 10 m. The soft w are whic h
w as implemen ted is fo cusing on the application to large data sets. It could b e
ac hiev ed that the costs of memory and running time dep end only on the n um-
b er of images. In the example of V esta with 5440 images, 164 hours (out of
168) are sa ved with resp ect to a reference adjustmen t soft w are. The metho d
is suitable to study and refine rotational mo dels of b o dies with solid surfaces.
adjustmen t, bundle blo c k adjustmen t ICRF, forced libration amplitude,
Phob os, V esta, parallel in v ersion
Zusammenfassung
Die Rotationselemen te eines planetaren K örp ers geb en seine Orien tierung im
Raum in Bezug auf den in ternationalen Himmelsreferenzrahmen (engl. ICRF)
an. Sie sind not w endig für die Berec hn ung v on K on trollpunktnetzen (KPN)
und für die Erstellung v on z.B. Karten o der Geländemo dellen des K örp ers.
Diese Arb eit b esc hreibt den Bündelblo ckausgleich im inertialen R efer enzr ah-
men . Das ist eine Metho de, um Rotationselemen te planetarer K örp er direkt
und analytisc h anhand photogrammetrisc her Messungen zu b estimmen. Si-
m ultan wird ein KPN b erec hnet und die externen Orien tierungen der K amera
w erden v erb essert. Im Gegensatz zum klassisc hen k örp erfesten Bündelblo c k-
ausgleic h ist hier die b estehende Abhängigk eit zwisc hen dem Rotationsmo-
dell des K örp ers und der Kameraorien tierung aufgelöst. Die Metho de wurde
anhand einer Sim ulation getestet und auf zw ei reale Datensätze angew andt.
Die Amplitude der erzwungenen Libration, die sic h auf die Orien tierung des
Hauptmeridians auswirkt, wurde für den Marsmond Phob os b estimm t. Mittels
Bilddaten aus den Missionen Viking und Mars Express ist eine Amplitude v on
1 , 13 ◦ ( ± 0 , 13 ) b erec hnet w orden, das zugehörige KPN mit 680 Punkten hat
eine Unsic herheit v on ∅ 55 m . Die Amplitude stimm t mit den Ergebnissen
v on Willner et al. (2010) und Ob erst et al. (2014), die durc h indirekte Me-
tho den erhalten wurden, üb erein. Die P olac hsenorien tierung von V esta wurde
mittels Bilddaten aus der aktuellen Da wn Missi on b estätigt. Das b erec hnete
KPN mit 82 806 Punkten wurde v erglic hen mit dem Ergebnis v on Preusk er et
al. (2012) aus einem k örp erfesten Bündelblo c k ausgleic h; b eide Lösungen sind
innerhalb des mittleren Sc hnittfehlers v on 10 m plausib el. Die Soft w are wurde
mit Sc h w erpunkt auf große Datensätze implemen tiert. Dab ei k onn te erreic h t
w erden, dass die K osten für Sp eic her und Laufzeit n ur v on der Anzahl der Bil-
der abhängig sind. Beim V esta-Beispiel mit 5440 Bildern w erden im V ergleic h
zu einer Referenz-Soft w are 164 v on 168 Stunden eingespart. Die Metho de ist
geeignet, um Rotationsmo delle v on planetaren K örp ern mit fester Ob erfläc he
zu studieren und zu v erb essern.
Ausgleic hsrec hn ung, Bündelblo c k ausgleic h ICRF, Librationsamplitude, Pho-
b os, V esta, parallele In v ersion
Con ten ts
1 In tro duction 1
1.1 Motiv ation and con ten t . . . . . . . . . . . . . . . . . . . . . . . 1
1 . 2 R e f e r e n c e f r a m e s .......................... 5
1.2.1 In ternational Celestial Reference F rame (ICRF) . . . . . 5
1.2.2 Hipparcos Celestial Reference F rame (HCRF) . . . . . . 7
1.2.3 Spacecraft and camera reference frame . . . . . . . . . . 7
1.2.4 Bo dy-fixed reference frame . . . . . . . . . . . . . . . . . 9
1.3 Rotational elemen ts . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Con trol p oin t net w ork analysis . . . . . . . . . . . . . . . . . . . 13
1 . 5 I m p l e m e n t a t i o n ........................... 1 5
1 . 6 M i s s i o n s a n d d a t a .......................... 1 8
2 Definitions and F oundations 23
2.1 Mathematical foundations and notation . . . . . . . . . . . . . . 23
2.2 Con version b et w een reference frames . . . . . . . . . . . . . . . 25
2.3 Bo dy-fixed bundle blo c k adjustmen t . . . . . . . . . . . . . . . . 28
2.3.1 The functional mo del . . . . . . . . . . . . . . . . . . . . 30
2.3.2 The sto c hastic mo del . . . . . . . . . . . . . . . . . . . . 34
3 Inertial F rame Bundle Blo c k A djustmen t 37
3.1 The extended functional mo del . . . . . . . . . . . . . . . . . . 37
3.2 Deriv ativ es of rotational elemen ts . . . . . . . . . . . . . . . . . 39
3.3 Deriv ativ es of classical parameters . . . . . . . . . . . . . . . . . 42
3.4 T est case - a sim ulation of Phob os . . . . . . . . . . . . . . . . . 43
3.5 Numerical stabilit y . . . . . . . . . . . . . . . . . . . . . . . . . 48
i
4 Application to Large Data Sets 51
4.1 Sparse matrix compression . . . . . . . . . . . . . . . . . . . . . 53
4.2 Optimisation of running time . . . . . . . . . . . . . . . . . . . 60
4.2.1 The splitting tec hnique . . . . . . . . . . . . . . . . . . . 60
4.2.2 The in verse decomp osition metho d . . . . . . . . . . . . 61
4.2.3 Sufficiency of the in v erse decomp osition . . . . . . . . . . 63
4.2.4 An iterativ e inv erse decomp osition algorithm . . . . . . . 66
4.2.5 P arallel computation . . . . . . . . . . . . . . . . . . . . 71
4.3 Implemen tation of the decomp osition algorithm in Op enCL . . . 73
4 . 3 . 1 M e m o r y d e s i g n ....................... 7 4
4 . 3 . 2 T i l e s i z e ........................... 7 4
4.3.3 W ork-groups and w ork-items . . . . . . . . . . . . . . . . 75
4.3.4 The parallel decomp osition algorithm . . . . . . . . . . . 76
5 Application to the science cases Phob os and V esta 81
5 . 1 P h o b o s ................................ 8 2
5.1.1 F orced libration amplitude . . . . . . . . . . . . . . . . . 83
5.1.2 Con trol p oin t net w ork for Phob os . . . . . . . . . . . . . 83
5 . 2 V e s t a ................................. 8 7
5.2.1 P ole axis orien tation . . . . . . . . . . . . . . . . . . . . 87
5.2.2 Comparison with previous CPN solution . . . . . . . . . 88
5.2.3 Surface spherical harmonics analysis . . . . . . . . . . . 90
6 Conclusions and Outlo ok 97
6.1 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . 97
6 . 2 O u t l o o k ............................... 9 9
Bibliograph y 101
A App endix 105
A.1 Pro of of the split tec hnique . . . . . . . . . . . . . . . . . . . . 105
A.2 Bac kw ard op eration in split mo de . . . . . . . . . . . . . . . . . 106
A . 3 P h o b o s t a b l e s ............................ 1 0 7
A . 4 C e n t r e o f fi g u r e ........................... 1 1 1
B Co de examples for Compressed Sparse Matrices 115
Chapter 1
In tro duction
1.1 Motiv ation and con ten t
Since the final decades of the 20th cen tury , spacecraft ha v e b een launc hed to
study ob jects in the solar system. In man y cases the spacecraft carries an imag-
ing system to gather optical information ab out the target. F or this thesis the
fo cus is set on planetary b o dies with solid surfaces - that means planets, dw arf
planets, mo ons, asteroids or comets whic h are suitable for a photogrammetric
surv ey . In the field of planetary geo desy images are used to study prop erties
of planetary b o dies, e.g. top ograph y , shap e and rotation. Ma jor ob jectiv es
of these studies are the definition of reference systems, the determination of
rotational parameters as w ell as mapping and mo delling of planetary surfaces.
This includes in particular the computation of con trol p oin t net w orks (CPNs)
whic h are a first-order realisation of the b o dy’s reference system and define
the b o dy-fixe d reference frame b y three-dimensional co ordinates. While a b o dy
mo v es in space, it c hanges its orien tation e.g. b y self-rotation, the precession
of the p ole axis or librations. These c hanges in orien tation are describ ed b y
the rotational mo del of the b o dy in terms of functions of time, form ulated with
resp ect to an inertial reference system. These functions indicate the direction
of the p ole axis as w ell as the orien tation of the b o dy’s prime meridian. Any
error in the rotational mo del con tributes to a misalignmen t b et w een optical
measuremen ts and the corresp onding ground co ordinates. Hence, in order to
create stable maps of high qualit y , a precise kno wledge of the rotational pa-
rameters is desired.
1
2
In the past cen tury spin p oles, rates and sense of rotation of the large asteroids
w ere mostly determined b y earth-based measuremen ts of ligh tcurv es and an
analysis whic h included a theoretical shap e mo del. Magn usson (1986) giv es a
go o d o v erview ab out the applied metho ds and the literature of that era. He
consolidated some of the common metho ds and prop osed a pro cedure whic h
is here referred to as (p ar ameter) r ange sc an metho d . There, a series of anal-
yses with trial v alues for the parameters to b e determined is p erformed, and
the data are statistically ev aluated suc h that a b est-fit yields the final result.
This approac h, gradually impro v ed e.g. b y De Angelis (1993), w as widely
used. With the a v ailable image data of the Hubble Space T elescop e Camera
and the spacecraft missions, the trial v alues are t ypically ev aluated with re-
sp ect to a computed CPN or the related impro v emen ts of image observ ations
(Thomas et al., 1995). A sp ecial effect, that has b een v erified for a few mo ons
in the solar system, is the forced libration of satellites. This small v ariation
in the mean self rotation rate dep ends on the satellite’s momen ts of inertia
and the gra vitational torque that a primary b o dy exerts on its satellite. The
amplitude of the forced libration con tributes to the orien tation of the prime
meridian and is therefore an imp ortan t rotational parameter. An angular error
in the mo del ab out the prime rotation axis will lead to a p ositional error of
the con trol p oin ts; this error is prop ortional to the size of the b o dy . F urther-
more, since the magnitude of this libration is related to the magnitude of the
gra vitational torque, conclusions ab out the in ternal structure of a b o dy can
b e dra wn (Murra y and Dermott, 1999). By applying the range scan metho d
with a bundle blo c k adjustmen t or similar t yp e of CPN analysis, amplitudes of
forced libration ha ve been obtained for the Martian mo on Phob os (Duxbury
and Callahan, 1989; Willner et al., 2010, a.o.) as w ell as for the Saturnian
mo ons Jan us, Epimetheus (Tiscareno et al., 2009) and Mimas (T a jeddine et
al., 2013). An alternativ e metho d has b een in tro duced b y Ob erst et al. (2014)
who used a rotation mo del of Phob os without longitudinal libration and ev al-
uated the spacecraft p osition residuals o v er the anomalistic p erio d 1 after a
bundle blo c k adjustmen t. With resp ect to Phob os also a theoretical approac h
has b een used, where the equations of motion are solv ed b y n umerical in tegra-
tion and the magnitude of forced libration could b e determined as one of the
1 p ericen ter to p ericen ter
3
orbit parameters (Jacobson, 2010; Ram baux et al., 2012).
It is imp ortan t to pa y atten tion to the in terdep endency of rotational elemen ts
and the computed CPN of a planetary b o dy . The initial result of the CPN,
that is deriv ed from the image measuremen ts, reflects errors in the measure-
men ts themselv es and in the camera’s external orien tation (i.e. the p osition
and p oin ting data of the camera). In order to obtain a reliable net w ork, the
camera orien tation parameters need to b e impro v ed. The classic approac h to
minimise suc h errors is the men tioned bundle blo c k adjustmen t, an iterativ e
pro cedure that finds a Least-Squares Regression solution for the underlying
mathematical mo del. This is a p o w erful and w ell-pro v en to ol in terrestrial
geo desy . In planetary geo desy the orien tation of the target is generally not
w ell-kno wn, esp. not fixed with resp ect to the observ er. The orien tation of the
camera reference frame, e.g. obtained b y star trac king, is giv en with resp ect
to an inertial reference frame. Also the p ositions are giv en in inertial co or-
dinates; they are obtained b y earth-based trac king metho ds (e.g. range and
Doppler measuremen ts) and ev en tually b y further n umerical in tegration of the
spacecraft’s tra jectory . Based on the assumed rotational mo del of the target
b o dy , the p osition v ectors and p oin ting angles are con v erted into orien tation
data with resp ect to the b o dy-fixed frame and this w a y , they are link ed to the
rotational elemen ts. The classical photogrammetric bundle blo c k adjustmen t,
designed for geo detic measuremen ts on Earth, cannot resolv e this fundamen-
tal dep endency since the underlying mathematical mo del is form ulated in the
b o dy-fixed reference frame only . Errors in the rotational mo del lead to large
p osition offsets and falsified p oin ting data in the b o dy-fixed frame. I.e. the
(b o dy-fixed) orien tation uncertain ties include the uncertain ties of rotational
elemen ts of the observ ed b o dy , to o. Therefore, a series of CPNs is computed
together with a set of statistic information whic h yield the basis of decision
with resp ect to the most reliable solution.
The metho ds to determine rotational elemen ts to da y are indirect metho ds.
The range scan metho d requires a large n um b er of adjustmen ts, one for eac h
assumed v alue of the rotational parameter in question. Hence, it is only suit-
able for small data sets of a few h undred images. As Thomas et al. (1997)
could determine the spin p ole of V esta with v ery few images, the precession
mo v emen t, recen tly indicated b y theoretical considerations of K onopliv et al.
4
(2014), could not b e detected b y this approac h. In order to co v er long-time p e-
rio ds of rotational b eha viour as w ell as the surface of larger planetary b o dies,
larger data sets need to b e formed and pro cessed. When this is done, the non-
linear increasing computational effort stresses the memory capacit y , running
time and hardw are exp enses in suc h a w a y , that ev en for a single adjustmen t
an effectiv e resource managemen t is required. The range scan metho d in these
cases is no longer practicable.
It is an aim of this w ork to present a method that allows the direct deter-
mination of rotational elemen ts based on an analysis of image data. This
metho d will b e referred to as inertial frame bundle blo c k adjustmen t, since
the underlying mathematical equations are form ulated with resp ect to the in-
ertial frame. In order to a v oid a p ossible misunderstanding the expression
dir e ct c omputation shall b e clarified. In con trary to the previously men tioned
tec hniques whic h yield their results indirectly and b y in terpretation, within
the inertial frame bundle blo c k adjustmen t the (time-indep enden t) rotational
parameters b ecome directly adjustable and it tak es only the iterations of a
single adjustmen t to obtain the result. T o exemplify the new metho d, t w o
planetary ob jects ha v e b een c hosen: Phob os, the inner mo on of Mars and the
protoplanet V esta, whic h is lo cated in the main asteroid b elt. The priorit y
here is the forced libration amplitude of Phob os and V esta’s mean p ole axis
orien tation. The second ob jectiv e of this w ork is the p erformance optimisation
of the implemen ted soft w are with resp ect to the computation of large CPNs.
In order to describ e planetary b o dies and their orien tation in space, the con-
cepts of reference frames and rotational elemen ts ha v e b een dev elop ed and
con tin uously impro v ed. The curren t c hapter pro vides an in tro duction to the
topic and its application to the reconstruction of a b o dy’s surface from image
data. It also inclu des an o v erview ab out the n umerical tasks, related existing
solutions and the in v olv ed missions and image data. More tec hnical informa-
tion and concrete form ula will b e giv en in c hapter 2 after the statemen t of the
mathematical notation and foundations. There also the classical metho d of a
photogrammetric bundle blo c k adjustmen t is describ ed. This metho d will then
b e extended to an inertial con text in c hapter 3, where the o wn w ork con tribu-
tion starts. Sub ject of c hapter 4 are all asp ects of implemen tation, hardw are
5
and soft w are considerations fo cusing the c hallenge of large data sets. The new
concepts and metho ds are then applied to the science cases Phob os and V esta
in c hapter 5. Chapter 6 includes the summary and an o v erview ab out future
goals and further soft w are dev elopmen t.
1.2 Reference frames
The motion and orien tation of b o dies in the solar system, b eing natural or
artificial, and sp ecific lo cations on a certain b o dy are form ulated in v arious
differen t reference systems. By con v en tion, in planetary science these are righ t-
handed orthogonal co ordinate systems. It pro v ed to b e practical to w ork with
3D cartesian co ordinates for lo cations and with angular expressions to describ e
orien tations at a giv en time. Hence, reference systems are defined b y sp ecifying
three principle axes (in righ t-handed order) and the origin. Co ordinate systems
of mo ving and rotating ob jects are asso ciated with a reference ep o c h. The
curren t standard ep o c h is J2000.0 (1st Jan uary 2000, 12:00:00 Barycen tric
Dynamical Time) and time is measured in SI seconds with resp ect to that
date. It is furthermore distinguished b etw een the abstract lev el of reference
systems and the concrete lev el of reference frames. While a reference system is
giv en b y definition, the resp ectiv e reference frame is its concrete realisation b y
co ordinate v alues. The co ordinate system whic h is giv en b y the frame resp ects
the definitions of the reference system as close as p ossible and is decisiv e for the
description of the ob ject in question. The follo wing subsections will describ e all
co ordinate systems whic h are relev an t for this w ork and ho w they are related
to eac h other. Figure 1.2 sho ws an exemplary sc heme of a spacecraft with a
moun ted camera and a target b o dy whic h are mo ving in space. The explicit
mathematical definition of the frame con v ersions follo ws in section 2.2.
1.2.1 In ternational Celestial Reference F rame (ICRF)
The inertial reference system whic h is tak en into accoun t for this w ork is the
In ternational Celestial Reference System (ICRS). Its origin is the solar system
barycen tre and the three principle axes are not rotating. In 1998 with the in-
tro duction of the In ternational Celestial Reference F rame (ICRF) this idealised
system w as realised first-time. Using radio in terferometry with v ery long base-
6
lines (VLBI), the ICRF is defined through the p ositions of extragalactic radio
sources (mainly quasars) that mo ve o nly marginal, relativ e to our solar system
(Ma et al., 1998). Out of the 608 listed sources, group ed according to differen t
criteria of qualit y , 212 defining sour c es w ere c hosen to sp ecify the frame (see
figure 1.1). A ccording to Ma et al., the alignmen t uncertain t y of the principle
axes is less than 0.02 milliarcseconds (compared with the ICRS axes). As a
Figure 1.1: Distribution of the ICRF defining sources on the celestial sphere
(Ma et al., 1998, fig. 10)
consequence the ICRF itself is ep o c hless. All other t yp es of reference frames
can b e related directly or in subsequen t order to the ICRF. The inertial frame
is used to describ e not only the motion of an ob ject within the frame, but
also its orien tation relativ e to the (stable) principle axes. A t a giv en time,
an y reference frame can b e describ ed with resp ect to the ICRF b y sp ecifying
the orien tation of b oth frames relativ e to eac h other. The In ternational Earth
Rotation Service (IERS) who monitors the radio sources released an up date
of the frame (Ma et al., 2009) whic h w as adopted b y the In ternational Astro-
nomical Union (IA U) as of 1st Jan uary 2010. The up date is officially referred
to as ICRF2 and the original frame as ICRF1. Since the up date is without
an y consequence to the rotational mo dels, the distinction is dropp ed here.
7
1.2.2 Hipparcos Celestial Reference F rame (HCRF)
The inertial reference system ICRS is also realised at optical w a v elengths b y
the Hipparcos Celestial Reference F rame (HCRF), consisting of 85% of the
stars in the Hipparcos catalog. A ccording to Kaplan (2005) the p osition errors
of these stars in ICRF co ordinates with resp ect to J2000.0 range b et w een 5
and 10 milliarcseconds (mas). The orien tation of the spacecraft and camera
reference frame with resp ect to the ICRF is t ypically obtained b y star trac king,
i.e. it is based on optical observ ation of stars. Hence, this frame is used b y
services whic h pro vide the orien tation data of spacecraft cameras, e.g. NASA’s
Na vigation and Ancillary Information Facilit y (NAIF).
1.2.3 Spacecraft and camera reference frame
Hierarc h y of reference frames with ICRF as fundamen tal frame
Figure 1.2: Orien tation and motion of spacecraft and b o dy are form ulated
within the ICRF. The con version betw een the frames is accomplished b y the
(p ossibly successiv e) application of rotation matrices. In this example, the
transformation b et w een spacecraft and camera frame is constan t.
The spacecraft motion is describ ed in inertial co ordinates, but it has an
attac hed lo cal frame whic h is pre-defined b y the mission team. The origin
in ICRF co ordinates is equal to the p osition of the prob e. Na vigation and
stabilisation of the fligh t system’s orien tation is ac hiev ed b y the attitude con-
8
trol subsystems on b oard (Do o dy, 2011). In the frame hierarc h y , the camera
reference frame is sub ordinate to the spacecraft frame (see figure 1.2). There
is a sp ecial platform (called instrumen t head) on whic h the camera system
is moun ted. It has a fixed orien tation with resp ect to the co ordinate system
used for the spacecraft. The camera itself has a certain alignmen t with resp ect
to the platform and hence to the spacecraft. This alignmen t is either fix or
dep ends on pre-defined rsp. commanded mo v es of the camera. In praxis, the
moun ting is often neglected, i.e. the p osition of camera and spacecraft is tak en
as iden tical.
Here only CCD frame cameras are considered and the ob ject whic h needs
to b e describ ed is the image that the camera has captured at a certain time.
The reference frame is defined b y the co ordinates of the CCD sensors (pixel)
whic h la y in a plane b ehind the camera lens. The origin of the camera frame
is the fo cal p oin t. The direction from the images’ principle p oin t to the lens is
the b ore side direction and the distance b et w een them is the (nominal) fo c al
length . The information ab out an instrumen t on b oard of a spacecraft is col-
lected in the instrument kernel (IK). The IK con tains all required information
to transform pixel co ordinates in to metric co ordinates (fo cal length, the n um-
b er of v alid pixels in line and sample direction, the co ordinates of the principle
p oin t, the pixel size etc.), including the definition of the three primary axes.
The assignmen t of the axes v aries for differen t instrumen ts rsp. missions, but
t ypically one axis con tains the b ore side v ector and the other axes are parallel
to the rectangular CCD b oundaries.
Example: + Z p oin ts in b ore side direction, then the X - Y -plane is parallel to
the CCD – all pixels ha v e the same Z -co ordinate whic h agrees with the neg-
ativ e fo cal length. If + X p oin ts from left to righ t (w atc hing from + Z ), + Y
p oin ts from top to b ottom.
The measuremen t of the image p oin t observ ations is accomplished within the
co ordinate system of the image. In order to compute three-dimensional (3D)
surface co ordinates of a target b o dy , they are transformed in to 3D cen tred
and metric co ordinates of the camera reference frame. F urthermore the ori-
en tation of the frame including its p osition needs to b e kno wn. All camera
orien tation data used in this w ork are either directly obtained b y the SPICE
k ernels of NAIF (A cton, 1996) or ha v e b een originally pro vided b y them and
9
where mo dified later on.
1.2.4 Bo dy-fixed reference frame
The description of the lo cation of a crater on a b o dy , constan tly mo ving and
c hanging its orien tation with resp ect to the inertial frame, requires a co ordinate
frame that is tied to the b o dy . Con ten t of this subsection are the recommen-
dations and con v en tions giv en b y the IA U W orking Gr oup on Carto gr aphic
Co or dinates and R otational Elements (Arc hinal et al., 2011).
An ob ject-cen tred reference system is defined with resp ect to its mean axis
of rotation and an additional c hoice of the zero-longitude that defines the di-
rection of the X-axis. The corresp onding reference frame is called b o dy-fixe d
reference frame and its main purp ose is to facilitate mapping. The frame is
defined b y con trol p oin ts with the origin commonly pinned to the cen ter of
mass.
F or planets and mo ons, the Z -axis is iden tified with the mean rotation axis
con taining the north p ole of rotation on the p ositiv e side. 2 If there is no other
information the axis is assumed to b e normal to the mean orbital plane. F or
the catagory of small b o dies the p ositiv e p ole of rotation is defined b y the
spinning direction of the b o dy: lo oking on to +Z the b o dy app ears to rotate
an ti-clo c kwise. The term north p ole is not used here, since there are cases
where the spin p ole mo v es in a short time b et w een north and south of the in-
v ariable plane. As a consequence only planets and mo ons can ha v e retrograde
rev olutions; Pluto b eing formerly a retrograde rotating planet is considered a
prograde rotating dw arf planet with its p ositiv e p ole south of the in v ariable
plane. A ccording to the recommendations for small irregular b o dies, the initial
definition of a b o dy-fixed co ordinate system should b e, if p ossible, based on a
shap e mo del and estimate of the moments of inertia (Arc hinal et al., 2011).
The prime meridian can b e c hosen in principle as arbitrary as on Earth, but
the IA U recommends to align it with the longest axis or the minim um momen t
of inertia. Often a landmark is used, but the prime meridian can b e defined
b y reference to another celestial ob ject, to o. On images top ographic features,
t ypically small craters, are iden tified and a candidate is c hosen that defines a
2 The p ole that la ys north of the in v ariable solar plane, as it is for Earth
10
sp ecific longitude. E.g. on Mercury the prime is defined with resp ect to Hun
Kal whic h marks the meridian 20 ◦ W (Robinson et al., 1999).
The b o dy-fixed co ordinates are considered as indep enden t of the self-rotation
and the b o dy’s orientation in space. A reorien tation of the spin axis in terms
of p olar motion, also kno wn as true p olar w ander (Matsuy ama et al., 2006;
Harada, 2012), is not considered in this con text. F urthermore, p ossible c hanges
of the top ograph y , e.g. b y tidal deformation, are not tak en in to accoun t since
they are exp ected small enough to b e neglected.
1.3 Rotational elemen ts
The rotational elemen ts of a planetary b o dy are used to describ e the orien tation
of its b o dy-fixed reference frame with resp ect to the ICRF at a giv en time.
The sp ecific directions of the principle axes are recorded with resp ect to the
standard ep o ch.
The celestial co ordinates α (righ t ascension) and δ (declination) describ e the
direction of the p ole axis, from the cen ter to the north p ole (rsp. the p ositiv e
p ole). This v ector is giv en in ICRF p olar co ordinates and is referred to as p ole
axis orien tation. In addition the angle W denotes the p osition of the b o dy’s
prime meridian, relativ e to the in tersection of the b o dy’s equatorial plane and
the ICRF equator (see figure 1.3). The no de Q = ( α + 90 ◦ , 0 ◦ ) coincides with
( W , 0 ◦ ) in the b o dy-fixed frame (measured as p ositiv e w est) and W is referred
to as prime meridian orien tation.
The angles α, δ, W are the time-dep ending functions of the b o dy’s r otational
mo del and include further mo dels of effects lik e precession, acceleration or
forced libration. Three subsequen t rotations align the ICRF with the reference
frame of the b o dy . A detailed description of this con v ersion is giv en in section
2.2 (page 27). It is w orth to p oin t out, that in praxis only the rotational
elemen ts are used to describ e the principle axes of the b o dy-fixed co ordinate
system although there migh t b e a defining CPN. The rotational mo del implies
the co ordinates of the feature that defines the zero-longitude. Hence, when
rotational elemen ts are impro v ed, the original definition of the prime meridian
should b e resp ected (Arc hinal et al., 2011). In the follo wing, the tw o rotational
mo dels of Phob os and V esta will b e giv en. It is t = s/ 86 400 the magnitude of
11
The orien tation of a b o dy with resp ect to the ICRF
Figure 1.3: The p ole axis orien tation ( α , δ ) states righ t ascension and declina-
tion of the b o dy’s Z-axis; W is the prime meridian orien tation w.r.t. the no de
Q . The longitude L of Q in b oth reference frames is: L I C RF = 90 ◦ + α (p os.
east) and L B F = W (p os. w est). Figure, according to Arc hinal et al. (2011),
is mo dified to matc h curren t notation.
12
da ys since J2000.0 (without unit) and T = 36 525 t equals one Julian cen tury;
dates b efore the standard ep o c h result in negative v alues.
Phob os (Arc hinal et al., 2011):
α ( t ) = α 0 − 0 . 108 ◦ T + 1 . 79 ◦ sin( M 1 ) (1.1)
δ ( t ) = δ 0 − 0 . 061 ◦ T − 1 . 08 ◦ cos( M 1 )
W ( t ) = 35 . 06 ◦ + 1128 . 844585 ◦ t + 8 . 864 ◦ T 2 − 1 . 42 ◦ sin( M 1 ) − λ 0 sin( M 2 )
with the time-indep enden t parameters
α 0 = 317 . 68 ◦ , δ 0 = 52 . 90 ◦ , λ 0 = 0 . 78 ◦ .
The function M 1 = M 1 ( t ) = 169 . 51 ◦ + 0 . 435764 ◦ t mo dels the precession mo-
tion of 826 da ys and M 2 = M 2 ( t ) = 192 . 93 ◦ + 1128 . 40967 ◦ t + 8 . 864 ◦ T 2 the
orbital motion. The similarit y of M 2 and W reflects that Phob os is in syn-
c hronous orbit around Mars. The second term in the definition of W is the
spin rate in degrees p er da y (one rev olution tak es ab out 7 h 39 min) and the
+ sign indicates the direct sense of rotation. The third term mo dels the spin
acceleration whic h is caused b y Phob os’ gradual approac h to Mars. The co-
ordinates ( α 0 , δ 0 ) describ e the axis of precession and λ 0 is the amplitude of
forced libration. In section 1.4 alternativ e results of λ 0 are giv en.
V esta (Arc hinal et al., 2013; Li and Mafi, 2013):
α ( t ) = 309 . 031 ◦ ± 0 . 01 (1.2)
δ ( t ) = 42 . 235 ◦ ± 0 . 01
W ( t ) = W 0 + 1617 . 3329428 ◦ t , W 0 = 285 . 39 ◦ .
The mo del ab o v e con tains a fixed p ole axis orien tation and the rotation rate of
5 h 20 m 31.662 s . The co ordinate system with W 0 = 285 . 39 ◦ is called Claudia
Double-Prime system, it assigns a longitude of 146 ◦ W to the crater named
Claudia suc h that the prime meridian passes through the feature named Olb ers
R e gio as defined b y Thomas et al. (1997). There is an alternativ e co ordin ate
system with W 0 = 75 . 39 ◦ used b y the Da wn team b efore. It is kno wn as
Dawn-Claudia system and assigns a longitude of 356 ◦ W to Claudia. Hence
b oth systems are rotated b y 210 degree with resp ect to eac h other. T able 1.1
sho ws a selection of earlier solutions of V esta’s rotational elemen ts.
13
Earlier rotational mo dels of V esta
publication α δ rotation rate
Gehrels (1967) 191 . 1 ◦ 75 . 1 ◦ 5 h 20 m 31 . 665 s pr op ose d
304 . 4 ◦ 48 . 7 ◦ 5 h 20 m 31 . 171 s dismisse d
Magn usson (1986) 307 . 3 ◦ 38 . 5 ◦ 5 h 20 m 31 . 646 s
Thomas et al. (1997) 301 . 0 ◦ 41 . 0 ◦ 5 h 20 m 31 . 664 s
Li et al. (2010) 305 . 8 ◦ 41 . 4 ◦ –
T able 1.1: Selected solutions of the decades prior to Da wn. The p ole co or-
dinates ( α, δ ) are w.r.t. J2000 (v alues b efore 1997 are con v erted). Gehrels
dismissed the 2nd solution b ecause of a larger error in the rotation rate.
1.4 Con trol p oin t net w ork analysis
The in tro duced concepts of reference frames and rotational elemen ts are re-
quired to obtain the 3D co ordinates of surface p oin ts, the so called con trol p oin t
net w ork (CPN) of a b o dy . As men tioned b efore, a CPN is the co ordinate-based
realisation of the b o dy-fixed reference system and corresp onds to a unique set
of rotational parameters. In fact, when the range scan metho d is applied with
a series of bundle blo c k adjustmen ts, b oth – the rotational elemen ts and the
CPN – are obtained sim ultaneously . In this section an o v erview ab out CPNs
and the related analysis is giv en. This includes a brief in tro duction in ho w they
can b e computed and statistically ev aluated. T o a v oid confusion, it is help-
ful to think of first-order CPNs, defining the b o dy-fixed reference frame, and
second-order CPNs whic h are obtained b y using this reference frame. Those
can b e considerable larger than and also b e completely different from the defin-
ing net w ork.
The co ordinates of a con trol p oin t (in the b o dy-fixed reference frame) are giv en
b y the in tersection of m ultiple ra ys whic h connect the p oin t with its image
represen tation (in the camera reference frame). The underlying c o-line arity
e quations inv olv e a rotation matrix whic h con v erts the camera reference frame
to the b o dy-fixed reference frame at image time. Hence, the orien tation of b oth
reference frames with resp ect to the ICRF need to b e kno wn at the sp ecific
time (see frame hierarc h y in figure 1.2). Merged to a single rotation matrix,
14
also the uncertain ties of b oth frames merge and p ossible errors affect the ini-
tial result of the in tersection p oin ts. By the pro cedure of the bundle blo c k
adjustmen t, where the equations are solv ed for all con trol p oin ts sim ultane-
ously , the in tersection errors and the camera orientation errors can b e reduced.
The kno wledge of the uncertain ties is v ery imp ortan t, b ecause of its ma jor in-
fluence on the result. It is tak en in to accoun t for the calculation in terms of a
sto c hastic mo del whic h is, briefly stated, a w eigh ting rule for the observ ations.
A differen t sto c hastic mo del will yield a differen t result. Hence, in order to
repro duce the results, the uncertain ties of all in v olv ed observ ations m ust b e
sp ecified.
An imp ortan t outcome of the analysis is the sto c hastic mo del ap osteori, whic h
states the uncertain ties of a) the computed con trol p oin ts, b) the impro ve d
orien tation data of the camera and c) the image p oin t observ ations. Based on
the w eigh ted residuals of all observ ations an o v erall system error is computed.
All of these quan tities can b e tak en in to accoun t for decision when a parameter
range scanning is applied. E.g. Tiscareno et al. (2009) analysed the difference
b et w een the original image p oin t measuremen ts and their predicted p ositions
b y the functional mo del after the adjustmen t and found a forced libration
amplitude ( λ 0 ) of 0 . 3 ◦ ± 0 . 9 for Jan us and 5 . 9 ◦ ± 1 . 2 for Epimetheus. T a jed-
dine et al. (2013) used a similar ev aluation metho d for Mimas and determined
λ 0 = 0 . 805 ◦ ± 0 . 022 . They applied differen t rotation mo dels for the bac k-
w ard pro jection in to the images, but here the camera p ositions w ere corrected
with the lim b fit metho d and the p oin ting angles w ere k ept fixed. Concerning
Phob os Willner et al. (2010) obtained the forced libration amplitude in cor-
resp ondence with the minimal mean error of the CPN; table 1.2 sho ws their
solution as w ell as those of other selected authors.
The follo wing t w o examples of CPN usage are related to a fixed ob ject-cen tred
reference frame. In the pro cess of creating a digital terrain mo del (DTM), con-
sisting of millions of p oin ts, also a bundle blo c k adjustmen t is deplo y ed, but to
a m uch smaller subset of con trol p oin ts. Here, the impro v ed camera orien tation
and their computed uncertain ties are the most relev an t results of the analysis
and the CPN is only a b y-pro duct. The selected con trol p oin ts w ork as inner
constrains of the net w ork to adjust the orien tation data of the camera. The
resulted p ositions and p ointing angles are k ept fix when the final n um b er of
15
The forced libration amplitude of Phob os
λ 0 Publication Metho d
0 . 80 ◦ ± 0 . 20 Duxbury and Callahan (1989) CPN
1 . 24 ◦ ± 0 . 15 Willner et al. (2010) CPN
1 . 09 ◦ ± 0 . 10 Ob erst et al. (2014) CPN
1 . 03 ◦ ± 0 . 22 Jacobson (2010) O
1 . 10 ◦ – Ram baux et al. (2012) O
T able 1.2: Selected results of Phob os’ forced libration amplitude λ 0 , obtained
either b y CPN analysis or b y analysing the orbital equation of motion (O)
p oin ts is computed subsequen tly . The co ordinates of the CPN can also b e used
to compute the co efficien ts of surface spherical harmonics, functions whic h ap-
pro ximate the whole surface and the principle axes of the frame. Apart from
the resulting uncertain t y , this alternativ e represen tation do es not dep end on
the original measuremen ts an y more, but yields a global shap e mo del. Large
net w orks can also directly b e triangulated and rendered to mo del the surface,
examples of the science cases Phob os and V esta are giv en in c hapter 5.
1.5 Implemen tation
Deplo ying a bundle blo c k adjustmen t to large net w orks requires a lot of com-
putation p o w er and the resources running time and memory need to b e con-
sidered. The memory restriction of a computer is a hard limit and decides
whether or not a certain program can b e executed. The running time also
con tributes to qualitativ e asp ects. Considering common blo c k adjustmen ts
with one fixed rotational mo del, computation times can tak e sev eral w eeks
and mon ths including pre- and p ost-pro cessing (priv ate comm unication with
F. Preusk er). Since time itself is a critical resource in scien tific pro jects, data
sets migh t b e reduced in order to publish results in time. Here, matrix pro d-
ucts and matrix in version a re the most time exp ensiv e parts of the program.
Compression tec hn iques and an in telligen t implemen tation design are k eys to
an efficien t managemen t of b oth resources. In c hapter 4 the topic of sparse
matrix compression and the adv an tage of symmetric matrices is discussed in
16
detail. This section giv es an o v erview ab out the n umerical tasks whic h need to
b e p erformed, related a v ailable solutions and the asp ect of computation time.
The reader shall get an idea of the concepts and elemen ts whic h will b e used
later on.
T ec hnically for a symmetric matrix N of dimension n and a giv en v ector b , the
equation N x = b is to solv e and n agrees with the n um b er of parameters that
are to adjust. The straigh t forw ard approac h is, to in v ert N and build the
pro duct with b subsequen tly . It is kno wn that the computation of x = N − 1 b
can b e realised m uc h more efficien tly (Dahmen and Reusk en (2008), c hapter
13), but the bundle blo c k adjustmen t also includes an ev aluation phase that
requires N − 1 . The handling of measuremen t errors and the sto c hastic mo del
ap osteori dep end on the in v erted matrix. Hence, a solution is sough t that re-
duces running time and main tains the statistical information. There are plen ty
approac hes to matrix in v ersion, and all inv olv e some transformation in to tri-
angular shap e and the in v ersion of the obtained triangular matrix. One of the
common metho ds to transform a symmetric matrix M in to triangular form,
is the Cholesky decomp osition. The algorithm computes a lo w er triangular L
with LL T = M or in a v arian t a lo w er triangular L and a diagonal D with
LD L T = M (Dahmen and Reusk en (2008), c hapter 3). The in v erse is then
giv en b y the pro duct ( L T ) − 1 D − 1 L − 1 .
The actual running time, i.e. the time whic h it tak es for a computer to ex-
ecute an algorithm (or program), dep ends highly on the sp ecifications of the
mac hine. In order to analyse the effectiv eness of an algorithm, a measure is re-
quired that is indep enden t of the hardw are and its p erformance sp ecifications.
Suc h a measure is the n um b er of required floating p oin ts op erations (flops),
e.g. m ultiplications or additions whic h are applied to floating p oin t n um b ers.
The n um b er of flops usually dep ends on v ariables of the algorithm whic h de-
termine the size of the problem, for instance the dimension parameters of a
matrix. In mo dern b o oks of computer science the term time c omplexity has
b een in tro duced for this p erformance-indep enden t consideration of the execu-
tion times, while in classic n umerical literature the term running time is used
(assuming that the con text of a theoretical consideration is clear). The sym b ol
O ( n 3 ) denotes a running time of cubic order, stating that the n um b er of flops
is b ound b y k n 3 for some real n um b er k . The v alue for n 3 is clearly dominan t,
17
but the larger n gro ws, the more the actual v alue of k needs to b e considered.
A standard matrix pro duct of t w o square matrices with dimension n tak es
O ( n 3 ) flops, so do es a standard in v ersion algorithm.
Considering six orien tation parameters p er image and three co ordinates p er
con trol p oin t as unkno wn parameters, n equals 30 000 if the data set consists of
1000 images and 8000 con trol p oin ts. It is n 3 = 27 000 · 10 9 flops, whic h is equal
to 27 000 Giga flops (1 Gflop = 10 9 flops). T o estimate the mac hine-dep ending
computation time, a v alue for x = k n 3 and the a v erage core p erformance y in
Gflops/s m ust b e sp ecified. The time in seconds is then appro ximated b y
t = x f l ops
y Gf l ops/s = x
y 10 − 9 s .
A core p erformance of 3.0 Gflops/s and a v alue of x = 1
3 n 3 yields t = 3000 s
rsp. 50 min utes. The structure of the matrix N allo ws the application of a
w ell-kno wn tec hnique (Niemeier, 2008) suc h that, in this case, the computa-
tion time shrinks to ab out six min utes. The adjustmen t problem is split in to
t w o partial problems of size n 1 = 6000 , n 2 = 24 000 where n 1 only dep ends
on the n um b er of images and n 2 only on the n um b er of con trol p oin ts. The
tec hnique is explained in detail in c hapter 4 (section 4.2.1). It basically reduces
the in v ersion time to O ( n 3
1 ) with the additional costs for a pro duct of ab out
O ( n 2
1 n 2 ) . It is o w ed to the sp ecial structure of N , esp. the sparse distribution
of non-zero elemen ts, that the use of the splitting metho d is of adv an tage. The
resulting p erformance impro v emen t is ev en for small data sets significan t, but
the c hallenge remains if the n um b er of images grows large. F or the V esta data
set, n 1 equals 32 640 and it tak es with the sp ecifications ab o ve 1.5 hours to
solv e the partial problem and more than 35 hours for the remaining pro duct.
Using a reference soft w are at DLR Berlin these are realistic n um b ers; the total
adjustmen t tak es four iterations and ab out 168 hours. F urthermore the de-
mand on memory dep ends on the n um b er of con trol p oin ts and exceeds 250
GB (Preusk er; priv ate comm unication).
While implemen ting the inertial bundle blo c k adjustmen t, to ols of an effectiv e
resource managemen t, e.g. sparse matrix compression and parallel computa-
tion, ha v e b een tak en in to account. All asp ects of implemen tation, hardw are
and soft w are considerations fo cusing the c hallenge of large data sets are sub ject
of c hapter 4, including the comparison of running times. It will b e sho wn that
18
b y a com bination of splitting tec hnique and matrix compression the pro duct,
whic h tak es ab out 95 % of the computation time, can b e eliminated suc h that
the requiremen ts for running time can b e reduced to O ( n 3
1 ) and for memory
to O ( n 2
1 ) . Hence, the n um b er of in v olv ed images b ecomes the most relev an t
factor with resp ect to resource limitations. A t a second lev el of optimisation
an algorithm has b een dev elop ed to b enefit from m ulti-core arc hitecture, ei-
ther CPU (compute pro cessing unit) or GPU (graphics pro cessing unit) using
parallel computation for the in v ersion task. This approac h com bines the in-
tro duced splitting tec hnique and the Cholesky decomp osition. Because of the
parallel design for the the single steps in the splitting tec hnique, the pro cedure
allo ws a large amoun t of parallel computation in con trary to other a v ailable
in v ersion tec hniques (section 4.2.4).
Since the p erformance of a program dep ends on differen t hardw are asp ects
and the en vironmen t of the op erational system, it w as tested on three differen t
arc hitectures, all equipp ed with Lin ux RedHat En terprises 6. In addition to
the w ork-station at TU Berlin whic h has b een used for soft w are dev elopmen t
(4 cores, 16 GB RAM), also computing facilities of DLR Berlin (32 cores, 256
GB RAM) and the North-German Sup ercomputing Alliance HLRN (24 cores,
64 GB RAM) w ere used. All core sp ecifications are related to In tel Xeon CPU
c hips, differing mainly in clo c k cycle and Gflops/s. The p erformance v alue of
eac h mac hine w as obtained based on the Whetstone b enshmark test (Curno w
and Wic hmann, 1976).
1.6 Missions and data
There are three differen t space missions con tributing to this w ork, t w o of them
are still on-going. NASA’s Viking mission to Mars and ESA’s Mars Express
mission pro vided the image data of Phob os. The image data of V esta is ob-
tained b y the Da wn mission of NASA. The next subsections pro vide brief in-
formation ab out these missions and describ es the selected data of the in v olv ed
framing cameras.
19
Viking
The Viking mission of NASA to w ards Mars w as represen ted b y tw o iden tical
spacecraft, Viking 1 and Viking 2, eac h consisting of an orbiter and a lander.
The primary mission ob jectiv es w ere to obtain high resolution images of the
Martian surface, c haracterise the structure and comp osition of the atmosphere
and surface, and search for evidence of life. During the whole op eration p e-
rio d, ab out 500 images of the Martian mo ons w ere tak en during flyb ys. Viking
Orbiter 1 (V O1) arriv ed at Mars on 19th June 1976 and concluded its mis-
sion after 1489 orbits of Mars on 7th August 1980. F rom 12th F ebruary to
11th Marc h 1977, V O1 approac hed Phob os and main tained an orbit that w as
sync hronised with the flyb y p erio d of the mo on. During th is close encoun ter
phase of ab out 30 martian rev olutions, Phob os could b e pictured from dis-
tances b et w een 90 to 650 km ab o v e the surface. Duxbury and Callahan (1988)
pro vide a detailed description of astrometric Viking observ ations that co v er the
complete op eration time, fo cusing on the p ossibilit y to impro v e the ephemeris
data of the satellites. Their study includes 166 images of Phob os and states
the uncertain ties of spacecraft p ositions and camera p oin ting angles, sho wn in
table 1.3. In this study only images of VO1 are considered and the ma jorit y of
them is tak en during the early encoun ter phase, at an a v erage distances of 460
km; only t w o of the images are captured in late 1977 at 1840 km and 3100 km
distance. The Viking Imaging System (VIS) consisted of t w o iden tical cam-
eras, VIS A and VIS B. There is one image of VIS B and 18 images of VIS A
included.
Viking orien tation uncertain ties
P osition 1 . 0 − 8 . 0 km errors in the early mission phase are smaller
(related to the p eriapsis passage)
P oin ting 0 . 03 − 0 . 3 km with star trac king metho d
up to 3 . 0 km when no star trac king w as a v ailable
T able 1.3: Uncertain ties ( 1 σ ) of p osition and p oin ting data of the Viking
Imaging System VIS, the true v alues are exp ected in 3 σ range (Duxbury and
Callahan, 1988).
20
Mars Express
Mars Express (MEX) is Europ e’s first mission to Mars. The spacecraft en tered
an orbit around Mars on 25th Decem b er 2003 carrying a lander and sev en
instrumen ts, including a High Resolution Stereo Camera (HRSC). The global
in v estigation of Mars and its t w o mo ons, Phob os and Deimos, is still on-going.
MEX p erformed sev eral mano euvres to capture Phob os during flyb ys. The
HRSC p oin ted to a predefined inertial p osition where Phob os w as exp ected to
en ter the field of view and the activ ated Sup er Resolution Channel (SR C) to ok
a series of images. A detailed rep ort ab out the imaging p erformance of the
SR C on Mars Express is giv en b y Ob erst et al. (2008). F or this study 53 frame
images of the SR C w ere selected, tak en at an av erage distance of 1887 km
with resp ect to the mo on’s centre of mass. P asew aldt et al. (2015) ev aluated
the image data with resp ect to the p osition and p oin ting uncertain ties; they
state that the p ositions ha v e an uncertain t y of ± 224 m (according to ESOC)
and the uncertain t y for p oin ting angles is giv en with 0.01 degree for most of
the images. F or some images, shifts of up to 160 pixel in sample direction
and 90 pixel in line direction are recorded (P asew aldt et al., 2015, fig. 2), and
the maxim um shift corresp onds to a p oin ting error of 0.084 degree. The SR C
image data and sto c hastic mo del used here is the same as in Willner et al.
(2010) and has b een attac hed in app endix A (table A.1).
Image data of Phob os
Viking (VIS) MEX (SR C)
Pixel Size 0.01176 mm 0.009 mm
F o cal Length 474.610 mm 984.76 mm
Distance ∅ 674 km (226 – 3078 km) ∅ 1877 km (579 – 5265 km)
Resolution ∅ 16.68 m (5.6 – 76.3 m) ∅ 17.12 m ( 5.3 – 48.4 m)
Images 19 53
T able 1.4: The nominal ground distance (km), assuming a radius of 11 km and
the resulting ground resolution (m) of Viking and MEX images. The a v erage
resolution of all 72 images together is 17 m.
21
Da wn
The launc h date of the Da wn mission w as 27th Septem b er 2007 and has b een
recen tly extended after the end of its prime mission on 30th June 2016. The top
lev el question of the mission is to understand, ho w size and w ater determined
the ev olution of the planets. Therefore t w o massiv e protoplanets, V esta and
Ceres, w ere c hosen as mission targets. Da wn is the first spacecraft whic h w en t
in orbit around an ob ject of the main asteroid b elt. It to ok ab out 69 000 images
and collected more than 132 GB of scien tific data, and it contin ues to orbit
its second mission target, the dw arf planet Ceres. The gian t asteroid V esta
w as explored from July 2011 to August 2012. There w ere t w o mission phases
in a High-Altitude Mapping Orbit (HAMO): 28th Septem b er 2011 to 2nd
No v em b er 2011 (HAMO-1) and 5th June 2012 to 25th July 2012 (HAMO-2).
The data set considered here, consists of 5440 images, com bined from HAMO-1
phase and HAMO-2 phase. Unlik e the Phob os data, here all images are tak en
b y the same spacecraft at distances whic h are close to the a v erage distance
of 944.5 km. Minim um and maxim um are within 35 km range of the mean,
implying a homogeneous ground resolution of ab out 88.1 meter (see table 1.5).
The distance is measured with resp ect to the cen tre of mass. F or all images
the p osition uncertain t y is giv en with 35 m and the p oin ting uncertain t y with
0.006 gon.
Image data of V esta
Pixel Size 0.0140 mm
F o cal Length 150.07 mm
Distance ∅ 944.5 km (915.2 – 979.1 km)
Resolution ∅ 88.1 m (85.40 – 91.3 m)
Images 5440
T able 1.5: The 5440 images, com bined from b oth of Da wn’s High-Altitude
Mapping Orbits, ha v e an a v erage nominal ground resolution of 88.1 metres.
22
Chapter 2
Definitions and F oundations
2.1 Mathematical foundations and notation
The metho ds whic h will b e describ ed and dev elop ed in this w ork, e.g. the bun-
dle blo c k adjustmen t (section 2.3), in v olv e theorems and subsequen t results of
Linear Algebra. In this section the notation for mathematical ob jects will b e
giv en first. Secondly , a collection of fundamen tal algebraic results, pro v en and
presen ted in detail b y Fisc her (2012), is stated here and tak en for gran ted in
all further considerations of matrix op erations and related algorithms.
• R denotes the real n um b ers and N the natural n um b ers including zero.
• M ∈ M n,m is an y matrix with n ro ws, m columns and co efficien ts in R .
• M n = M n,n is the subset of square matrices, i.e. rows equal co lumns; the
restriction to only one dimension parameter will imply that the matrix in ques-
tion is square.
• I n is the unit matrix in M n , i.e. a diagonal matrix with n ones along the
diagonal; the dimension parameter is omitted if the con text is clear.
• O ( n x ) is an asymptotic b ounding condition of a n umerical series that dep ends
on the v ariable n , x ∈ R arbitrary . The definition is:
S = O ( n x ) ⇐ ⇒ lim
n →∞
S
n x = 0 ⇐ ⇒ S
n x < k for some constan t k .
This notation is called big-O notation (reading S is of order n x ). If x = 1 , S
is of linear order, for x = 2 (rsp. x = 3 ) S is of quadratic (rsp. cubic) order.
23
24
• Definition 1 (blo ck diagonal matrix): A square matrix A with dimension
n is called a blo ck diagonal matrix with blo c k dimension s if the s × s blo c ks
along the diagonal
B k = { a ij ∈ A : ( k − 1) s < = i, j < k s } , k = 1 ,..., n
s
con tain the only p ossible non-zero en tries of A . That is
| a ij | > 0 = ⇒ a ij ∈
n/s
[
k =1
B k .
Lik e the blo c k diagonal matrix is an extension of the diagonal matrix, the
concept of a triangular matrix is extended in an analog w a y .
• Definition 2 (blo ck triangular matrix): A is called a blo ck triangular matrix
if there exist a blo c k diagonal matrix B and a triangular matrix T suc h that
A = B + T .
More precisely: Let B k ∈ M s,s b e defined as ab o v e and s a factor of n s.th.
n/s = n s ∈ N . A matrix A ∈ M n with the structure
A =
B 1
B 2 U
· · ·
L · · · B n s
is called upp er blo ck triangular matrix (with blo c k size s ) if a ij = 0 ∀ a ij ∈ L
and lower blo ck triangular matrix if a ij = 0 ∀ a ij ∈ U . It is
L =
n s
[
k =2 { a ij ∈ A : ( k − 1) s < = i < k s ; j < ( k − 1) s } (lo wer part)
U =
n s − 1
[
k =1 { a ij ∈ A : ( k − 1) s < = i < k s ; j > = k s } (upp er part)
The structure of blo c k triangular matrices will b e relev an t for the implemen-
tation of the algorithm, describ ed in 4.2.4.
• The determinan t of a matrix is a function det : A 7→ R that assigns a
real n um b er to the matrix A. F or all A ∈ M n the follo wing holds:
25
det ( A ) = 0 ⇐ ⇒ there is no in v erse of A. Consequen tly , if det ( A ) differs from
zero, the in v erse A − 1 exists and satisfies A A − 1 = I n .
• F or all A, B ∈ M n with | det ( A ) | > 0 and | det ( B ) | > 0 it is | det ( AB ) | > 0
and ( AB ) − 1 = B − 1 A − 1 .
• A 3 × 3 matrix M with real co efficien ts corresp onds unique to set of 3 func-
tions that are defined on R 3 . M then op erates on v ∈ R 3 b y m ultiplication
M v =
a 11 a 12 a 13
a 21 a 22 a 23
a 31 a 32 a 33
X
Y
Z
=
a 11 X + a 12 Y + a 13 Z
a 21 X + a 22 Y + a 23 Z
a 31 X + a 32 Y + a 33 Z
.
The v ector v is mapp ed to another v ector, where eac h comp onen t is a linear
com bination of the original v ector comp onen ts. If M has a determinan t equal
to +1 , the matrix acts as a rotation. The columns are orthonormal v ectors,
whic h are the co ordinates of the three base v ectors e 1 , e 2 , e 3 after the rotation.
• The transp osed of a matrix A ∈ M n,m , denoted with A T , is obtained b y
exc hanging ro ws with columns, hence A T ∈ M m,n . The pro ducts AA T ∈ M n
and A T A ∈ M m are alw a ys symmetric. F urther is ( AB ) T = B T A T .
• If all column v ectors of a matrix are linear indep enden t, it is said to ha v e ful l
r ank . An y square matrix with full rank has a determinan t differen t from zero,
hence its in v erse exists. If A ∈ M n,m with full rank, then N = A T A ∈ M m
has full rank, to o. Hence N − 1 exists.
Note: It follo ws, that the definition of N can b e extended to N = A T P A if P
is a diagonal matrix with strictly p ositiv e diagonal en tries. T o see this, P is
written as pro duct P = P sq r t P sq r t with P sq r t ( ii ) = p P ( ii ) and A is replaced
with P sq r t A .
• The trigonometric functions sin( x ) , cos( x ) and tan( x ) are tak en as unit
sensitiv e, i.e. units are transformed in to radian t automatically b y
x 7→ π x
t with t =
180 x in degree
200 x in gon
π x is radian t .
2.2 Con v ersion b et w een reference frames
As stated in the in tro duction, the con v ersion of one reference frame in to an-
other is accomplished b y rotation, i.e. the application of a rotation matrix M
26
to the frame’s base v ectors. If a v ector is given with respect to frame A and
shall b e transformed in to a v ector of frame B , A is called the lo c al fr ame and
B the tar get fr ame .
There are sev eral equiv alen t options to realise suc h a rotation, whic h are all
presen ted b y M . One p ossibilit y is to split M in differen t primary rotations,
that are rotations around a main axis of the co ordinate system. They are
defined at first, then t w o wa ys of rotation sequences, that matter for the im-
plemen tation of the bundle blo c k adjustmen t, are distinguished.
Primary Rotations
Considering a righ t hand cartesian co ordinate system, the three primary ro-
tations R 1 = R (1 , 0 , 0) around the X -axis, R 2 = R (0 , 1 , 0) around the Y -axis and
R 3 = R (0 , 0 , 1) around the Z -axis are defined suc h that for θ = 90 ◦ the follo wing
transformation of base v ectors is realised:
R 1 ( θ ) : (0 , 1 , 0) 7− → (0 , 0 , 1) , Y + 7→ Z + an ti-clo c kwise
R 2 ( θ ) : (1 , 0 , 0) 7− → (0 , 0 , 1) , X + 7→ Z + clo c kwise
R 3 ( θ ) : (1 , 0 , 0) 7− → (0 , 1 , 0) , X + 7→ Y + an ti-clo c kwise
Giv en this, the single matrices that rotate a v ector b y θ ◦ accordingly ha v e the
form
R 1 ( θ ) =
1 0 0
0 cos( θ ) sin( θ )
0 − sin( θ ) cos( θ )
, R 2 ( θ ) =
cos( θ ) 0 − sin( θ )
0 1 0
sin( θ ) 0 cos( θ )
R 3 ( θ ) =
cos( θ ) sin( θ ) 0
− sin( θ ) cos( θ ) 0
0 0 1
.
Note: Since sin( − x ) = − sin( x ) and cos( − x ) = cos( x ) the matrix R T
i ( θ ) =
R i ( − θ ) realises the in v erse rotation, that is the rotation in the rev ersed direc-
tion. Hence R T
i ( θ ) R i ( θ ) = I is for all θ and for all i ∈ { 1 , 2 , 3 } the unit matrix.
In general an y pro duct of t w o rotational matrices is a rotation matrix again
and, hence, an y sequence of pro ducts, to o.
27
2-1-3 Rotation
M = M 2 − 1 − 3 ( φ, ω , κ ) := R 3 ( κ ) R 1 ( ω ) R 2 ( φ ) will b e called shortly a 2-1-3 rota-
tion. The term sp ecifies the order of the rotation sequence:
◦ 1st rotation: φ ◦ around Y -axis, 2nd rotation: ω ◦ around X -axis, 3rd
rotation: κ ◦ around Z -axis
The 2-1-3 rotation is used e.g. to rotate a camera reference frame in to a b o dy-
fixed reference frame. The angles φ and ω reflect the relativ e orien tation of the
p ole axis to the Z -axis of the camera, so the first t w o rotations will align b oth
axes with eac h other. Then, κ is the offset b et w een the remaining axes. The
order of the rotation sequence is relev an t for the computation of the p ointing
angles’ deriv ativ es, M is defined b y the en tries
M 11 = cos( φ ) cos( κ ) + sin( ω ) sin( φ ) sin( κ );
M 12 = cos( ω ) sin( κ );
M 13 = − sin( φ ) cos( κ ) + sin( ω ) cos( φ ) sin( κ );
M 21 = − cos( φ ) sin( κ ) + sin( ω ) sin( φ ) cos( κ );
M 22 = cos( ω ) cos( κ );
M 23 = sin( φ ) sin( κ ) + cos( κ ) sin( ω ) cos( φ );
M 31 = sin( φ ) cos( ω );
M 32 = − sin( ω );
M 33 = cos( ω ) cos( φ ); (2.1)
and the angles can b e receiv ed b y the follo wing relationship
φ = tan M 31
M 33 , ω = sin( M 32 ) , κ = tan M 21
M 22 . (2.2)
The v ector ( M 31 , M 32 , M 33 ) is the p ole axis v ector e 3
e 3
e 3 in camera co ordinates.
3-1-3 Rotation
The second imp ortan t rotation sequence is the 3-1-3 rotation whic h is related
to the rotational mo del of a planetary b o dy B . The ICRF is the lo cal reference
frame here and the b o dy-fixed reference frame is the target frame. The rotation
matrix, defined b y the angles ( α, δ , W )
R B = R α,δ ,W := R 3 ( W ) R 1 ( π
2 − δ ) R 3 ( π
2 + α ) (2.3)
28
con v erts lo cal ICRF co ordinates in to b o dy-fixed co ordinates. This sequence
of rotation matc hes the IA U con v en tion to describ e the orien tation of the
b o dy-fixed reference frame with resp ect to the ICRF b y means of the b o dy’s
rotational elemen ts.
The in verse rotation R T con v erts b o dy-fixed co ordinates in to ICRF co ordi-
nates. F rom there they can b e further con v erted into co ordinates with resp ect
to another reference frame (e.g. camera or primary b o dy) b y application of
the corresp onding rotation matrix (see eq. 2.4).
Implicit Rotation
Let C and B t w o reference frames and R C , R B the rotation matrix that rotates
the ICRF in to C rsp. B . then the pro duct
R C B = R B R T
C (2.4)
defines the transformation from C to the target frame B . The lo cal frame C
is rotated bac k in to the ICRF via R T
C and from there rotated in to the target
frame B . Using (2.2) for R C B , the relativ e frame orien tation can b e receiv ed.
On the other hand, if R B and R C B are kno wn, the matrix that con v erts ICRF
to C-cen tered co ordinates is obtained b y
R C = R T
C B R B . (2.5)
2.3 Bo dy-fixed bundle blo c k adjustmen t
Here, the classical metho d of a photogrammetric bundle blo c k adjustmen t
whic h is form ulated in the b o dy-fixed reference frame will b e describ ed, as
used in planetary geo desy b y e.g. Willner et al. (2010) or Preusk er et al.
(2012). As stated b efore, b y applying the metho d, the orien tation parameter
of the camera can b e significan tly impro v ed. The resulting CPN is cleaned
from errors in image p oin t measuremen ts and the o verall in tersection error is
reduced. The functional mo del is based on the follo wing con text: A surface
p oin t that w as captured m ultiple times and iden tified in n ≥ 2 images, can
b e expressed as the in tersection p oin t of n lines. Kno wing the p osition of the
camera and the orien tation of the reference frames at the time the picture w as
tak en, the 3D co ordinates of the p oin t can b e calculated.
29
In table 2.1 are listed the parameters whic h are estimated b y the classical
(b o dy-fixed) bundle blo c k adjustmen t. The first t w o columns sho w the t yp e
of parameter and the corresp onding sym b ol used for it. The third columns
sho ws the n um b er of parameters for eac h group, there are three co ordinates
for eac h control p oin t and of eac h camera p osition as w ell as three p oin ting
angles. Columns four and fiv e hold the coun ting v ariables for eac h t yp e rsp.
for the total n um b er of parameters. F or eac h of these parameters a starting
v alue is required that will b e up dated during the pro cess. T able 2.2 lists the
observ ation parameters and is structured in the same w a y . P arameters listed
in b oth tables are additional observations , the image p oin t observ ations are
k ept constan t.
T able of Unkno wns
parameter sym b ol dim coun t total
Con trol P oin t ( X , Y , Z ) 3 nCP 3nCP
Camera P osition ( X 0 , Y 0 , Z 0 ) 3 nIm 3nIm
Camera P oin ting ( ω , φ, κ ) 3 nIm 3nIm
T able 2.1: P arameters whic h are estimated in the bundle blo c k adjustmen t,
their sym b ols and coun ting v ariables
T able of Observ ations
parameter sym b ol dim coun t total
Image P oin t ( ξ , η ) 2 nIP 2nIP
Camera P osition ( X 0 , Y 0 , Z 0 ) 3 nIm 3nIm
Camera P oin ting ( ω , φ, κ ) 3 nIm 3nIm
T able 2.2: P arameters whic h function as observ ations in the bundle blo c k
adjustmen t. Since the camera parameters also function as unkno wns (table
2.1) they are called additional observ ations.
30
2.3.1 The functional mo del
The functional mo del for a photogrammetric bundle blo c k adjustmen t is de-
riv ed from the mathematical equation of a line in the three dimensional space
R 3 . A line L is defined b y t w o p oin ts ( P 0 , P 1 ) or a p oin t and a direction v ector
P , the direction can b e obtained b y P = P 1 − P 0 .
L = P 0 + m P =
x 0
y 0
z 0
+ m
x
y
z
, m ∈ R (2.6)
Lik e P 0 = ( x 0 , y 0 , z 0 ) corresp onds with m = 0 and P 1 with m = 1 , an y p oin t
of this line b elongs to a sp ecific v alue of m .
Considering the example for a camera reference frame giv en in subsection 1.2.3
and a planetary b o dy that is pictured b y this camera, P 0 is the p oin t that
represen ts the camera lens - the origin of the camera’s co ordinate system.
Equation 2.6 describ es a line that connects a p oin t P on the CCD with a
surface p oin t of the pictured ob ject (for some unknown but y et existing m ).
T o express this p oin t in co ordinates that are cen tred within the target b o dy ,
the rotation matrix R C T B = R 2 − 1 − 3 ( φ, ω , κ ) , whic h dep ends on the p oin ting
information of the camera, is applied as discussed in 2.2 (p. 27).
The surface p oin t of the target b o dy T B is then giv en b y the functional relation
X
Y
Z
T B
= m R C T B
ξ
η
z f
C amer a
+
X 0
Y 0
Z 0
T B
(2.7)
for some real n umber m where ( ξ , η , z f ) is the image p oin t v ector in camera
co ordinates and ( X 0 , Y 0 , Z 0 ) T B is P 0 in T B -co ordinates. Solving for the image
p oin t v ector leads to
R T
C T B
X − X 0
Y − Y 0
Z − Z 0
T B
= m
ξ
η
z f
C amer a
. (2.8)
The nine parameters on the left side are referred to as unknowns , where as the
image p oin t v ector is called observation . Here, ( ξ , η ) are the co ordinates in the
image plane and z f is the (negativ e) fo cal length. Ob viously this parameter is
31
constan t in all images of the same camera. The parameter z f is used to further
transform the equation and find a form, that will no longer in v olv e m .
Equation 2.8 is a system of three single linear equations, therefore sometimes
referred to as co-linearit y equation. Denoting the left hand side of 2.8
X 0
Y 0
Z 0
= R T
C T B
X − X 0
Y − Y 0
Z − Z 0
T B
the three lines can b e written shortly as
X 0 = mξ , Y 0 = mη , Z 0 = mz f .
Because eac h line i ∈ { 1 , 2 , 3 } dep ends on the same set of parameters u =
( ω , φ, κ, X 0 , Y 0 , Z 0 , X , Y , Z ) resulting from the pro duct of ro w i of R T
C T B with
the difference v ector, one can set
X 0 = F 1 ( u ) = mξ
Y 0 = F 2 ( u ) = mη
Z 0 = F 3 ( u ) = mz f
.
Dividing eac h line b y m = Z 0 /z f the represen tation of the image p oin t obser-
v ation do es not dep end on m an y more and
ξ = z f
X 0
Z 0 = z f
F 1 ( u )
F 3 ( u ) =: F ξ ( u ) (2.9)
η = z f
Y 0
Z 0 = z f
F 2 ( u )
F 3 ( u ) =: F η ( u ) (2.10)
can b e written as functions of unkno wns.
Applying the T a ylor dev elopmen t to first order giv es
ξ = F ξ ( u 0 ) + F 0
ξ ( u 0 )( u − u 0 ) + w ξ (2.11)
η = F η ( u 0 ) + F 0
η ( u 0 )( u − u 0 ) + w η . (2.12)
with F 0
ξ ( u 0 ) = ( p 0
1 , . . . , p 0
9 ) the v ector of partial deriv ativ es, that are obtained
b y using the c hain and quotien t rule:
p 0
k = z f
F 0
1 ( u 0 ) k F 3 ( u 0 ) k − F 0
3 ( u 0 ) k F 1 ( u 0 ) k
( F 3 ( u 0 ) k ) 2 .
32
By replacing F 1 with F 2 the partial deriv ativ es F 0
η ( u 0 ) = ( p 0
10 , . . . , p 0
18 ) are
obtained. The equation ab o v e corresp onds to exactly one image p oint obser-
v ation. The v ariable u 0 will b e d ifferen t for an y other observ ation, since it
either corresp onds with a differen t con trol p oin t or the same p oin t in a differ-
en t image. Therefore, a subscribt i is added to all v ariables indicating that
they corresp ond to observ ation L i = ( ξ i , η i ) . In order to resp ect the stan-
dard notation in common literature ab out adjustmen t theory (Niemeier, 2008,
c hapter 4), let
A i = p 0
i 1 . . . p 0
i 9
p 0
i 10 . . . p 0
i 18 ! , v i = − w ξ i
w η i ! , ˆ x i = u i − u i 0 (2.13)
and
L 0
i = ˆ
L i = F ξ i ( u i 0 )
F η i ( u i 0 ) ! , l i = L i − ˆ
L i .
Equations 2.11 and 2.12 can then b e rewritten in matrix form
l i + v i = A i ˆ x i , (2.14)
whic h yields the linearised functional mo del for image p oin t observ ation L i .
Considering all image p oin t observ ations and com bining them in to one v ector,
a system of 2 nI P linear equations is built analogue to (2.14), yielding
l + v = A ˆ x .
V ector l holds the differences of all the actual observ ations and their appro x-
imations according to equations 2.9 and 2.10, v is built analogue to 2.13.
F urther ˆ x is the difference of all the unkno wn parameters and their starting
appro ximations – nU = 6 nI m + 3 nC P in total (compare table 2.1). A is the
enlarged Jacobi matrix with nU columns, where the en tries for unin v olv ed pa-
rameters are set to zero. F or the observ ed unkno wns (additional observ ations)
of eac h image j , the functional mo del is already linear.
( L j 1 , . . . , L j 6 ) = ( ω , φ, κ, X 0 , Y 0 , Z 0 ) j
denotes the a priori v alues of the observed image parameters, whic h are k ept
fixed during the adjustmen t. ˆ
L j k ( k = 1 ,..., 6 ) denotes the estimated or
33
impro v ed v alue, whic h c hanges during the course. Hence l j k = L j k − ˆ
L j k leads
to the functional mo del
l j k + v j k = L j k − ˆ
L j k = 1 · ˆ x j k
for the observ ed unkno wns. Therefore, in addition to the 2 nI P rows of A , for
eac h of the additional observ ations (see table 2.2) a line is added that has an
en try of +1 in the corresp onding column and zero elsewhere. Consequen tly
the v ectors l and v are extended b y the corresp onding elemen ts of l j k , v j k . By
design the columns of A are related to the unkno wn parameters in the order
giv en in table 2.3.
Column order of unkno wn parameters
Image 1 Image 2 . . . CP 1 CP 2 . . .
( ω , φ, κ, X 0 , Y 0 , Z 0 ) 1 ( ω , φ, κ, X 0 , Y 0 , Z 0 ) 2 ... ( X , Y , Z ) 1 . . . . . .
Image parameter columns 1 : 6 nI m CP col. 6 nI m + 1 : nU
T able 2.3: Corresp ondence of unkno wn parameters and the columns of the
design matrix A ; the total n um b er of parameters is nU = 6 nI m + 3 nC P
where nI m is the coun t of images and nC P the coun t of con trol p oin ts.
The system A ˆ x = l + v has infinitely man y solutions. If it is solv ed suc h
that v is minimal in length, the functional represen tation of the observ ations
is optimal. A Least Squares Metho d will lead to this result.
F or the momen t let N = A T A , then
N ˆ x = A T l + A T v (2.15)
is called the normal e quation . If ˆ x solv es N ˆ x = b ob viously A T v = 0 . The
Least Squares algorithm solv es ˆ x = N − 1 A T l and up dates u 0 = u 0 + ˆ x .
It terminates if ˆ x → 0 whic h is equiv alen t to u = u 0 or, more lik ely , v T v →
> 0 . This ensures v T v is minimal at the end and the v ector u 0 con tains the
impro v ed parameters from table 2.1.
This, so far, explains the functional mo del and the mathematical bac kground.
It is p ossible to further con trol the outcome of the adjustmen t, if a sto c hastic
34
mo del is in tro duced in addition. That means, the use of statistical information,
that are a v ailable for the set of observ ations (table 2.2). If no information are
a v ailable, the statistic quan tities ( w eig hts ) can b e calculated.
2.3.2 The sto c hastic mo del
In praxis, observ ations come with a measure of uncertain t y . F or image p oin ts
this v alue can b e deriv ed from the pixel size, p oin ting and p osition information
are assigned with an assumed deviation ( σ ) that migh t dep end on the metho d
of trac king.
The sto c hastic mo del is giv en b y the co v ariance matrix Σ ll . If σ i is b elongs to
observ ation i and n is the total n um b er of observ ations, a n × n matrix Σ l l
is built that satisfies Σ ll ( i, i ) = σ 2
i . F or large systems, co v ariances are often
neglected and a diagonal matrix is used instead. This praxis is follo w ed in this
w ork, to o. F rom there the w eigh t matrix P is constructed as
P = σ 2
0 Σ − 1
ll
for some reasonable n umerical constan t σ 0 > 0 . The v alue ensures n umerical
stabilit y and can b e set to the mean 1
n P σ i or simply to the v alue that corre-
sp onds to the uncertain t y of an image p oin t observ ations.
Redefining N as N = A T P A c hanges the normal equation to
N ˆ x = A T P l + A T P v =: b . (2.16)
Of course, it is preferable to main tain the statistical information during the
pro cess. If the impro v emen ts shall b e ev aluated after eac h iteration step, tec h-
nically the in v erse of the normal matrix N , denoted with N − 1 , is required.
In the literature also Q xx is used to denote the in v erse. The diagonal of this
matrix con tains at the end the impro v ed w eigh ts for th e unkno wn parameters.
T o ev aluate the image observ ations the w eigh ts are calculated from the HA T -
matrix
H = AN − 1 A T .
The dimension of H agrees with the total n u m b er of observ ations n = | l | .
Hence, if n is large H requires for the n 2 entries a large amoun t of memory .
T ypically only the diagonal en tries of H are used to compute the w eigh ts for
35
observ ations l and impro v emen ts v . Setting s 2
0 = v T P v / ( n − nU ) the new
w eigh t for observ ation i is obtained b y
s 0 p H ( i, i ) = σ i . (2.17)
Since the additional observ ations are con tained here, the new w eigh ts of the
camera orien tation parameters are computed implicitly in this step. These
v alues are not b e up dated during the iterations of the program. By
s 0 p P − 1 − H ( i, i ) = σ v i
the standard deviation of the impro v emen t v i is obtained. The qualit y v ariable
sigma a p osteriori s 0 estimates the system error. Since σ 2
0 is replaced with s 2
0
in the next iteration, the factor σ 0 /s 0 con v erges to +1 .
The v ector obtained b y ¯ v i = v i /σ v i holds the normed impro v emen ts. A sp ec-
ified threshold σ max is used to ev aluate the magnitude of the impro v emen ts.
Observ ations for whic h | ¯ v i | > σ max are eliminated from the data set. See
Niemeier (2008) for further information.
An effectiv e w a y to calculate the HA T-diagonal without building H is giv en in
c hapter 4 (see page 65).
36
Chapter 3
Inertial F rame Bundle Blo c k
A djustmen t
In this c hapter the classic bundle blo c k adjustmen t, describ ed in section 2.3,
will b e extended and form ulated within an inertial reference frame. This ref-
erence frame is the ICRF. Figure 3.1 sho ws a sc heme of the extended metho d.
Compared with the b o dy-fixed context, almost all asp ects here are the same.
The ma jor difference is, that the rotational mo del of the b o dy is tak en in to
accoun t in form of a second rotation matrix. As a consequence the orien tation
of the camera is no longer giv en with resp ect to the b o dy-cen tred reference
frame but with resp ect to the ICRF. This affects also the computation of the
deriv ativ es of the classical parameters whic h are giv en in section 3.3.
3.1 The extended functional mo del
The matrix R C T B in equation (2.7) on page 30 transforms the image co ordi-
nates directly in to b o dy-fixed co ordinates. The principle of the bundle blo c k
adjustmen t in the inertial frame is the split of the rotation matrix R C T B in to
t w o single rotations whic h in v olv e the inertial reference frame (ICRF). F rom
the analytical rotational mo del the angles α , δ and W are calculated for the
time asso ciated with the image and the 3-1-3 rotation matrix R T B is built ac-
cording to equation (2.3) on page 27. If a ligh t time correction was applied to
obtain the p osition v ector of the camera, the corrected time is used here, to o.
The orien tation of the camera with resp ect to the ICRF is giv en b y the matrix
37
38
Figure 3.1: Ov erview of the inertial frame bundle blo c k adjustmen t
R C suc h that R C con v erts ICRF co ordinates in camera co ordinates and R T
C
vice v ersa. The con v ersion b et w een camera reference frame and b o dy-fixed ref-
erence frame can b e written as R C T B = R T B R T
C . Hence, co-linearit y equation
(2.7) on page 30, is equiv alen t to
X
Y
Z
T B
= m R T B R T
C
ξ
η
z f
C amer a
+ R T B
X 0
Y 0
Z 0
I C R F
(3.1)
The p osition v ector of the camera is no w giv en with resp ect to the ICRF. The
solving for the image p oin t v ector leads to
m
ξ
η
z f
C amer a
= R C
R T
T B
X
Y
Z
T B
−
X 0
Y 0
Z 0
I C R F
(3.2)
where the righ t hand side can b e shorten again with ( X 0 , Y 0 , Z 0 ) and the re-
presentation
ξ = z f
X 0
Z 0 , η = z f
Y 0
Z 0
39
of the image p oin t observ ation remains the same as b efore. Since the equations
for eac h en try in R T B are kno wn, it is p ossible to build the partial deriv ativ es
for sp ecified rotational elemen ts and include them in the bundle blo c k adjust-
men t. As a consequence, the v ector of unkno wns is extended and lik ewise the
Jacobi matrix A whic h holds no w in addition the deriv ativ es with resp ect to
these new parameters. Let nR E the n um b er of rotational elemen ts whic h are
to determine. Since the new unkno wns are time indep enden t, they simply ap-
p ear once in the unkno wn v ector and result in only nRE additional columns
of A (see figure 3.2).
P attern of the normal matrix for
n images ( I m ),
nC P con trol p oin ts ( P t ),
nRE rotational elemen ts.
The blank areas are zero en tries.
The columns for the rotational el-
emen ts are dense, all other columns
con tain mostly zeros.
Figure 3.2: Distribution pattern of non-zero elemen ts of normal matrix N
T w o adv an tages follo w: First, this set up leads to a larger coherence of the
whole net w ork - via the rotational parameters eac h observ ation is no w con-
nected with an y other. Secondly , the computational effort will not significan tly
increase. Nevertheless the new approac h requires a completely new implemen-
tation of the adjustmen t algorithm.
3.2 Deriv ativ es of rotational elemen ts
The p ossible parameters, that are to include in the functional mo del, dep end
on the complexit y of the functions, that define the rotational elemen ts. A
general represen tation is giv en b y
α ( t ) = α 0 + a ( t ) , δ ( t ) = δ 0 + d ( t ) , W ( t ) = W 0 + w ( t )
40
where a, d, w are time-dep enden t functions and the other v ariables are the
initial co ordinates for the three angles with resp ect to the reference ep o c h. In
case that a, d and w are zero for t = 0 , they giv e exactly the orien tation of
the b o dy at J2000.0 . Since the observ ations are functions of α, δ and W and
furthermore the angles are argumen ts of trigonometric functions, it b ecomes
necessary to apply the c hain rule for differen tiating comp osite functions t 7→
f ( g ( t )) , whic h is
f ( g ( t )) 0 = g 0 ( t ) f 0 ( g ( t )) .
F or the b o dy V esta it is a = d = 0 and λ 0 = 0 . In this simple case, only the
inertial v alues are included as rotational elemen ts. Since
∂ α
∂ α 0
= ∂ δ
∂ δ 0
= ∂ W
∂ W 0
= 1 ,
the c hain rule giv es
∂ f ( g )
∂ g 0
= ∂ g
∂ g 0
∂ f ( g )
∂ g = 1 · ∂ f
∂ g = f 0 ( g ) , g ∈ { α, δ , W } . (3.3)
F or a b o dy with precession and a forced libration lik e Phob os the functions
a, d, w can b e simplified written as
a ( t ) = p α sin P t , d ( t ) = p δ cos P t , w ( t ) = r t + p W sin P t + λ 0 sin O t
where P t dep ends on the precession, O t on the orbit p osition and r is the mean
spin rate. In this case λ 0 can b e included as a unkno wn parameter. Because
of ∂ W /∂ λ 0 = sin O t the c hain rule leads to
∂ f ( W )
∂ λ 0
= ∂ W
∂ λ 0
∂ f ( W )
∂ W = sin O t f 0 ( W ) .
These examples are sufficien t to demonstrate the principle and to sho w that
the deriv ativ es are sp ecific for a c hosen target b o dy .
There is a pro cedure whic h is indep enden t of the c hoice of rotational elemen ts,
the required building of the deriv ativ es f 0 ( α ) , f 0 ( δ ) , f 0 ( W ) for the set of func-
tions f that define the en tries of the rotation matrix R T B . T o do this, equation
3.2 is rewritten as
m
ξ
η
z f
= R C
r 11 X + r 21 Y + r 31 Z
r 12 X + r 22 Y + r 32 Z
r 13 X + r 23 Y + r 33 Z
− R C
X 0
Y 0
Z 0
(3.4)
41
with r ij en tries in R T B = R 3 − 1 − 3 ( α + 90 , 90 − δ , W ) as defined in 2.3. The
en tries r ij are linear com binations of trigonometric functions:
r 11
r 21
r 31
=
− cos( W ) sin( α ) − sin( W ) sin( δ ) cos( α )
sin( W ) sin( α ) − cos( W ) sin( δ ) cos( α )
cos( δ ) cos( α )
r 12
r 22
r 32
=
cos( W ) cos( α ) − sin( W ) sin( δ ) sin( α )
− sin( W ) cos( α ) − cos( W ) sin( δ ) sin( α )
cos( δ ) sin( α )
r 13
r 23
r 33
=
sin( W ) cos( δ )
cos( W ) cos( δ )
sin( δ )
.
Since sin 0 ( x ) = cos( x ) and cos 0 ( x ) = − sin( x ) , it is ∀ a ∈ R
∂ ( a sin( x ))
∂ x = a cos( x ) , ∂ ( a cos( x ))
∂ x = − a sin( x ) . (3.5)
F or the sak e of space sa ving let c denote the cosine and s the sine function
in the follo wing three matrices, obtained b y building the deriv ativ es for eac h
en try of R T B :
R 0
α =
− c ( W ) c ( α ) + s ( W ) s ( δ ) s ( α ) − c ( W ) s ( α ) − s ( W ) s ( δ ) c ( α ) 0
s ( W ) c ( α ) + c ( W ) s ( δ ) s ( α ) s ( W ) s ( α ) − c ( W ) s ( δ ) c ( α ) 0
− c ( δ ) s ( α ) c ( δ ) c ( α ) 0
R 0
δ =
− s ( W ) c ( δ ) c ( α ) − s ( W ) c ( δ ) s ( α ) − s ( W ) s ( δ )
− c ( W ) c ( δ ) c ( α ) − c ( W ) c ( δ ) s ( α ) − c ( W ) s ( δ )
− s ( δ ) c ( α ) − s ( δ ) s ( α ) c ( δ )
R 0
W =
s ( W ) s ( α ) − c ( W ) s ( δ ) c ( α ) − s ( W ) c ( α ) − c ( W ) s ( δ ) s ( α ) c ( W ) c ( δ )
c ( W ) s ( α ) + s ( W ) s ( δ ) c ( α ) − c ( W ) c ( α ) + s ( W ) s ( δ ) s ( α ) − s ( W ) c ( δ )
0 0 0
.
42
R 0
α and R 0
W can b oth b e built from columns (rsp. ro ws) of matrix R T B it-
self:
∂ r ij /∂ α : r 0
i 1 = − r i 2 , r 0
i 2 = r i 1 , r 0
i 3 = 0; i = 1 , 2 , 3
∂ r ij /∂ W : r 0
1 i = r 2 i , r 0
2 i = − r 1 i , r 0
3 i = 0; i = 1 , 2 , 3
Let
u ( t ) := ( ω , φ, κ, X 0 , Y 0 , Z 0 , α, δ , W ) t
the subset of the unkno wn vector that corresp onds with the image time t , then
m
ξ
η
z f
= F ( u ( t ) , X , Y , Z )
holds for the observ ation of p oin t ( X , Y , Z ) at time t . T o get the partial
deriv ativ es with resp ect to g ∈ { α , δ , W } , the en tries r ij in equation (3.4) are
replaced with the corresp onding en tries of the deriv ativ e matrix; the righ t term
is ignored, since it is indep enden t of the rotational elemen ts. This leads to the
general form ula
∂ F ( u ( t ) , X , Y , Z )
∂ g = R C R 0
g
X
Y
Z
, g ∈ { α, δ, W } . (3.6)
F or a particular rotational elemen t p , equation (3.6) b ecomes
∂ F ( u ( t ) , X , Y , Z )
∂ p = ∂ g
∂ p R C R 0
g
X
Y
Z
, g ∈ { α, δ, W } .
3.3 Deriv ativ es of classical parameters
P ositions: The p osition v alues are obtained as in the b o dy-fixed case, but
here R T
C T B is replaced with R C . Hence, if col i is column i ∈ { 1 , 2 , 3 } of R C ,
then ∂ F
∂ X 0
= col 1 , ∂ F
∂ Y 0
= col 2 , ∂ F
∂ Z 0
= col 3
with F as b efore. It is clearly to see, that the determination of the p osition
v ector no longer dep ends on the rotational mo del.
43
Con trol P oin ts: In the b o dy-fixed case, the v alues for the con trol p oin ts
are simply the negativ e p osition deriv ativ es. Here, the deriv ativ es are deriv ed
from
R C R T
T B
X
Y
Z
= R C
r 11 X + r 21 Y + r 31 Z
r 12 X + r 22 Y + r 32 Z
r 13 X + r 23 Y + r 33 Z
as it w as the case for the rotational elemen ts. It follo ws for the X -co ordinate
∂ F
∂ X =
r c 11 r 11 + r c 12 r 12 + r c 13 r 13
r c 21 r 11 + r c 22 r 12 + r c 23 r 13
r c 31 r 11 + r c 32 r 12 + r c 33 r 13
,
where r c ij represen t the en tries of R C . Equally the deriv ativ es for Y are ob-
tained b y replacing r 1 i with r 2 i rsp. for Z b y replacing with r 3 i , i ∈ { 1 , 2 , 3 } .
P oin ting angles: It is con v enien t to build th e deriv ativ e matrix of R C with
resp ect to eac h angle, denoted with R 0
ω , R 0
φ and R 0
κ . Let
P = R T
T B
X
Y
Z
T B
−
X 0
Y 0
Z 0
I C R F
the difference of the con trol p oin t and the p osition in ICRF co ordinates. then
∂ F
∂ ω = R 0
ω P
is the deriv ativ e v ector with resp ect to ω and equally R 0
φ P rsp. R 0
κ P for
the remaining angles. The 2-1-3 rotation w as defined on page 27 and the
deriv ativ es can b e obtained with rule (3.5) on page 41 as in the classical case.
3.4 T est case - a sim ulation of Phob os
The principle w as first tested on a sim ulation, b ecause this pro cedure allo ws
maximal con trol and ev aluation of an algorithm. Another consequence is a
b etter understanding of the b eha viour with resp ect to the sto c hastic mo del. It
is imp ortan t to find out, ho w w ell the observ ations need to b e kno wn in order to
44
reconstruct the giv en top ograph y successfully . It is furthermore of adv an tage
to mo del preferably realistic errors. Here, a realistic mo del of the Martian
Mo on Phob os was created, based on an existing data set of 680 con trol p oints
whic h are distributed in 73 images with giv en frame parameters (p ositi on and
p oin ting of the camera).
Creating the syn thetic data
With the giv en orien tation data the initial co ordinates of eac h con trol p oin t
are obtained indep en tly b y a simple adjustmen t minimising the forw ard in ter-
section error. No w these v alues are k ept fix as the defined top ograph y of the
sim ulated ob ject. The next step is, to c hange the co ordinates of the image
measuremen ts in suc h a w a y , that they exactly matc h the defined p oin ts. This
can b e done b y applying equation 2.8 to eac h p oin t ( X , Y , Z ) . The v ector
from the origin of the camera’s reference frame to the p oin t on the surface is
obtained b y
X 0
Y 0
Z 0
C amer a
= m
ξ
η
z f
C amer a
.
Then ξ = z f X 0 / Z 0 , η = z f Y 0 / Z 0 are calculated to eliminate the scaling factor
m . If the unit of fo cal length z f is millimetres, the unit of the image co ordinates
will b e millimetres, to o. In this w a y image p oin t co ordinates are sim ulated
that p erfectly matc h the defined con trol p oin ts. F urthermore, the sim ulation
resp ects the ph ysical prop erties and rotational elemen ts giv en in the IA U rep ort
(Arc hinal et al., 2011) as they are stated on page 12.
F or later reference it is defined
p α = 1 . 79 , p δ = 1 . 08 , p W = 1 . 42 , λ 0 = 0 . 78 (3.7)
and emphasised that λ 0 is the forced libration amplitude whic h is fo cused here
while the other three parameter are related to the precession cone. F urther-
more are α 0 = 317 . 68 ◦ and δ 0 = 52 . 90 ◦ the time-indep enden t co ordinates of
the mean p ole axis orien tation.
F alsification of the rotational mo del
T o test functional mo del and soft w are, in a first step only parameters of
the rotational mo del ab ov e w ere c hanged and all other parameters k ept un-
c hanged. Rotational parameters lik e λ 0 , p α but also α 0 and δ 0 w ere set to
45
differen t starting v alues; the results of the adjustmen t w ere then compared to
the original input v alues: the difference of b oth is the reconstruction error,
sho wn in table 3.1. T o giv e an example, λ 0 ∈ {− 1 , 0 , 5 } (degree) means, that
three differen t adjustmen ts w ere p erformed and the maximal deviation of the
final v alue from 0.78 ◦ lies under 0.00012 degree. All other lines in the table
represen t a single adjustmen t for a giv en pair of parameters, = 10 − 4 denotes
the magnitude of the reconstruction error. The results are obtained with the
Reconstruction of rotational parameters
parameter start v alue true v alue error iterations
λ 0 {− 1 , 0 , 5 } 0 . 78 < 1 . 2 2 − 4
p α , λ 0 (0 , 0) (1 . 79 , 0 . 78) (5 . 4 , 1 ) 22
(1 , − 0 . 1) (3 . 0 , 1 ) 17
α 0 , δ 0 (316 . 8 , 51 . 9) (317 . 68 , 52 . 9) < 2 . 4 4
(315 , 55) < 1 . 0 5
(300 , 40) (0 . 5 , 3 ) 9
T able 3.1: Differen t adjustmen t cases for a sim ulation with falsified rotational
parameters: forced libration amplitude λ 0 , precession cone parameter p α and
mean p ole axis orien tation ( α 0 , δ 0 ). Columns t w o to four sho w the false start
v alues, the true sim ulated v alues and the reconstruction errors; = 10 − 4 ,
v alues are in degree. Column fiv e shows the n um b er of iterations that the
algorithm needed to p erform.
corresp onding sto chastic model, i.e. negligible errors for camera orien tation.
Other sto c hastic mo dels, assuming higher p oin ting and p osition errors induce
sligh tly higher reconstruction errors and up to four more iterations. The maxi-
m um error of 26 corresp onds to a mo del that falsely assumes p osition errors
of 30 m.
F alsification of the orien tation data
In a next step, errors w ere also added to the orien tation data of the camera.
The image p oin t observ ations themself w ere not c hanged.
A pseudo random n um b er 0 ≤ r ≤ mag and a random sign s = ± 1 w ere
46
obtained in C/C++ language b y setting
r = dr and 48() ∗ mag
s = dr and 48() < 0 . 5? − 1 : 1 .
So sr is either p ositiv e or negativ e within magnitude mag . The pseudo random
generator ab o v e creates uniformly distributed n um b ers. This w a y , a total of
24 differen t data sets with falsified initial v alues w ere created:
• 4 differen t data sets with p oin ting errors (gon) :
mag ∈ { 0 . 01 , 0 . 05 , 0 . 1 , 0 . 8 }
• 4 differen t data sets with p osition errors (m) :
mag ∈ { 1 , 15 , 100 , 300 }
• 16 differen t com binations of the p ointing and position error magnitudes
ab o v e, e.g. (0.01 gon, 1 m); (0.05 gon, 1 m); (0.1 gon, 1 m); (0.8 gon, 1
m) etc.
No w for this falsified data sets, the rotational elemen ts w ere c hanged as de-
scrib ed b efore. The adjustmen t for all eigh t cases (listed in table 3.1) w as
rep eated for all the 24 data sets with differen t sto c hastic mo dels. Because this
complexit y lead to far more than 300 test runs, the results are summarised
rather then ha ving all of them listed:
• In all constructed data sets, the forced libration amplitude could b e
reconstructed within t w o to fiv e iterations at an a v erage accuracy lev el
of 4 . 5 .
• P oin ting data could b e reconstructed up to an offset of 0 . 1 gon with
accuracy of 3 . 5 gon.
• P osition offsets could not b e successfully reconstructed, since the algo-
rithm is more sensitiv e to p oin ting corrections. The resulting CPN is
shifted in the magnitude of the mean p osition error.
Summarising it is to sa y , that the successful reconstruction of the rotational
elemen ts did not dep end on the p osition or p oin ting accuracy . It also did
not dep end on the successful reconstruction of the top ograph y . The latter
47
is unexp ected and could result from the uniform distribution of pseudo ran-
dom n um b ers, so that this indep endency migh t not hold in a real data case.
F urthermore, the p osition accuracy must be smaller than th e pixel resolution
error, in order to successfully reconstruct the CPN. F or the data sets whic h
satisfy this condition, the original CPN could b e reconstructed within 15 m.
The algorithm itself is functional and reasonab e results for data sets within
the determined error b ounds can b e exp ected.
F unctional dep endency of the precession amplitudes
There is a functional dep endency of the precession amplitudes p α = 1 . 79 ,
p δ = 1 . 08 , p W = 1 . 42 and the declination of the precession axis δ 0 = 52 . 9
whic h shall b e discussed briefly . The line that connects the ICRF-Origin with
the p oin t ( α 0 , δ 0 ) is the precession axis of the p ole, α 0 = 317 . 69 in the curren t
example. The plane whic h is orthogonal to it con tains the complemen tary
precession axes of the other t w o frame axes. The p ole axis describ es a circle of
radius p δ around the precession axis, suc h that the declination δ ranges within
[ δ 0 + p δ , δ 0 − p δ ] . Sim ultaneously , the X -axis describ es a circle of radius p δ
around the complemen tary precession axis whic h affects the v alue of W . F or
δ 0 = 0 the tra jectory of the p ole in ICRF co ordinates w ould b e circular, but
if | δ 0 | > 0 it has the shap e of an ellipse. Because of the declination ab o v e
the ICRF equator the p ole axis passes through a larger range of longitudes
and the c hange in righ t ascension is larger than p δ . The radius of the sphere
at latitude δ 0 is giv en b y cos( δ 0 ) . The induced tra jectory of + X around the
complemen tary precession axis has an elliptic shap e, to o. This axis has a
declination of δ 0 − 90 and the sphere radius at this lev el is cos( δ 0 − 90) = sin( δ 0 ) .
This leads to the follo wing relationship b et w een the precession constants:
p α = p δ
cos( δ 0 ) and p W = p α sin( δ 0 ) = p δ tan( δ 0 ) , (3.8)
where δ 0 is the time indep enden t declination of the precession axis o ver the
ICRF equator and p δ the radius of the precession cone. This dep endency needs
to b e considered e.g. when the partial deriv ativ es are built with resp ect to δ 0 .
48
3.5 Numerical stabilit y
It has b een noticed that the determinan t of the normal matrix equals zero
in man y cases. This is caused b y the large n umerical range of the en tries,
i.e. the magnitude for p oin ting parameters differs greatly from the magnitude
for p osition parameters. Common implemen tations of in v ersion algorithms
either refuse the matrix in this case or compute a pseudo-in v erse based on
rounded v alues. Using Matlab there will b e an error message, that the result
is a pseudo-in verse. T o a v oid an in v olv ed pseudo-in v erse, a scaling metho d
has b een dev elop ed that smo oths the n umerical range and ensures, that the
diagonal en tries are dominan t in magnitude. The application of the scaling
increases the n umerical stabilit y of the in v ersion pro cedure.
Let N the normal matrix and denote with N ii the diagonal en tries, 1 ≤ i ≤ n .
The matrix D , defined b y
D ij =
√ N ii
− 1 i = j ,
0 else
is the scaling matrix for N . D is applied b y m ultiplication from left and righ t
to N , suc h that
N = D N D (3.9)
is the scaled normal matrix. The m ultiplication from left scales the ro ws of
N and the m ultiplication from righ t the columns. Hence, equation (3.9) is
equiv alen t to the definition
N ij = N ij D ii D j j .
If N is in compressed sparse format with nnz non-zero elemen ts, the scaling
can b e quic kly realised in O ( nnz ) (in c hapter 4, page 53, the compression
is explained). When tested with Matlab for a defect N , there w as no error
message for the scaled matrix.
The in v erted matrix N − 1 needs to b e rescaled for the statistical ev aluation. It
is
N − 1 = D − 1 N − 1 D − 1
and a rep eated application of the transformation (3.9) giv es N − 1 . T o only
calculate the solution x = N − 1 b , the matrix do es not need to b e rescaled. It is
49
con v enien t to scale the v ector b , to o and solv e instead of N x = b the equation
N y = D b .
The solution is giv en b y
y = N − 1 D b
= D − 1 N − 1 b
= D − 1 x
and hence, it follo ws
x = D y .
50
Chapter 4
Application to Large Data Sets
The data set for the test case in the previous c hapter is so small, that consider-
ations ab out memory and running time requiremen ts w ere not necessary so far.
If it comes to large data sets, the computational effort increases significan tly .
Therefor the demands on memory and computation time will b e analysed in
this c hapter. Starting with the general structure of the adjustmen t problem,
three differen t levels of resource optimisation will b e discussed. The reference
cost lev el is giv en b y a soft w are and a computing mac hine of the DLR Berlin
for whic h the resource requiremen ts are a v ailable (Preusk er; priv ate comm uni-
cation). The data set of V esta with 5440 images and ≈ 83000 con trol p oin ts
is used to exemplify the sa ving in resources b y applying the metho ds of this
c hapter. The reference mac hine in all corresp onding figures is the cluster listed
in table 4.1.
The costs with resp ect to memory and computing time using the reference
facilities dep end on the n um b er of images as w ell as on the n um b er of con trol
p oin ts. The metho d, describ ed in section 4.2.2, reduces the main cost factor
to the n umber of images. The metho d rsp. algorithm, describ ed in section
4.2.4, can b e implemen ted b y taking parallel programming in to accoun t. The
parallel design is discussed in section 4.3 and further reduces the running time
b y using appropriate parallel computation arc hitecture. Since the p erformance
of the parallel algorithm is mac hine-dep ending, additional computing facilities
w ere used for p erformance tests. The op erating system of all mac hines is Lin ux
Redhat 6, the individual configuration is sho wn in table 4.1.
In c hapter 2, page 34, the adjustmen t problem is form ulated in terms of the
51
52
Configuration of the testing mac hines
HLRN Cluster W ork-station
Chip (In tel Xeon) E5-2670 0 X 7560 E5-1607 0
2.60 GHz 2.27 GHz 3.0 GHz
Gflops/s (p er core) 2.99 2.14 3.04
Cores 24 32 4
RAM 64 GB 128 GB 16 GB
T able 4.1: The v alues for the pro cessor p erformance in Gflops/s ha v e b een
obtained based on the Whetstone b enshmark test Curno w and Wic hmann
(1976).
normal equation N ˆ x = b . Recall, that N is built from the Jacobi matrix A
of the functional mo del and the w eigh t matrix P ,deriv ed from the sto c hastic
mo del, b y N = A T P A . The solution consists of t w o comp onen ts,
1) functional solution: computation of ˆ x
2) sto c hastic solution: computation of the diagonal elemen ts of N − 1 and
the diagonal of AN − 1 A T .
It shall b e emphasised again that the sto c hastic solution is necessary for obtain-
ing the sto c hastic mo del ap osteori and to ev aluate the computed impro vemen ts
of the functional mo del.
The follo wing general assumption on data sets is made: The data set to
whic h the bundle blo c k adjustmen t is applied, consisting of nI m images, nC P
con trol p oin ts, and nO bs image p oin t observ ations, resp ects the relations
nO bs
nC P nI m nC P . (4.1)
The assumption states that the a v erage observ ation rate of the con trol p oin ts is
small against the n um b er of images and the latter is small against the n um b er
of con trol p oin ts. T ypically (4.1) is satisfied, but there migh t b e exceptions
whic h are to exclude from further consideration.
In addition to the notation ab o v e, throughout the c hapter, N will denote the
normal matrix and dim its dimension.
53
4.1 Sparse matrix compression
By construction, the Jacobi matrix A con tains mainly zeros in eac h line and
N has only non-zero en tries for correlated parameters. A matrix that has a
ma jorit y of elemen ts whic h are zero is called a sp arse matrix , b ecause of its
sparse distribution of non-zero elemen ts ( nnz ). Sparse matrices are compressed
and used in compressed form since the b eginning of programming. The b est
example is a diagonal matrix D with dimension n . In this case the lo cation
of all nnz is precisely kno wn, hence the only thing that needs to stored is a
v ector of length n . This is a v ery efficien t compression of D . No w for an y
m × n matrix M it is true that the more elemen ts are zero the more memory
w ould b e w asted if the space for all mn matrix elemen ts w ould b e allo cated.
Hence a represen tation of a matrix that sa v es this memory space is of great
adv an tage. The sparsit y S of a matrix M ∈ M n,m is defined as
S ( M ) := siz e
n m , siz e = n um b er of nnz elemen ts . (4.2)
Mathematically S is the densit y of the nnz , it will b e zero for the zero-matrix
and one for a dense matrix. Hence, the smaller S ( M ) is, the less memory
space is required to store M and the more meaningful a compression b ecomes.
Since the structure of a matrix is often kno wn b eforehand, the densit y can
help a programmer to decide, whether or not a compression is useful.
Description of a typic al c ompr ession for a gener al matrix: T ec hnically the only
information required is where in the matrix the nnz are lo cated. A v ector I 1
stores the index of the ro w of eac h nnz elemen t and another v ector I 2 k eeps
the information where in this v ector a new column starts.
Example:
M =
1 0 4
2 3 0
0 0 5
nnz = { 1 , 2 , 3 , 4 , 5 }
I 1 = { 0 , 1 , 1 , 0 , 2 }
I 2 = { 0 , 2 , 3 , 5 }
n = 3
m = 3
siz e = 5
Ro ws and columns are indexed with 0, 1, 2, suc h that I 1 reflects the information
that "1" is in ro w 0, "2, 3" are in ro w 1, "4" is in ro w 0 and "5" is in ro w
2. This v ector has alw a ys the same n um b er of elemen ts as the nnz v ector.
The v ector I 2 has alw a ys one elemen ts more than the n um b er of columns, here
m + 1 = 4 . The n um b ers "1", "3" and "4" are the first nnz of column j=0,1,2
54
// A csm in C-co de
1 t yp edef struct csm = {
2 double *nnz;
3 in t * I 1 , * I 2 ;
4 in t n,m,size;
5 b o ol ccf;
6 };
T able 4.2: The data t yp e csm , a compressed sparse matrix, consists of a p oin ter
to the non-zero elemen ts (2), t w o p oin ters to index elemen ts (3), v ariables to
store dimension parameter and size (4), the format sp ecification (5); defined
in C-language.
and their index inside the nnz v ector is giv en b y I 2 [ j ] . Therefore if j 1 = I 2 [ j ]
and j 2 = I 2 [ j + 1]
nnz [ j 1 ] ≤ x < nnz [ j 2 ] , I 1 [ j 1 ] ≤ i < I 1 [ j 2 ]
yields all non-zero elemen ts x of column j with the corresp onding ro w index i .
T ogether with the n um b er of ro ws n and columns m and the v ariable siz e that
k eeps the length of the nnz -v ector a compressed sparse matrix (csm) system
that represen ts M is giv en. The en try at I 2 [ m ] = siz e ensures a sa v e access
of elemen ts on the implemen tation lev el. T able 4.2 sho ws an example of the
structure in C-co de. The length of I 1 is the same as that of nnz , but its
elemen ts are in teger v alues. It tak es 4 B yte to store an in teger v alue and 8
Byte to store a double v alue. Neglecting the minor requiremen ts of O ( n ) for
I 2 and the data t yp e itself, it tak es 3
2 siz e · 8 Byte to store the csm. Since 8 nm
Byte are required to store the whole matrix and
3
2
8 siz e
8 nm = 3
2 S ( M ) ,
memory is only sa v ed, if S ( M ) < 2
3 .
F ormats: In the giv en example the compression format is the c olumn c om-
pr esse d format (ccf ) represen ting a column ma jor order of M . If the column
n um b ers are stored in I 1 and the starts of the next ro w in I 2 , this results in
a ro w compressed format corresp onding with a ro w ma jor order of M . Both
55
// Example for a simple transp osition metho d
1 csmT ransp ose (csm* M) {
2 in t sw ap = M → m ;
3 M → m = M → n ;
4 M → n = sw ap ; // n,m are sw app ed
5 M → ccf = ! ccf ; // format is flipp ed;
6 };
T able 4.3: C/C++ co de example for a transp osition of a compressed sparse
matrix M ; lines 2 to 4 sw ap the v ariables for ro ws n and columns m , in line
5 the b o olean v ariable ccf is negated whic h c hanges tr ue to f al se and vice
v ersa.
formats are useful, but the most imp ortan t asp ect is that one format is the
transp osed v ersion of the other. T ransp ositions are used to a large exten t
within the adjustmen t program. Therefore the b o olean v ariable ccf w as added
to the csm system, suc h that the order of the represen tation can b e changed
quic kly . Setting the v ariable ccf of a column compressed csm from tr ue to
f al se c hanges M to M T (whic h is no w ro w compressed). T able 4.3 sho ws the
co de example. If S ≥ 2
3 , no memory space is sa v ed an y more, but there is
another imp ortan t asp ect of csm-represen tation, that still do es carry w eigh t –
the running time acceleration during matrix m ultiplication.
The pro duct AB = C of t w o square matrices A and B can b e realised b y:
1 in t i,j,k;
2 for ( i = 0 , i < n , i + + )
3 for ( j = 0 , j < n , j + + )
4 { double a ij = A [ i ][ j ] ;
5 for ( k = 0 , k < n , k + + )
6 C [ i ][ k ] + = a ij B [ j ][ k ] ;
7 } .
No w for an y a ij = 0 in line 4 the follo wing t w o lines will ha v e no mathemat-
ical effect, they only cons ume run time of O ( n ) . If A is a csm, a ij runs only
through the nnz-v ector p erforming siz e · n m ultiplications. Hence the gain
in sp eed is already ( n 2 − siz e ) n flops and the n um b er of op erations required
56
shrinks further if B is also compressed. Therefore the m ultiplication sp eeds
up considerably . In the extreme case, where A, B and hence C are diagonal,
the pro duct is done in just n flops instead of O ( n 3 ) .
F ast csm-pro ducts are p ossible b et w een the follo wing compress format com-
binations: [ ccf , ccf ] , [ cr f , cr f ] and [ ccf , cr f ] . If b oth matrices ha v e the same
format, the pro duct itself can b e built as a csm in this format, to o. Since
here, the columns (rsp. ro ws) are built one after another, b oth v ersions are
also suitable for parallel computing. F or the remaining [ cr f , ccf ] com bination
the pro duct can b e realised to o, but not with the same sp eed up factor. A
b enefit of this v ersion is, that the output compression format can b e c hosen,
whic h impro v es the flexibilit y in the algorithm’s design. Co de examples for
csm-pro duct routines are giv en in app endix B.
A square matrix with dimension dim requires 8 dim 2 Bytes to b e allo cated
in double p oin t precision. This means the memory need increases, dep ending
on the n um b er of parameters to adjust, in quadratic order O ( dim 2 ) . Because
the normal matrix N is symmetric, the allo cation can b e restricted to elemen ts
of the upp er triangle of the matrix without lo osing information. This reduces
the memory space to 8 dim ( dim + 1) / 2 Bytes, whic h is still of quadratic order
but sa ves almost half of the memory . This reduction will b e referred to as
triangular al lo c ation . Figure 3.2 on page 39 sho ws the distribution pattern of
the nnz in N . The matrix can b e split in to four parts
" N 11 N 12
N T
12 N 22 # with N 11 ∈ M dim 1 , N 22 ∈ M dim 2 and N 12 ∈ M dim 1 ,dim 2 ,
and all required parts can b e stored in sep erated memory blo c ks, as sho wn
on figure 4.1. The size of dim 2 is giv en b y the n um b er of con trol p oin ts, i.e.
dim 2 = 3 nC P , and dim 1 = dim − dim 2 is the n um b er of remaining parameters
whic h agree with the additional observ ations (compare section 2.3).
In the follo win g is sho wn ho w m uc h memory is required for the single parts,
dep ending on dim 1 , dim 2 and the observ ation rate of eac h con trol p oin t. Based
on the concrete v alues of the V esta data set, figure 4.2 sho ws the need for the
triangular allo cation in total (lev el 0) and the reduced need if N 22 is compressed
(lev el 1) resp ectiv e if N 22 and N 12 are compressed (lev el 2). The data set
consists of nIm = 5440 images and nCP = 82 829 con trol p oin ts with an a v erage
rate of 9.3 observ ations p er con trol p oin t, hence nObs = 770 310 observ ations.
57
Figure 4.1: The splitting of normal matrix N
Out of the 294 GB RAM for the triangular allo cation, ab out 290 GB can b e
sa v ed, since:
N 11 : only dep ends on camera parameters: dim 1 = 6 × nIm
memory: 3.97 GB
N 22 : only dep ends on con trol p oin ts: dim 2 = 3 × nCP
6 non-zero en tries p er con trol p oin t with triangular allo cation
memory: 5.69 MB
N 12 : the correlation b et w een the t w o groups of unkno wn parameters
6 × 3 × nObs non-zero en tries, memory : 105.8 MB
N T
12 : the transp osed N 12 matrix do es not need to b e sa v ed .
The example corresp onds to a bundle blo c k adjustmen t in the b o dy-fixed for-
m ulation, describ ed in section 2.3. In this case a csm-represen tation reduces
the memory cost from 290.4 GB to just 111.5 MB (Mega Byte = 1024 2 Byte).
In the inertial frame con text N 12 also con tains the correlations of the rotational
parameters and the con trol p oin ts. In this case, the n um b er of nnz increases
for eac h rotational parameter b y dim 2 for N 12 and b y dim 1 for N 11 . Since the
rotational parameters are v ery small in n um b er and eac h only yields a v ery
lo w con tribution to the resource demands (ab out 2 MB in the example ab o v e),
58
Figure 4.2: Memory need for the single parts of normal matrix N considering
a triangular represen tation – total (lev el 0), with compressed N 22 part (lev el
1), with compressed N 22 and N 12 parts (lev el 2). The need for the compressed
matrices ( < 0 . 2 GB ) is b elo w ey e resolution.
they will not b e considered an y further in this con text. It is b ecause of the
sp ecial structure of the adjustmen t problem, that N 22 con tains only nnz on a
3 × 3 blo c k diagonal. The assumption, that the relations (4.1) hold, ensures
that S ( N 12 ) is v ery small, to o. The sparsit y , defined in (4.2), for b oth matrices
is giv en b y
S ( N 22 ) = 2
3
1
nC P , S ( N 12 ) = 1
nC P
nO bs
nI m .
The t w o tables on page 59 sho w ho w the memory requiremen ts increase with
gro wing data sets. There are three real cases (Phob os, Mercury and V esta)
and one general example listed. In table 4.4 the memory need for a general
triangular allo cation is compared with the need for the N 11 part of the matrix;
table 4.5 sho ws the memory sa vings, if the triangular allo cation metho d and
sparse matrix compression are com bined, including the requiremen ts for the
non-zero elemen ts of N 12 and N 22 for differen t data sets.
Remark: Considering 30000 images one w ould exceed 120 GB of RAM, so
there will b e a natural hardw are limit at some p oin t. There is some extra
59
Memory requiremen ts to store normal matrix N
N
N
Images Complete N
N
N P art N 11
N 11
N 11 of N
N
N
Data Set dim memory dim 1 memory
Phob os 72 2496 23.7 MB 432 0.71 MB
Mercury 3446 48192 8.65 GB 20676 1.59 GB
V esta 5440 281127 294.42 GB 32640 3.97 GB
Example 10000 1532640 8750.6 GB 60000 13.4 GB
T able 4.4: Memory need to store the complete normal matrix N (triangular
allo cation) and the N 11 part of N (see figure 4.1), in four differen t cases. The
requiremen ts increase with the n um b er of images in quadratic order.
A dv an tage of using the splitting tec hnique and compression
Con trol P art N 12
N 12
N 12 and N 22
N 22
N 22 T otal Memory
Data Set P oin ts no csm csm needed sa v ed
Phob os 688 23 MB 1 MB 1.78 MB 21.92 MB
Mercury 9172 7 GB 13 MB 1.60 GB 7.05 GB
V esta 82829 290 GB 112 MB 4.08 GB 290.34 GB
Example 500000 8737 GB 641 MB 14.02 GB 8736.6 GB
T able 4.5: A ccum ulated memory requiremen ts for N 12 and N 22 : without com-
pression ( no csm ) and with sparse matrix compression ( csm ) and the total
memory to store N , if the splitting tec hnique is used together with compression
(incl. N 11 part, see table 4.4 ab o v e). In the case Example a p oin t observ ation
ratio of 9:1 is assumed, v alues are rounded.
60
space required for the data structure of the compressed matrix. Of course,
also N 11 could b e compressed, but in order to solv e the adjustmen t problem,
there is space required for matrix op erations. The metho d whic h is going to
b e describ ed no w, uses the memory region of N 11 to p erform the required
transformations. Hence, with resp ect to the memory resource, the n um b er of
in v olv ed images is the most significan t factor.
4.2 Optimisation of running time
Ha ving completed the considerations of the memory resource, the same ev al-
uation is done with resp ect to running time. As describ ed in c hapter 1, it is
measured in flops and the concrete times dep end on the sp ecifications of the
computing mac hine.
4.2.1 The splitting tec hnique
Using the sub division sc heme for N as b efore and assuming non-zero deter-
minan ts of N , N 11 and N 22 , the in v erse of N is according to Bronstein and
Semendja jew (2005) giv en b y:
N − 1 = " N 0
11 − N 0
11 N 12 N − 1
22
− N − 1
22 N T
12 N 0
11 N 0
22 # (4.3)
with
N 0
11 = ( N 11 − N 12 N − 1
22 N T
12 ) − 1 ∈ M dim 1 ,
N 0
22 = N − 1
22 N T
12 N 0
11 N 12 N − 1
22 + N − 1
22 ∈ M dim 2 .
In Niemeier (2008) the principle is applied within a b o dy-fixed bundle ad-
justmen t; since b oth sources use it without pro of, a pro of is attac hed in the
app endix (see A.1). The computation of N − 1 according to this pattern, re-
quires t w o in v ersions and four pro ducts 1 . The in v ersions and the first t w o
pro ducts form a task sequence that will b e called forwar d op er ation . The sin-
gle tasks are listed in table 4.6 with their magnitude in flops, the final column
sho ws the reduction in costs if sparse compression is used. The sequence of
the remaining t w o pro ducts will b e called b ackwar d op er ation .
The direct in v ersion of N requires O ( dim 3 ) flops and for dim = dim 1 + dim 2
1 A dditions are neglected here.
61
without with Gain
T ask Compression Compression F actor
N − 1
22 in v. O ( dim 3
2 ) O ( dim 2 ) 1 /dim 2
2
N 12 N − 1
22 pro d. O ( dim 1 dim 2
2 ) O ( nO bs ) S ( N 12 ) S ( N 22 )
N 12 N − 1
22 N T
12 pro d. O ( dim 2
1 dim 2 ) O ( dim 2
1 ) S ( N 22 )
N 0
11 in v. O ( dim 3
1 ) O ( dim 3
1 ) 1
T able 4.6: The task sequence of the forw ard op eration to compute N − 1 in split
mo de (see eq. 4.3). Columns tw o and three sho w the magnitude of required
flops without and with sparse matrix compression, column four the factor of
flop cost reduction using compression.
the iden tit y
dim 3 = ( dim 1 + dim 2 ) 3 = dim 3
1 + dim 3
2 + 3 dim 2
1 dim 2 + 3 dim 2
2 dim 1
holds, suc h that the cost gain is not ac hiev ed b y the splitting tec hnique itself,
but b y the sparsit y of N 12 and N 22 .
Since N 22 is a blo c k matrix, its in v erse is obtained fast in O ( nC P ) and can
b e ev en done in parallel. Using N 22 in compressed form and optimising the
forw ard pro ducts reduces the costs for the whole forw ard op eration to O ( dim 3
1 )
flops. It remains the exp ensiv e bac kw ard op eration of O ( dim 2
1 dim 2 ) . The ap-
plication of the splitting tec hnique with a partial compression is implemen ted
within the reference soft w are and reduces a running time of 40 da ys to just 1.5
da ys, but this optimisation can b e impro v ed further.
4.2.2 The in v erse decomp osition metho d
T o a symmetric matrix M ∈ M n a pair ( L, D ) of matrices, where D is diagonal
and L lo wer triangular, that satisfies M = LD L T is called the Cholesky LDL T
de c omp osition of M . It is named after the founder of the widely implemen ted
and used algorithm that transforms M in to L (Dahmen and Reusk en, 2008,
c hapter 3 p. 87). Since L is triangular, it can b e in v erted quic kly with only
n ( n + 1) / 2 flops.
Claim: F or a giv en c holesky decomp osition ( L, D ) of M , it is
M − 1 = ( L T ) − 1 D − 1 L − 1 = ( L − 1 ) T D − 1 L − 1 . (4.4)
62
Pro of : It is b y definition M − 1 M = I , whic h is equiv alen t to
M − 1 LD L T = I
⇐ ⇒ M − 1 LD = ( L T ) − 1
⇐ ⇒ M − 1 L = ( L T ) − 1 D − 1
⇐ ⇒ M − 1 = ( L T ) − 1 D − 1 L − 1 .
This is the first equalit y , to see the second w e use I = I T and replace
I = ( L L − 1 ) T = ( L − 1 ) T L T | × ( L T ) − 1
⇐ ⇒ ( L T ) − 1 = ( L − 1 ) T
This principle will no w b e com bined with the split tec hnique. Let
¯
N 11 := N 11 − N 12 N − 1
22 N T
12 ,
the matrices whic h need to b e in v erted can b e written as
L 1 D 1 L T
1 = ¯
N 11 (4.5)
L 2 D 2 L T
2 = N 22 (4.6)
using their Cholesky decomp ositions. Consequen tly
¯
N − 1
11 = ( L − 1
1 ) T D − 1
1 L − 1
1 (4.7)
N − 1
22 = ( L − 1
2 ) T D − 1
2 L − 1
2 . (4.8)
Replacing N 0
11 = ¯
N − 1
11 in equation (4.3) leads to
N − 1 = " ( L − 1
1 ) T D − 1
1 L − 1
1 − ( L − 1
1 ) T D − 1
1 L − 1
1 N 12 N − 1
22
− N − 1
22 N T
12 ( L − 1
1 ) T D − 1
1 L − 1
1 N − 1
22 + N − 1
22 N T
12 ( L − 1
1 ) T D − 1
1 L − 1
1 N 12 N − 1
22 #
whic h can b e split in to the pro duct
N − 1 = " ( L − 1
1 ) T D − 1
1 0
− N − 1
22 N T
12 ( L − 1
1 ) T D − 1
1 ( L − 1
2 ) T D − 1
2 # " L − 1
1 − L − 1
1 N 12 N − 1
22
0 L − 1
2 # .
By defining
S := " L − 1
1 − L − 1
1 N 12 N − 1
22
0 L − 1
2 # , D := " D − 1
1 0
0 D − 1
2 #
63
a decomp osition of N − 1 is found, that satisfies
N − 1 = S T D S . (4.9)
This is of great adv an tage, since the decomp osition is sufficien t to solv e the
adjustmen t problem. The pro of of sufficiency will b e giv en in the next sub-
section. It can b e seen in the definition of S , that N 0
11 do es not need to b e
computed fully and the large matrix N 0
22 do es not need to b e computed at
all. The pro duct of the triangular matrix L − 1 with N 12 N − 1
22 can b e realised
in O (18 nO bs · dim 1 ) flops, if b oth matrices are giv en as csm, but additional
memory is required here. It will b e sho wn, that these additional costs can b e
sa v ed, to o, b y writing S as
S = " L − 1
1 0
0 I dim 2 # " I dim 1 − N 12 N − 1
22
0 L − 1
2 # . (4.10)
In fact, there are no bac kw ard pro ducts required with this metho d. The tasks
whic h are to p erform are
• decomp osing N 22 and getting L − 1
2 , D − 1
2
• computing − N 12 N − 1
22 and ¯
N 11
• decomp osing ¯
N 11 and getting L − 1
1 , D − 1
1 .
Figure 4.3 sho ws ho w effectiv e this metho d is; for the large V esta data set the
metho d here (lev el 2) is compared with the previous lev el of optimisation (lev el
1). Ab out 36 hours, 96 % of running time in this example, are sa v ed.
4.2.3 Sufficiency of the in v erse decomp osition
Giv en the matrix sc heme of the in v erse normal matrix N − 1 = N 0
11 , N 0
12 ; ( N 0
12 ) T , N 0
22
and its decomp osition ( S, D ) as b efore, it is to show that the functional and the
sto c hastic solution (see page 52) of the adjustmen t problem can b e obtained.
T o solv e the normal equation for ˆ x , the v ectors ˆ x and b are split in to t w o parts
of [ dim 1 , dim 2 ] accordingly to the split of N . then ˆ x = N − 1 b b ecomes
ˆ x 1
ˆ x 2 ! = N 0
11 N 0
12
( N 0
12 ) T N 0
22 ! b 1
b 2 !
64
Figure 4.3: Optimisation of in v erting N with split tec hnique – lev el 1: reference
soft w are with partial compression, lev el 2: metho d describ ed in subsection
4.2.2, lev el 3: metho d describ ed in subsection 4.2.4 using parallel computation.
The in v ersion of N 22 tak es only a few seconds.
65
and hence b y definition of N − 1
ˆ x 1 = N 0
11 b 1 − N 0
11 N 12 N − 1
22 b 2
ˆ x 2 = − ( N 12 N − 1
22 ) T N 0
11 b 1 + ( N 12 N − 1
22 ) T N 0
11 N 12 N − 1
22 b 2 + N − 1
22 b 2 .
The v ariables in ˆ x 2 (prin ted in b old) equal − ˆ x 1 , s uc h that the equations ab o v e
can b e simplified as follo ws
ˆ x 1 = N 0
11 ( b 1 − N 12 N − 1
22 b 2 ) (4.11)
ˆ x 2 = N − 1
22 b 2 − ( N 12 N − 1
22 ) T ˆ x 1 (4.12)
F rom the equations ab o v e it can b e seen, that the computation of N 0
12 is not
necessary , further on equation (4.11) can b e realised in t w o steps
y 1 = D − 1
1 L − 1
1 ( b 1 − N 12 N − 1
22 b 2 )
ˆ x 1 = ( L − 1
1 ) T y 1 .
Hence, the decomp osition of N 0
11 is sufficien t. The same holds for N − 1
22 , since
the parts of its decomp osition can b e applied to b 2 sequen tially in the same
w a y . The pro duct with v ectors is obtained quic kly , esp. if the matrix on the
left hand side is sparse and compressed. Also recall, that in this case the in-
v olv ed transp ositions only require zero-run time 2 (compare section 4.1 on page
55).
It remains to sho w that the diagonal of the HA T-matrix H := AN − 1 A T can
b e computed. By asso ciativ e la w it is
H = A ( S T D S ) A T = ( AS T ) D ( S A T ) .
Since ( S A T ) T = AS T , a similar decomp osition is found for H . Setting L H :=
S A T yields to H = L T
H D L H . No w let h i b e column i of L H , then
H ii = h T
i D h i = X
j
D j j · h i ( j ) 2
is the i -th diagonal en try of H .
Hence, the calculation of L H is sufficien t. Denoting with n = | l | the length of
2 i.e. less than 1 second
66
v ector l , the matrix L H has dim × n elemen ts. If L H is computed columnwise,
only the space for a v ector of length dim needs to b e allo cated, whic h can b e
reduced ev en further.
As men tioned earlier, the computation of L − 1
1 ( − N 12 N − 1
22 ) can b e a v oided. Us-
ing equation (4.10) leads to
L H = L − 1
1 0
0 I dim 2 ! " I dim 1 − N 12 N − 1
22
0 L − 1
2 ! A T #
and the righ t hand pro duct with A T in v olv es only v ery sparse matrices. Once
again, eac h column a ( i ) of A T is split in to [ a 1 ( i ) , a 2 ( i )] of dimension dim 1 and
dim 2 ; the first part corresp onds with additional observ ation parameters only
and the second con tains the remaining con trol p oin t deriv ativ es. The equation
ab o v e b ecomes
h i = L − 1
1 0
0 I dim 2 ! a 1 ( i ) − N 12 N − 1
22 a 2 ( i )
L − 1
2 a 2 ( i ) !
= L − 1
1 ( a 1 ( i ) − N 12 N − 1
22 a 2 ( i ))
L − 1
2 a 2 ( i ) !
and the lo w er part of the righ t hand v ector L 22 a 2 ( i ) consists of only 3 non-zero
elemen ts for all i . Since the upp er part is b ound b y dim 1 , h i can b e stored in
compressed form of maximal dim 1 + 3 elemen ts. Hence, again memory as w ell
as running time is sa v ed b y taking in to accoun t the sparse nature of A .
4.2.4 An iterativ e in v erse decomp osition algorithm
The decomp osition of the partial matrix ¯
N 11 consumes still running time of
cubic order, although the dimension of the problem has b een reduced to dim 1 .
F or large data sets this is a relev an t cost factor, esp. since a bundle blo c k ad-
justmen t consists of more than just one iteration. T able 4.7 shows the running
times on all testing mac hines with resp ect to the in v ersion of ¯
N 11 for the pre-
viously considered data sets of Phob os, Mercury and V esta (table 4.4 on page
59); the v alues are based on n 3 / 3 flops with dimension parameter n = 6 nI m .
It can b e seen, that for 10 000 images just four iterations tak e 1–2 da ys of com-
putation time. Therefore an algorithm w as dev elop ed whic h mak es use of the
m ulti-core arc hitecture of mo dern computing mac hines and high p erformance
67
Running time for in v ersion of partial matrix ¯
N 11
¯
N 11
¯
N 11
Data Set Images HLRN cluster w ork-station
Phob os 72 00:00:00 00:00:00 00:00:00
Mercury 3446 00:16:25 00:22:57 00:16:09
V esta 5440 01:04:37 01:30:17 01:03:33
Example 10000 06:41:20 09:20:45 06:34:44
T able 4.7: Running times (hh:mm:ss) for differen t data sets based on n 3 / 3
flops for partial matrix ¯
N 11 , n is sixfold n umber of images. See table 4.1 on
page 52 for description of the three mac hines.
computers of the North-German Sup ercomputing Alliance. Dep ending on the
hardw are that is used a sa ving factor of v 1:7 (cluster) rsp. 1:8 (HLRN) could
b e ac hiev ed for this part of the adjustmen t program. The testing mac hines are
describ ed on page 52.
The algorithm transforms a symmetric matrix M iterativ ely i n to a matrix S
and computes sim ultaneously a diagonal matrix D , suc h that S T D S = M − 1 ;
S is a blo c k triangular matrix and the dimension of eac h blo c k can b e b ound
b y an arbitrary giv en blo c k size.
Compared with the definition of a blo c k triangular matrix on page 2.1, the
condition that all blo c ks need to ha v e the same dimension is relaxed here in
suc h a w a y , that the blo c k structure is b ound b y s . That means, the dimen-
sion of all blo c ks is less or equal to s . The generalisation is o wing to the fact,
that the dimension of an arbitrary matrix is not necessarily a m ultiple of an
arbitrary giv en blo c k size s . But if for a matrix M ∈ M dim the dimensions
of the diagonal blo c ks are b ound b y s , this matrix can b e naturally em b edded
in to a larger matrix M 0 , that has the follo wing prop erties:
dim = n s + r , r < s = ⇒ M 0 ∈ M ( n +1) s with M 0 = M 0
0 I s − r ! .
Since I s − r is a unit matrix, it follo ws
( M 0 ) − 1 = M − 1 0
0 I s − r ! .
68
Both, M 0 and ( M 0 ) − 1 , are blo c k triangular with blo c k size s . Suc h an em-
b edding can alw a ys b e done and hence, the blo c k size can b e c hosen freely .
Although the algorithm will b e applied esp ecially to ¯
N 11 , the pro of here is for
a general symmetric M for whic h M − 1 exists. In case the in v erse do es not
exist, the algorithm stops with an error message.
F ollo wing the steps of a complete induction, the algorithm is describ ed for
an arbitrary blo c k size s ≤ dim with n = dim/s n um b er of diagonal blo c ks.
W.l.o.g. dim/s ∈ N holds.
Algorithm for n = 1
n = 1
n = 1 and s = dim
s = dim
s = dim :
This is simply equation 4.4 with S = L − 1 and D = D − 1 .
Algorithm for n = 2
n = 2
n = 2 and s = dim 1 = dim 2 = dim/ 2
s = dim 1 = dim 2 = dim/ 2
s = dim 1 = dim 2 = dim/ 2 :
This is the split tec hnique and requires only one iteration. The algorithm
p erforms the follo wing steps as a forw ard op eration :
0: extract M 22 blo c k and compute its LDL T form: M 22 = L 2 D 2 L T
2
in v ert the UTM L T
2 and replace M 22 with the transp osed L − 1
2
1: compute B 2 = − M 12 M − 1
22 and replace M 12 with B 2 ( 3 )
2: compute ¯
M 11 = M 11 − M 12 M − 1
22 M T
12 (up dating M 11 )
3: rep eat step 0 on ¯
M 11 : ( L − 1
1 ) T D − 1
1 L − 1
1 = ¯
M − 1
11 = M 0
11
⇒ replace ¯
M 11 with L − 1
1
After this steps the bac kw ard op eration is to p erform
• replace B 2 with B 0
2 = L − 1
1 B 2
The obtained matrices lo ok lik e this:
S = L − 1
1 B 0
2
0 L − 1
2 ! , D = D − 1
1 0
0 D − 1
2 ! .
3 Assuming that a cop y of M 12 exists; this assumption will b e drop ed later on.
69
Keeping the definition of B 2 , B 0
2 in mind, it follo ws that
S T D S = ( L T
1 ) − 1 D − 1
1 L − 1
1 ( L T
1 ) − 1 D − 1
1 B 0
2
( B 0
2 ) T D − 1
1 L − 1
1 ( B 0
2 ) T D − 1
1 B 0
2 + ( L − 1
2 ) T D − 1
2 L − 1
2 !
= M 0
11 M 0
11 B 2
B T
2 M 0
11 B T
2 M 0
11 B 2 + M − 1
22 !
= M − 1 (4.13)
Algorithm for arbitrary n
n
n and s = dim/n
s = dim/n
s = dim/n :
Setting the original matrix M = M n , a brief description is giv en b y:
Start with calculation of L − 1
n , D − 1
n . Complete the forw ard op eration and set
¯
M 11 = M n − 1 . Rep eat the forw ard pro cedure un til M 1 , the upp er left 16 × 16
blo c k, is reac hed and compute the last elemen ts L − 1
1 , D − 1
1 .
Detail le d description : F or general n ∈ N the forw ard steps are
0: ( n − 1) times - extract diagonal blo c k and compute L − 1
k , D − 1
k , n ≥ k ≥ 2
1: ( n − 1) times - compute B k = − ( M 12 M − 1
22 ) ( k ) , n ≥ k ≥ 2
2: ( n − 1) times - building ¯
M 11 =: M k − 1 , n ≥ k ≥ 2
3: rep eat step 0 on M 1
After the forw ard op eration terminates, M is
L − 1
1 B 2 . . . ∗ ∗
0 L − 1
2 . . . B n − 1 ∗
. . . ∗ B n
0 L − 1
n − 1 ∗
0 L − 1
n
(4.14)
and d := ( D − 1
1 , D − 1
2 , . . . , D − 1
n ) is the connected diagonal.
F or the n − 1 steps of the bac kw ard op eration, write D ( k ) for the diagonal
matrix with diagonal d k = ( D − 1
1 , . . . , D − 1
k ) and set again S 1 = L − 1
1 . Starting
with the square in the upp er left corner of the matrix
70
• replace B 2 with B 0
2 = S − 1
1 B 2 and set S 2 = S − 1
1 B 0
2
0 L − 1
2 !
F or the case n = 2 it w as sho wed that S T
2 D (2) S 2 = M − 1
2 . Analog
• replace B k with B 0
k = S k − 1 B k and set S k = S k − 1 B 0
k
0 L − 1
k !
for the remaining k ≤ n . This w a y , the matrix (4.14) c hanges step wise to
S 2 B 0
3 . . . ∗
0 L − 1
3 . . . B n
. . . ∗
0 L − 1
n
,...,
S n − 2 B 0
n − 1 ∗
0 L n − 1 B n
0 0 L − 1
n
, S n − 1 B 0
n
0 L − 1
n !
(4.15)
and S T
k D ( k ) S k = M − 1
k holds for eac h k . The final step n − 1 7→ n is ev aluated:
S := S n = S n − 1 B 0
n
0 L − 1
n ! , D := D ( n )
and it follo ws
S T D S = S T
n − 1 D ( n − 1) S n − 1 S T
n − 1 D ( n − 1) B 0
n
( B 0
n ) T D ( n − 1) S n − 1 ( B 0
n ) T D ( n − 1) B 0
2 + ( L − 1
n ) T D − 1
n L − 1
n !
= M − 1
n − 1 M − 1
n − 1 B n
B T
n M − 1
n − 1 B T
n M − 1
n − 1 B n + ( L − 1
n ) T D − 1
n L − 1
n !
= M 0
11 − M 0
11 M 12 M − 1
22
− M − 1
22 M T
12 M 0
11 M − 1
22 M T
12 M 11 M 12 M − 1
22 + M − 1
22 !
= M − 1 . (4.16)
The equalit y of line 2 and line 3 is obtained after renaming the v ariables ac-
cording to the case of the single split scenario n = 2 whic h holds for arbitrary
dimensions dim 2 > 0 , dim 1 = dim − dim 2 .
Because of the iterativ e structure, the bac kw ard op eration can not b e omitted
here, but matrix pro ducts are suitable for parallel computation. The decomp o-
sition and in v ersion of the diagonal blo c ks can b e ac hiev ed quic kly if the blo c k
71
size is small enough. A size of s = 16 w as c hosen whic h will b e referred to as
the tile size . Because of the sp ecification regarding the memory transfer on a
computing unit, this is curren tly the optimal tile size in parallel programming
(Hsu and Kremer, 2004).
Applying the decomp osition algorithm to ¯
N 11 , the running time reduces fur-
ther if the n um b er of parallel pro cesses is sufficien tly large. On figure 4.3 the
time for lev el 3 corresp onds with this optimisation, the 1.5 hours are reduced
to 13 min utes of computation time. Subsection 4.2.5 is dedicated to the topic
of parallel computation in general and section 4.3 explains the implemen tation
design of the algorithm in detail.
4.2.5 P arallel computation
On hardw are that is equipp ed with m ultiple core pro cessors, sev eral tasks can
b e p erformed in parallel at the same time. A problem is suitable for parallel
computation, if it can b e brok en in tasks whic h are indep enden t of eac h other
(task parallel) and if the memory parts that need to b e c hanged b y the tasks
are disjoin t (data parallel); it migh t b e required to sync hronise certain tasks
whic h need to b e p erformed on the same memory region successiv ely (Munshi
and Gaster et al., 2012).
The resp onsibilit y to a v oid conflicts and to sync hronise o v erlapping tasks lies
with the Soft ware designer. The programming language and the compiler need
to b e capable of managing parallel programming threads. A simple w a y of us-
ing parallel computing in C language, is the inclusion of the pthr ead library .
F or more complex tasks that require a deep er con trol of thread-administration,
there are the application programming in terfaces Op enMP and Op enCL. Both
are capable of task parallelism and data parallelism, but with Op enCL one can
execute co de on Graphics Pro cessing Units (GPU), to o.
F or the in v ersion of triangular matrices parallel computation with pthr ead is
used within the implemen tation of the bundle blo c k adjustmen t. Here, the
columns can b e computed indep enden tly of eac h other. The part of the pro-
gram where the 3 × 3 blo c k matrix N 22 is in v erted also in v olv es pthr ead . Since
eac h blo c k can b e in v erted indep enden tly , the task is v ery suitable for parallel
computing, to o. The sp eed-up factor matc hes the n um b er of a v ailable parallel
threads nT hr eads , if the problem size is an (integral) m ultiple of nT hr eads .
72
Another field, for whic h parallel computation is used within the soft ware, is
matrix m ultiplication: T o compute the pro duct AB for A ∈ M n,m , B ∈ M m
one needs m 2 op erations for eac h line, nm 2 op erations in total. Since eac h line
can b e computed indep enden tly , one can divide the task in differen t groups
whic h act parallel. In the case, w ere the problem size n agrees with the n um-
b er of a v ailable threads, all lines can b e computed at the same time. More
t ypical is a realisation where the memory region of the matrices is structured
in m ultiple blo c ks of a certain dimension, the so called tile size , and eac h blo c k
is computed in parallel. Not all matrix pro ducts here are done in parallel, but
the principle is used within a parallel v ersion of the iterativ e decomp osition al-
gorithm (4.2.4). The implemen tation w as realised in Op enCL and is discussed
in the next section. This section closes with a comparison of the running times
for the V esta using the in tro duces decomp osition metho ds. The running time
of one complete iteration of the bundle adjustmen t, including the set up of the
sparse system and the statistic ev aluation is ab out 34 min utes on the sup er-
computer (HLRN). On the m ulti-core cluster it tak es ab out 60 min utes and on
the w ork-station ab out 90 min utes. Figure 4.4 shows the running times of one
iteration with the metho d of subsection 4.2.2 alone (righ t bar) and with the
parallel v ersion of the iterativ e decomp osition algorithm of subsection 4.2.4
in addition (left bar). All other asp ects are the same. The sp eed up factor
due to the parallel computation shrinks to 1:3 for HLRN and to 1:2 for the
cluster. The acceleration factor in the previous c hapter is giv en with resp ect
to the part of the program, whic h formerly consumed most of the running
time, the computation of N − 1 rsp. its decomp osition. Considering the whole
iteration on the HLRN, the ev aluation tak es ab out half of the runni ng time
no w and the decomp osition only a quarter. The adjustmen ts for V esta to ok
four iterations with a total running time of 2h:16min on HLRN, ab out four
hours on the cluster and ab out six hours on the w ork-station. In this example
the factor of memory reduction is 1:16, with resp ect to running time ab out
1:20 at optimisation lev el 2 and ab out 1:40 at lev el 3 (using the cluster).
73
0
20
40
60
80
100
120
140
HLRN Cluster Work-Station
Minutes
Machines
Comparison of the parallel with the single-thread decomposition algorithm
parallel
single-thread
Figure 4.4: Running times for one iteration of the inertial frame bundle blo c k
adjustmen t, pro cessing the V esta data set on differen t hardw are – using the
parallel decomp osition algorithm (left bar) and the single-thread decomp osi-
tion (righ t bar).
4.3 Implemen tation of the decomp osition algo-
rithm in Op enCL
The algorithm in 4.2.4 requires the use of b oth, m ultiplication and in v ersions
of triangular matrices, on a large scale. Sp ecial efforts w ere tak en to ensure
a balanced w orkload on the computation device whic h is recommended in
Munshi and Gaster et al. (2012). It could b e ac hiev ed, that the forw ard and
the bac kw ard op eration can b e p erformed at the same time (task parallelism)
– after the p oin t of sync hronisation for b oth op erations has b een passed for
eac h iteration.
74
4.3.1 Memory design
The algorithm is applied to a symmetric matrix M ∈ M dim transforming it
in to a blo c k triangular matrix S . The dimension of the blo c ks of S resp ectiv ely
the tile size is denoted with T . F or all dim ∈ N there exist natural n um b ers
n and r ≤ T suc h that dim = ( n − 1) T + r . As stated on page 67, M can b e
em b edded in to a larger M 0 and M 0 ∈ M nT . Hence, if necessary M is enlarged
to dimension nT suc h that it consists of n 2 blo c ks of size T × T . In a w orse case
scenario this w ould lead to an o v erallo cation of ( n + T − 1) ∗ T − n = ( T + n )( T − 1)
elemen ts whic h is of O ( n ) and can b e neglected.
A con tinuous memory region, that can hold the non-zero elemen ts of a blo c k
triangular matrix with blo c k size T and dimension nT , is allo cated. This
corresp onds with 1
2 n ( n + 1) T 2 elemen ts. The consequence of sa ving the memory
of the zero-blo c ks is, that it m ust b e though t of a memory design in order to
access eac h elemen t of M . The T × T sub-blo c ks are sequen tially n um b ered
from 0 to 1
2 n ( n + 1) − 1 and a co ordinate system, comparable to that of a
matrix, is used to access them. There are ordered in n column-groups cg and
n ro w-groups r g , as it is sho wn on figure 4.5.
The n um b ers of the first blo c ks in eac h column-group j , form a n umerical
sequence 0,1,3,6,. . . giv en b y the rule
id j =
j
X
i =1
i = 1
2 j ( j + 1) , j = 0 , 1 , . . . , n − 1
and the final blo c k for eac h group is giv en b y the n um b er
id j + j, j = 0 , 1 , . . . , n − 1 .
Hence, M [ id j T 2 ] will access the first en try of cg j , while M [ ( id j + j ) T 2 ]
refers to the first en try of the diagonal blo c k. This allo ws to na vigate through
the matrix whic h is partitioned in to equal sized small T × T blo c ks, optimised
for parallel programming. The v alues of M are transferred to the memory
blo c k according to this pattern.
4.3.2 Tile size
Mean while a lot of publications describ e the topic of the tile size in detail. Con-
sidering the publication Hsu and Kremer (2004), whic h examines the influence
75
Figure 4.5: Memory design of a blo c k triangular matrix. The sub-blo c ks are
n um b ered from 0 to 1
2 n ( n + 1) − 1 and ordered in a matrix-lik e structure. The
co ordinates C G and RG are used to iden tify eac h id via C G ( C G − 1) / 2 + R G .
of the tile size on the p erformance of parallel m ultiplication, the term tile refers
to the dimension of the sub blo c ks of the pro duct whic h are computed b y a
single thread. It can b e translated in to blo ck size within the curren t con text.
A ccording to the result there, 16 is an optimal tile size, based on some relev an t
hardw are prop erties of the computing device. A p erformance analysis of the
algorithm for differen t v alues for T, rev ealed that for T = 6 the p erformance is
w orse and confirmed that T = 16 is an optimal c hoice. Despite the fact, that
T = 6 w ould b e an in tuitive c hoice, the tile size w as therefore set to 16 .
4.3.3 W ork-groups and w ork-items
The functions that are executed in parallel on the Op enCL device are called
k er nel s . Eac h instance of a k ernel is called work-item and has a unique global
ID. The n umber of requested w ork-items can b e larger than the n um b er of
76
factual parallel cores on the device; in this case the surplus of requests is
queued un til an item has finished its w ork and a new instance can b e called.
Hence, a w ork-item corresp onds with a thr ead of the pthr ead libraries. The
w ork- items are organised in work-gr oups whic h ensure that all items of the
group execute on the same compute unit. In the simplest case the n um b er of
items in eac h w ork-group is just one. F or the CPU-v ersion of the algorithm,
whic h is fo cused and describ ed here, this is the recommended configuration.
A ccording to the memory design (figure 4.5) eac h w ork-group is, dep ending on
the task whic h is to p erform, assigned to a column-group cg or a ro w-group
r g and op erates on the s p ecified memory region. The n um b er of w ork-groups
nW G is set to
nW G = n − 1
and corresp onds with the n um b er of images via
6 nI m = ( n − 1) T + r = ⇒ nW G = 6
T nI m − 1 .
This implies that there are less w ork-groups required than images are giv en.
Since the n umber of wo rk-groups is limited to 8892 (for In tel) ab out 23 700
images can b e pro cessed without increasing the tile size T . If T m ust b e
increased, it should b e doubled to 32 .
4.3.4 The parallel decomp osition algorithm
The n um b er of required iterations is n − 2 . F or eac h iteration k consider the
forw ard steps according to detaile d description on page 69. The follo wing tasks
within eac h step can b e separated in indep enden t (data parallel) groups:
0: in v erting the triangular [16 , 16] matrix L k
groups: maximal 16, eac h in v erting one column
1: computing B k : [16 , ( k − 1)] × [16 , 16]
groups: k − 1 , eac h computing one blo c k ( b k i ) of B k
2: computing the 1
2 ( k − 1) k blo c ks of M 12 M − 1
22 M T
12 : [16 , ( k − 1)] × [16 , ( k − 1)]
groups: k − 1 , eac h group computes k / 2 (resp. ( k − 1) / 2 ) blo c ks
design: group i = 0 , . . . , k − 2 op erates on cg i (and k − 2 − i , if i < k
2 )
77
The algorithm starts at k = n − 1 and stops at k = 1 , step 0 is then rep eated
separately on the remaining blo c k L 0 . Hence the n um b er of the required w ork-
groups decreases in eac h iteration.
Note (1:) If step 1 is p erformed at once, one needs to store B k temp orarily
b eforehand. The w ork-item of w ork-group i computes the pro duct in t w o steps:
1a) ˜
b k i = ( blo c k i of M 12 ) × ( L − 1
k ) T (4.17)
1b) b k i = ˜
b k i × ( D − 1
k L − 1
k ) . (4.18)
There is no lac k of p erformance, if (4.18) is done after step 2 has finished.
The storage of B k b ecomes unnecessary , the pro duct in step 2 simplifies to
˜
B k D k ˜
B T
k and the completion of step 1b) completes the forw ard op eration.
Note (2:) There is only one column-group j = k
2 − 1 that has exactly k
2 blo c ks. 4
The column-groups left of it ha v e less and the column-groups to the righ t con-
tain more blo c ks. F or each cg < k
2 there is a partner cg 0 = k − 2 − cg , suc h that
their n umber of blo c k adds up to k . Hence, b oth can share the w orkload easily .
Bac kw ard op eration: The transformation of the bac kw ard pro ducts, see (4.15)
on page 70, requires n − 1 iteration steps. As describ ed b efore, they would
need to b e p erformed after all steps of the forw ard op eration are completed.
Recall that B 0
k = S k − 1 B k has to b e computed in eac h iteration for k = 1 , . . . , n .
Since with gro wing k the dimensions of S k − 1 and B k increase, the workload
increases for eac h iteration. It is true that b y definition eac h of the S k is a
factor of all sequen tially follo wing S j , j > k . Indeed
S 1 = " S 0 , 0
0 , I # " I , B 1
0 , L − 1
1 # = " S 0 , S 0 B 1
0 , L − 1
1 #
S 2 = " S 1 , 0
0 , I # " I , B 2
0 , L − 1
2 # = " S 1 , S 1 B 2
0 , L − 1
2 #
. . . (4.19)
S n = " S n , 0
0 , I # " I , B n
0 , L − 1
n # = " S n − 1 , S n − 1 B n
0 , L − 1
n # .
No w eac h B k consists of k blo c ks b k 0 , . . . , b k ( k − 1) , placed ab ov e the diagonal
blo c k of column-group k whic h holds L − 1
k . In particular it is B 1 = b 10 , B 2 =
4 It needs to distinguish b et w een o dd and ev en k .
78
( b 20 , b 21 ) T and S 0 = L − 1
0 . If the S k are sequen tially replaced b y their definition,
m ultiplying ev erything out leads to
S 1 = " L − 1
0 , L − 1
0 b 10
0 , L − 1
1 #
S 2 =
L − 1
0 , L − 1
0 B 1 , L − 1
0 ( b 20 + b 10 b 21 )
L − 1
1 , L − 1
1 b 21
0 L − 1
2
S 3 =
L − 1
0 , L − 1
0 b 10 , L − 1
0 b 0
20 , L − 1
0 ( b 0
30 + b 10 b 0
31 )
L − 1
1 , L − 1
1 b 21 , b 0
31
0 L − 1
2 , L − 1
2 b 32
L − 1
3
.
This iden tit y can is used for an alternativ e approac h to realise the bac kw ard
op eration. T o explain the idea, consider S 3 ab o v e: after the transformation
is done the first time for the final column, the memory blo c k b 32 is no longer
required and can b e replaced with L − 1
2 b 32 . Similar in the next iteration, the
blo c ks b 21 and b 0
31 are no longer in v olv ed in further calculations; the can b e
replaced with L − 1
1 b 21 and L − 1
1 b 0
31 . This w a y the matrix S is built from b ottom
to top and b oth op erations can b e done in parallel. As a consequence, the
decomp osition is obtained faster if enough parallel cores are a v ailable. The
transformation is explained in greater detail within the app endix (page 106).
Hence, the bac kw ard op eration is obtained b y
b j i + = b k i b j k for all column-groups j > k , ∀ i = 0 , . . . , k
F or eac h j > k a w ork-group can b e set up and launc hed parallel to the forw ard
op eration after B k is computed (step 1 b on page 77). If k = n − 1 the n um b er of
bac kw ard w ork-groups is zero, if k = n − 2 the n um b er is one and so on. Since
the n um b er of forw ard w ork-groups decreases from n − 2 to one sim ultaneously ,
there is a constan t n um b er of n − 2 w ork-groups throughout the time. This
w a y an optimal balance of the w orkload is ensured for parallel computation.
Figure 4.6 illustrates the sp ecific design that realises bac kw ard computation on
column-groups j ≥ k and the next iteration of forw ard calculation on column-
groups j < k (see figure 4.6). Hence, this is an example of task parallelism
and step 1 b w orks as a sync hronisation p oin t b et w een b oth.
79
Figure 4.6: Bac kw ard and forw ard op eration in task-parallelism. The bac k-
w ard op eration is p erformed on the column-groups j > k and the forw ard
op eration on j < k . The p oin t of sync hronisation b et w een b oth tasks is the
completion of the transformations on column-group C G = k .
80
Figure 4.7: P arallel decomp osition algorithm on differen t computing mac hines
(sp ecifications on page 52); the horizon tal axis sho ws the dimension of the
quadratic matrix.
Figure 4.7 sho ws the running times of the Op enCL algorithm on the three
mac hines for matrices with dimensions from 6000 to 60 000. This corresp onds
with data sets of 1000 to 10 000 images. F or the quad-core PC no acceleration
could b e noticed - the times are almost iden tical to the non-iterativ e decomp o-
sition approac h describ ed in 4.2.2. Here the n um b er of cores is not sufficien t to
b enefit from the iterativ e and parallel algorithm. The pro cessor p erformance
of the HLRN with ab out 3 Gflops/s p er core and the 24 ph ysical cores lead
to a sp eed up factor of appro ximately 1:8. The DLR cluster is equipp ed with
more cores but the pro cessor p erformance is only 2.14 Gflops/s p er core; it
tak es almost t wice the time of the HLRN and the lo cal sp eed factor of 1:7 is
lo w er. The comparison sho ws that b oth quan tities, the n um b er of cores and
the p erformance factor need to b e considered.
Chapter 5
Application to the science cases
Phob os and V esta
Credit: The data sets describ ed and ev aluated here, w ere pro duced b y Dr.
K onrad Willner (Phob os) and F rank Preusk er (V esta) who ha v e carried out
the photogrammetric measuremen ts as w ell as the tie p oin t matc hing across
the image bundle. A data set is a list of image ob jects as sho wn in table 5.1,
the p osition and p oin ting data of the in v olv ed cameras w ere obtained from
NASA and ESOC.
Image observ ation Camera Orien tation
Image n um b er F o cal length Image time
Contr ol p oint Image p oint
ID Sample Line
... ... ...
P osition X Y Z
P oin ting φ ω κ
T able 5.1: T emplate of an image ob ject within a data set. Image n um b er,
time and fo cal length as w ell as p osition and p oin ting data of the camera are
assigned as meta-data to eac h ob ject; the list of image p oin t observ ations in
camera co ordinates together with a unique ID of the pro jected con trol p oin t
completes the format.
81
82
5.1 Phob os
The Phob os data set consists of 689 con trol p oin ts and 8010 image p oin t mea-
suremen ts, distributed ov er 19 Viking Orbiter and 53 Mars Express images.
Willner et al. (2010) computed a con trol p oin t net w ork of 665 p oin ts with an
a v erage accuracy of 40 m and a forced libration amplitude of 1 . 24 ◦ . The image
data has b een describ ed in the in tro duction.
The analysis of the heterogeneous data is c hallenging due to the h uge differ-
ences b et w een the uncertain ties of the cameras’ orien tations.
W eigh ting mo del and pre-adjustmen t
W eigh ts: Since the MEX p ositions are far more accurate than the Viking
p ositions, the sto c hastic mo del is not homogeneous. F urthermore the qualit y
of SR C images is m uc h higher, esp. with resp ect to the con trast v alues. T o
comp ensate the latter the uncertain t y for MEX observ ations w as set to one
pixel (0.009 mm) and the uncertain t y for Viking observ ations to the t w ofold
pixel size (0.02352 mm).
The w eighting model for the SR C orien tation data is tak en from Willner et
al. (2010), see A.1 on page 108. The uncertain ties range from 1 m to 1000
m for p ositions and from 0.01 gon to 0.5 gon for the p oin ting angles. F or the
Viking images, the p osition w eigh ts w ere initially set to 5 km for close range
images and to 24 km for the t w o images at far distance from late 1977. F or the
p oin ting uncertain t y a constan t v alue of 0.4 gon w as adopted. This completes
the initial w eigh ting mo del.
Pre-adjustmen t: Firstly , the Viking orien tation w ere adjusted in a prepa-
ration step. F or the pre-adjustmen t the parameters α 0 and δ 0 w ere included
with a uncertain t y of 0 . 2 ◦ and the libration amplitude λ 0 with a uncertain t y
of 1 ◦ in a join t adjustmen t. By this means of con trol, the initial v alues for
the rotational elemen ts did not c hange. The Viking orien tation w ere replaced
with the new v alues and the resulting w eigh ts w ere replaced in the sto c hastic
mo del for the Viking frames only . This w a y an up dated data set with an up-
dated w eigh ting mo del w as obtained; the app endix con tains b oth as table A.2
(w eigh ts) and table A.3 (orien tation).
83
5.1.1 F orced libration amplitude
Recalling the rotation mo del of Phob os (as giv en on page 12):
α ( t ) = α 0 − 0 . 108 T + 1 . 79 sin( M 1)
δ ( t ) = δ 0 − 0 . 061 T − 1 . 08 cos( M 1)
W ( t ) = 35 . 06 ◦ + 1128 . 8445850 t + 8 . 864 T 2 − 1 . 42 sin( M 1) − λ 0 sin( M 2)
with the time indep enden t parameters
α 0 = 317 . 68 ◦ , δ 0 = 52 . 90 ◦ , λ 0 = 0 . 78 ◦ .
After the preparation three differen t t yp es of adjustmen ts w ere p erformed,
with the w eighting model resulted from the pre-adjustmen t. In the first case,
only the forced libration amplitude λ 0 w as included in the functional mo del:
the giv en start v alue of 0 . 78 ◦ w as adjusted to 1 . 13 ◦ ( ± 0 . 013 ).
In order to test the indep endence of this result from other parameters, t w o
additional cases ha v e b een studied. In a second case, only the constan t p ole
co ordinates ( α 0 , δ 0 ) w ere resolv ed: a change in righ t ascension of 0 . 07 ◦ and in
declination of only 0 . 003 ◦ w as noticed. In the third case all three parameters
ha v e b een adjusted join tly . Here ˆ
λ 0 w as computed with 1 . 132 ◦ ( ± 0 . 0133 ),
confirming the result of the first case. The results of these three adjustmen ts
are listed in table 5.2. In all three cases the rotational elemen ts are included as
complete unkno wn parameters lik e the con trol p oin ts and unlik e the orien tation
parameters whic h undergo a w eigh t con trol.
The result for ˆ
λ 0 = 1 . 13 ◦ is in agreemen t with 1 . 09 ◦ ± 0 . 1 ◦ (Ob erst et al., 2014)
and 1 . 24 ◦ ± 0 . 15 ◦ (Willner et al., 2010), and v ery close to 1 . 1 ◦ , the result of
the theoretical consideration of Ram baux et al. (2012).
5.1.2 Con trol p oin t net w ork for Phob os
The computed CPN of the adjusted data set consists of 680 p oin ts. Considering
the first adjustmen t case, the a v erage norm of the w eigh t v ector ( σ X , σ Y , σ Z )
is ab out 55 m and the forw ard in tersection error of the con trol p oin ts is ab out
13 m (table 5.3 sho ws the CPN statistics for all three cases).
Figure 5.2 sho ws a three dimensional mo del of Phob os, whic h w as created from
679 con trol p oin ts (Burmeister, 2016a). The p oin t cloud is triangulated and
84
Results for rotational elemen ts (pre-adjusted Viking)
case param. start v al. end v alue diff. σ
σ
σ
Libration λ 0 0 . 78 ◦ 1 . 13 ◦ +0 . 35 ◦ 0 . 013 ◦
P ole axis α 0 317 . 68 ◦ 317 . 75 ◦ +0 . 07 ◦ 0 . 018 ◦
δ 0 52 . 9 ◦ 52 . 897 ◦ − 0 . 003 ◦ 0 . 01 ◦
Join t adj. λ 0 0 . 78 ◦ 1 . 132 ◦ +0 . 352 ◦ 0 . 0133 ◦
α 0 317 . 68 ◦ 317 . 654 ◦ − 0 . 026 ◦ 0 . 0130 ◦
δ 0 52 . 9 ◦ 52 . 885 ◦ − 0 . 015 ◦ 0 . 01 ◦
T able 5.2: Results for three separate adjustmen ts (case) of rotational elemen ts
(param.) with initial v alues (start v alue) are sho wn in col. 4 (end v alue); col.
5 sho ws the difference and col. 6 the uncertain t y ( σ ). In the first case only
libration amplitude λ 0 is adjusted, in the second case only the mean p ole axis
orien tataion ( α 0 , δ 0 ) , case 3 is a join t adjustmen t of all the three parameters.
The join t adjustmen t affects the c hanges in ( α 0 , δ 0 ) but not the c hange in λ 0 .
CPN Statistics (for results in table 5.2)
case s 0
s 0
s 0 mean error fwi error
Libration 5 . 09 · 10 − 11 55 . 65 m 13 . 35 m
P ole axis 5 . 50 · 10 − 11 60 . 39 m 14 . 10 m
Join t adj. 5 . 06 · 10 − 11 55 . 55 m 13 . 35 m
T able 5.3: Con trol p oin t net w ork statistic: Col. 1 as in table 5.2, col. 2 holds
the o v erall system error s 0 , the CPN mean error in col. 3 is computed from
|| ( σ X , σ Y , σ Z ) || of the adjusted p oin ts, col. 4 sho ws the forw ard in tersection
error. In all three cases, the net w ork consists of 680 p oin ts.
rendered with IDL; shading and ligh t source sim ulation whic h are needed to
mak e top ographic features visible w ere done with CloudCompare View er. The
large Stic kney crater on the w estern hemisphere is clearly to see in the upp er
left image (leading side) and partly to see in the lo w er left image (northern
hemisphere).
The 3D con trol p oin ts ha v e b een con v erted to p olar co ordinates and their
distribution is sho wn in figure 5.1. The figure also sho ws the differen t ranges
85
-80
-60
-40
-20
0
20
40
60
80
-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180
Latitude
Longitude
Distribution of Phobos Control Points
Sigma range < 40 m 40-60 m 60-120 m 120-180 m 180-350 m
Figure 5.1: Distribution of con trol p oin ts on Phob os (unit sphere pro jection)
with sigma v alues from 30 m to 345 m. The ma jorit y of p oin ts has v alues
under 60 m, the CPN mean error is 55.65 m; in the region b et w een 60 ◦ and
100 ◦ East on the northern hemisphere Viking and MEX images o v erlap.
of σ = || ( σ X , σ Y , σ Z ) || for eac h con trol p oint separated in four classes: σ 0 = 60
m and smaller, 2 σ 0 , 3 σ 0 and ab o v e. There are 15 p oin ts o v er 180 m, most of
them lo cated in the north-eastern quadran t. The larger sigma v alues o ccur in
the region where Viking and MEX images o v erlap and for the CPs in images
close to that. The CPN solution is consisten t with the previously obtained
b y Willner et al. (2010). The difference in the libration amplitude λ 0 of 0.11
degree leads to a maximal p ositional offset of ab out 100 m for the MEX images,
measured in the b o dy-fixed reference frame. The a v erage effect on the p ositions
is ab out 30 m, if all images are considered. As far as comparable, b oth CPNs
differ in parts up to 260 m. The accum ulated uncertain t y of the p oin ts is ab out
100 m; further deviations can b e explained with a p ossibly differen t sto c hastic
mo del for the Viking images (whic h is not a v ailable for comparison). The
solution obtained here with the inertial frame metho d is closer to the result of
a b o dy-fixed adjustment that only in v olv es MEX SR C images.
86
W estern hemisphere sho wing the
craters 1) T o dd, 2) Drunlo and 3)
Stic kney
View cen tred at 180 ◦ (v ertical) and
equator (horizon tal), with T o dd on
the righ t half b elo w the equator
Northern hemisphere with four
longitudes ( 0 ◦ , 90 ◦ E , 180 ◦ E , 90 ◦ W )
Southern hemisphere with four
longitudes ( 0 ◦ , 90 ◦ W , 180 ◦ E , 90 ◦ E )
Figure 5.2: Reconstructed shap e of Phob os from 679 con trol p oin ts, triangula-
ted and rendered with sim ulated ligh t source. The four differen t p ersp ectiv es
are pro jections in to fundamen tal planes with the view cen tred at the third
axis. +X in tersects the prime meridian, +Y in tersects 90 ◦ E and +Z con tains
the northern p ole.
87
5.2 V esta
There are 82 806 con trol p oin ts distributed o v er 5440 images and the a v erage
observ ation rate is 9.3 p er p oin t. Unlik e the Phob os data, here all images
are tak en by the same spacecraft at distances whic h are close to the a v erage
distance of 944.5 km (table 1.5 on page 21). The distance is measured with
resp ect to the cen tre of mass. The w eigh ting mo del is pro vided b y Preusk er
(priv ate data con tribution). F or all images the p osition uncertain t y is 35 metres
and the p oin ting uncertain t y is 0.006 gon. The images do not completely co v er
V esta around the northern p ole region.
5.2.1 P ole axis orien tation
The curren t rotational mo del for V esta is simple, the only dynamical term is
the self rotation p erio d of ab out 5 h 20 min. P arameters of in terest are here
the co ordinates ( α, δ ) of the p ole axis orien tation. T w o differen t mo dels will b e
considered in this subsection. The first agrees with the Da wn-Claudia system
stated on page 12. The uncertain t y of the p ole axis orien tation is giv en with
0 . 01 ◦ (Arc hinal et al., 2013). The second mo del, prop osed b y Preusk er (priv ate
comm unication), w as applied during the prepro cessing of the data set whic h
is used here:
(Preusk er) α ( t ) = 309 . 03312 ◦
δ ( t ) = 42 . 22623 ◦
W ( t ) = W 0 + 1617 . 3331237 ◦ t , W 0 = 74 . 6625 ◦ .
The inertial frame bundle blo c k adjustmen t w as p erformed for b oth mo dels.
The uncertain t y for α and δ w as set to 0.03 degree ( 3 σ ). The rotation rate
and the v alue for W 0 remained constan t.
T able 5.4 lists the results and the difference with resp ect to the starting v alues
for b oth adjustmen ts . F or the first mo del, the c hanges for b oth, α and δ are
less than 0 . 01 ◦ . F or the second mo del, the c hange in righ t ascension is sligh tly
ab o v e 0 . 01 ◦ but closer to the IA U v alue for α . Both results confirm the curren t
IA U p ole axis orien tation of (309 . 031 ◦ , 42 . 235 ◦ ) within the giv en range of 0 . 01 ◦ .
Compared with eac h other, b oth results differ b y 0 . 0019 ◦ (alpha) and 0 . 0074 ◦
(delta), whic h is within the computed error of ± 0 . 008 ◦ . The o v erall system
error σ apost of b oth inertial frame solutions is ab out 1 . 4 · 10 − 6 .
88
A djusted p ole axis orien tation for V esta
init. mo del Da wn-Claudia Preusk er
parameter result c hange result c hange
α 309 . 0246 ◦ − 0 . 0064 ◦ 309 . 0227 ◦ − 0 . 0104 ◦
δ 42 . 2296 ◦ − 0 . 0054 ◦ 42 . 2222 ◦ − 0 . 0040 ◦
difference of b oth results : ∆ α = 0 . 0019 ◦ , ∆ δ = 0 . 0074 ◦ < 0 . 008 ◦
T able 5.4: A djusted p ole axis orien tation of V esta ( r esult ) ± 0 . 008 ◦ for t w o
differen t initial mo dels, change indicates the difference to the initial v alues.
Both results are, compared with the IA U mo del (309 . 031 ◦ , 42 . 235 ◦ ) , within
the uncertain t y range of 0 . 01 ◦ and, compared with eac h other, within 0 . 008 ◦ .
5.2.2 Comparison with previous CPN solution
Since here, all input and output data of the previous bundle blo c k adjustmen t
in the b o dy-fixed reference frame are a v ailable, the solution can b e compared
in detail with the adjustmen t solution form ulated in the inertial frame. Hence,
Preusk er’s b o dy-fixed solution and the results, obtained here b y the inertial
frame metho d, are compared with resp ect to the camera orien tation data and
the co ordinates of the con trol p oin ts. The difference in righ t ascension and
declination of b oth solutions is ∆ α = 0 . 0104 ◦ , ∆ δ = 0 . 004 ◦ . The effect of this
difference on the p ositions of the b o dy-fixed solution is in a v erage 76 m and
ab out 0.009 gon on the p oin ting angles. Hence, the difference of these parame-
ters to the parameters of the inertial frame solution should not b e significan tly
larger. The adjusted p ositions differ b y ∅ 42 m and the p oin ting angles b y ∅
0.0042 gon. Finally , the co ordinates of the con trol p oin ts only differ b y 15 m in
a v erage, resp ecting the giv en error range of ∅ 10 m for b oth CPNs. This sho ws,
that b oth net w orks are plausible. It has to b e emphasised that the assumed
uncertain t y v alues of (30 m, 0.006 gon) for the camera orien tation data can
not comp ensate a difference of 0 . 01 ◦ in the p ole axis orien tation. T aking the
uncertain t y of the rotational mo del in to accoun t, the b o dy-fixed orien tation
uncertain ties w ould need to b e increased.
Figure 5.3 sho ws three p ersp ectiv es of a 3D mo del that has b een created from
the adjusted CPN (Burmeister, 2016b). The views are pro jections in to stan-
89
View cen tred at the prime meridian
(v ertical) and equator (horizon tal)
Left view rotated b y 180 ◦ (v ertical
axis), sho wing the "sno w man"
The 3D mo del w as created from
82 806 con trol p oin ts, obtained b y
the forw ard in tersection metho d af-
ter the orien tation parameters of the
camera ha v e b een adjusted. The
p oin t cloud is triangulated and ren-
dered with a sim ulated light source,
in order to mak e the top ographic
features visible.
Southern hemisphere
Figure 5.3: Reconstruction of V esta from 82 806 con trol p oin ts - the p ersp ec-
tiv es ab o v e are pro jections in to Y-Z plane, the far side view is 180 ◦ rotated
around the Z-axes (w.r.t leading side). The p ersp ectiv e b elo w sho ws the X-Y
plane view ed from ab o v e the south p ole. See also the reference DTM from
Preusk er (Figure 5.4)
90
Figure 5.4: Reference DTM of V esta (Preusk er et al., 2012, fig. 2)
dard planes, suc h that the great circle con taining the prime falls on to the Z-axis
(the upp er t w o images) or on the X-axis (the southern hemisphere). The p oin t
cloud w as triangulated and rendered with the soft w are GeoMagic, the shading
and view settings w ere done with CloudCompare View er. A reference DTM
based on a b o dy-fixed solution (figure 5.4), published in Preusk er et al. (2012),
has b een included.
5.2.3 Surface spherical harmonics analysis
The surface spherical harmonics of order j = 0 , 1 , . . . in R 3 are giv en b y
Y j ( θ , φ ) =
j
X
m =0
P j m (sin θ ) ( C j m cos( mφ ) + ¯
C j m sin( mφ )) , C j m , ¯
C j m ∈ R
when θ is the spherical latitude and φ the longitude. P j m is called L egendr e
p olynomial for m = 0 and asso ciate d L e gendr e p olynomial for m > 0 with
degree j (see definition Bronstein and Semendja jew (2005) 3.3.1.3.4). Since
the co efficien ts C m , ¯
C m are arbitrary real n um b ers, Y j is as ev ery p olynomial
uniquely defined b y the c hoice of its co efficien ts.
Remark: The typic al definition in mathematics states cos θ as ar gument of
P j m . It is imp ortant to note, that in this c ase θ is not the latitude. Inste ad,
one me asur es the angle fr om the north p ole (0 ◦ ) down to the e quator (90 ◦ ) and
91
further to the south p ole (180 ◦ ) . Henc e, 0 ≤ θ ≤ 180 is the spheric al c o-latitude
w.r.t. the north p ole. Sinc e cos θ agr e es with the sine value of the latitude on
b oth hemispher es, the definition ab ove is obtaine d.
The analysis of the top ograph y f based on spherical harmonics relies on the
serial dev elopmen t
f ( θ , φ ) = r =
∞
X
j =0
Y j ( θ , φ ) , ∀ surface p oin ts P = ( θ , φ, r ) (5.1)
That means, all con trol p oin ts are pro jected to the unit sphere ( θ , φ ) and f
simply assigns r , the distance to the barycen tre, to each point. This assignmen t
m ust b e unique, so ob viously it is to assume
|| P − P 0 || > 0 ⇐ ⇒ || ( θ , φ ) − ( θ 0 , φ 0 ) || > 0 .
With this necessary restriction to the top ology f is defined on the sphere
and can b e represen ted b y equation (5.1) (Bronstein and Semendja jew, 2005,
3.3.2). The equation implies, that f can b e describ ed b y a set of co efficien ts
that define the Y j . Since this set is infinite, an appro ximation of f is to use
instead. The truncation
f d =
d
X
j =0
Y j satisfies lim
d →∞
( f − f d ) = 0 (5.2)
where d ∈ N is called the degree of dev elopmen t.
The co efficien ts of the surface spherical harmonics ha ve b een computed based
on equation (5.2) with d = 60 for all 82 806 con trol p oin ts. The system of
equations can b e written in matrix form as Ac = b with the v ector of radii
on the righ t hand side and the unkno wn co efficien ts c . W eigh ted with the
sto c hastic mo del ap osteori P C P for the con trol p oin ts, the solution is obtained
b y in v ersion as
c = ( A T P C P A ) − 1 A T P C P b .
In the app endix section A.4 the pro cedure is explained for j = 0 and j =
1 , in total 61 2 = 3721 co efficien ts ha v e b een computed. T able 5.5 lists the
unnormalised co efficien ts up to order j = 10 , that are the first 121 co efficien ts
of the series. Figure 5.5 sho ws a larger sp ectrum (400 v alues), normalised to
92
the unit sphere radius b y applying C j m /C 00 . The v ertical lines mark the C j 0
co efficien ts, they are follo w ed b y C j 1 , ¯
C j 1 , C j 1 , ¯
C j 1 , . . . , C j j , ¯
C j j ; the plot sho ws
the magnitudes in b oth directions of the zero-line. A figure sho wing all of the
co efficien ts has b een dismissed b ecause of its lo w w eigh ting information.
-0.04
-0.02
0
0.02
0.04
0 50 100 150 200 250 300 350
Magnitude
Spectrum of surface spherical harmonics up to order 19
j=10 j=19
Figure 5.5: Zo om in to sp ectrum of the surface spherical harmonics co efficien ts
for d = 60 and j = 0 ,..., 19 sho wing the first 400 v alues. The v ertical lines
mark the legendre co efficien ts C j 0 , follo w ed b y C j m , ¯
C j m ( 1 ≤ m ≤ j ) suc-
cessiv ely . V alues are normalised suc h that C 00 = 1 , | C 20 | = 0 . 23 (the second
large impulse) is the only other v alue ab o v e 0.04; magnitude is mirrored b elo w
zero-line for b etter visibilit y . See table 5.5 for concrete n um b ers.
93
F rom the co efficien ts of the spherical harmonics, the shap e of V esta w as
reconstructed. F or d = 60 the result in kilometres is 559.57 x 560.02 x 456.69.
Compared with the originally obtained CPN there is a difference in X of 30 m,
in Y of 1.3 km and in Z of 10 m, table 5.6 sho ws the rounded v alues (full km).
The IA U gives the diameter of V esta with 578 x 560 x 458 with an uncertain t y
of ± 10 km eac h (Arc hinal et al. (2011), table 6). Hence, there is a significan t
deviation for the semi-ma jor axis (18 km). The co ordinates of the ellipsoid’s
cen tre are (1.9, -4.6, 4.6). The dense distribution of measuremen ts in the south-
ern p ole region mirrors the obtained offset b et w een the cen ter of mass and the
cen ter of figure. The images of the curren t data set do not sufficien tly co v er the
northern p ole region and there is a strongly un balanced distribution of mea-
suremen ts, fa v ouring the southern hemisphere. The section ab out the cen tre
of figure (A.4, page 111) con tains the w ell-kno wn connection b et w een the first
order co efficien ts and the cen tre of figure; it is included to emphasise, that the
distribution of con trol p oin ts pro jected on to the sphere should b e uniform and
esp. cen tred. This can b e obtained if e.g. a DTM is used to complete the CPN.
Finally , the CPN has b een reconstructed from the co efficien ts. Figure 5.6
sho ws the same p oin t cloud as b efore, appro ximated b y a series of surface
spherical harmonics of degree and order 60. The views are the same as for
the original CPN. The a v erage appro ximation error is 378 m (min. 5 m, max.
4900 m).
94
Surface spherical harmonics co efficien ts for V esta
j
j
j C j 0
C j 0
C j 0 j
j
j C j 0
C j 0
C j 0 j
j
j C j 0
C j 0
C j 0 j
j
j C j 0
C j 0
C j 0 for m = 0
m = 0
m = 0
0 260357.3 3 9368.7 6 2217.0 9 -293.0
1 1953.5 4 2673.9 7 -1399.1 10 813.1
2 -60315.6 5 -3574.2 8 555.0
j
j
j m
m
m C j m
C j m
C j m ¯
C j m
¯
C j m
¯
C j m j
j
j m
m
m C j m
C j m
C j m ¯
C j m
¯
C j m
¯
C j m for m > 0
m > 0
m > 0
1 1 -266.2 2601.7 8 1 992.9 -1782.3
2 1 1441.5 1820.1 8 2 229.7 -2840.4
2 2 1517.7 8450.9 8 3 -2448.8 -374.9
3 1 -8080.6 -7028.8 8 4 1827.7 -631.6
3 2 2453.5 -3070.3 8 5 -878.6 1494.5
3 3 -6568.8 -356.7 8 6 2132.7 -881.2
4 1 3372.9 725.8 8 7 -715.4 -239.2
4 2 -2641.9 228.6 8 8 155.1 -715.2
4 3 3448.0 -271.9 9 1 -265.1 -554.0
4 4 1081.6 386.1 9 2 513.0 -992.7
5 1 2176.9 919.3 9 3 553.3 2150.1
5 2 1095.8 553.6 9 4 1318.1 -861.1
5 3 1095.3 3315.0 9 5 207.5 -240.4
5 4 1668.0 1333.8 9 6 210.9 719.4
5 5 -2454.7 1095.9 9 7 888.9 -1396.5
6 1 929.1 -5969.6 9 8 947.4 -1038.0
6 2 -1612.9 -247.8 9 9 531.1 -225.7
6 3 -689.1 -562.7 10 1 182.1 800.0
6 4 1084.3 -232.0 10 2 -374.9 389.4
6 5 -1023.2 1564.4 10 3 2783.1 -150.1
6 6 -2105.4 -2112.5 10 4 -372.9 95.6
7 1 -2555.7 7738.4 10 5 586.7 -398.0
7 2 1246.9 4713.2 10 6 -706.5 508.7
7 3 175.8 -1322.0 10 7 -22.6 -635.5
7 4 -1664.8 -1164.4 10 8 48.3 123.1
7 5 532.4 1551.1 10 9 -1237.4 -943.5
7 6 -209.0 461.5 10 10 112.2 -1337.6
7 7 -1241.1 -1538.0
T able 5.5: Unnormalised co efficien ts of surface spherical harmonics up to order
10 for degree of dev elopmen t d = 60 , obtained b y in v ersion with the sto c hastic
mo del for adjusted con trol p oin ts
95
Size parameter of V esta
Source Diameter COE
CPN (b est-fit) 560 x 559 x 457 (2, -5.5, 3.5)
SSH ( d = 60 ) 560 x 560 x 457 (2, -5.0, 4.5)
IA U 578 x 560 x 458 —
T able 5.6: Size parameter of V esta (in km); col. 2 and 3 sho w the diameters
and cen tre of the ellipsoid (COE) in X,Y,Z co ordinates for a b est-fit ellipsoid
using the CPN and computed b y surface spherical harmonics of degree and
order 60 (SSH). IA U v alues ( ± 10 km for eac h dimension) ha v e b een included
from Arc hinal et al. (2011) for comparison.
View cen tred at 180 ◦ (v ertical) and
equator (horizon tal)
Southern hemisphere
Figure 5.6: The 3D mo del w as created with 82 806 con trol p oin ts, obtained b y
a series of surface spherical harmonics of degree d = 60 . Latitude and longi-
tude are sp ecified b y the p oin ts of figure 5.3. The p oin t cloud is triangulated
and rendered with a sim ulated ligh t source, in order to mak e the top ographic
features visible.
96
Chapter 6
Conclusions and Outlo ok
6.1 Summary and conclusions
The bundle blo c k adjustmen t in the inertial reference frame is a metho d to
obtain rotational elemen ts in a direct and analytical w a y from image p oin t
measuremen ts. It is an extension of the classical metho d in the b o dy-fixed ref-
erence frame that resolv es the in terdep endence of the rotational mo del and the
camera’s external orien tation (p osition and p oin ting). The functionalit y of the
principle w as tested in a case study with sim ulated data. The new approac h
turned out to impro ve the inner constrains of the con trol p oin t net w ork. It
furthermore allo ws to analyse the dep endencies of rotational elemen ts and to
iden tify parameters whic h are either w eakly determined or not y et mo delled.
Applied to the real data set of Phob os, the forced libration amplitude λ 0 =
1 . 13 ◦ ± 0 . 013 w as computed with a CPN of 680 p oin ts. The result is in agree-
men t with previous results and hence, supp orts the assumption of a uniform
densit y of Phob os (Willner et al., 2010). The mean uncertain t y of the CPN is
ab out 55 m and the forw ard in tersection error is ab out 13 m. The deviation in
the p ole axis orien tation from the join t adjustmen t (table 5.2) indicates, that
an impro v emen t of Phob os’ rotational mo del is p ossible and should b e sub ject
to a subsequen t analysis. The metho d can b e used for ev aluation of further
rotational elemen ts suc h as the precession p erio d or the acceleration term.
F or V esta, the p ole axis orien tation w as computed together with a large CPN
of ≈ 83 000 con trol p oin ts. The resulting v alues (309 . 0246 ◦ ± 0 . 008 , 42 . 2296 ◦ ±
0 . 008) are only sligh tly smaller then the curren tly adopted v alues of (309 . 031 ◦ ,
97
98
42 . 235 ◦ ) and w ell within the exp ected range of 0 . 01 ◦ . Also a comparison to a
b o dy-fixed adjustmen t w as made sho wing that the new metho d is consisten t
with the previous results. On one hand, this confirms the curren t p ole axis
orien tation of the IA U mo del (Arc hinal et al., 2013) and on the other hand, it
sho ws the consistency of the inertial frame adjustmen t metho d. A comparing
analysis of the b o dy-fixed solution (Preusk er; priv ate data con tribution) and
the inertial frame solution with resp ect to the camera orien tation data yielded
another criteria of reliabilit y: the actual differences of the corrected camera
parameters are smaller than the offsets caused b y the differen t p ole axis ori-
en tation. F urthermore, the co ordinates of the con trol p oin ts only differ b y 15
m in a v erage with a giv en forw ard in tersection error of 10 m for b oth CPN
solutions.
It has b een sho wn, that the implemen ted soft w are can handle large data
sets; the costs for the resources running time are O ( n 3 ) and for memory O ( n 2 )
with 1
6 n b eing the n um b er of images. Hence, the capacit y limit dep ends on
the n um b er of images and not on the n um b er of con trol p oin ts. This is a
significan t impro v emen t compared with the reference adjustmen t soft w are: the
computation time of 168 hours (sev en days) for the adjustmen t of the V esta
data set is reduced to ab out eigh t hours in a first optimisation step, where
the ma jor w ork is done b y a single core of the CPU (sequen tial computation).
In a second optimisation step a decomp osition algorithm w as dev elop ed to
b enefit from m ulti-core arc hitecture (parallel computation): the running time
could b e further reduced b y factor 2 to four hours only on the same mac hine.
P erformance tests of the parallel algorithm sho w ed that the c hoice of hardw are
mak es a significan t difference with resp ect to the sp eed-up factor – on other
testing mac hines, a of factor 3 (maxim um) and a factor of v 1 (minim um)
w ere obtained. The comparison sho w ed, that four cores are not sufficien t to
b enefit from this new algorithm. On the other hand it turned out, that the
n um b er of parallel cores is not the most relev an t criteria, to o. Instead, the
c hip p erformance (in Gflop/s) and the n um b er of cores together need to b e
considered. A 3-CPU no de of the Xeon E5-2670 (24 cores) w as sho wing the
b est p erformance.
Hence, ev en for large data sets, rotational elemen ts can b e computed in
acceptable time. By means of a w eigh t con trol, the rotational elemen ts can b e
99
k ept fixed and a b o dy-fixed bundle blo c k adjustmen t is p erformed as a conse-
quence. It is also p ossible to directly c ho ose a b o dy-fixed op eration mo de, but
the additional resource requiremen ts for the rotational elemen ts are negligible.
The new adjustmen t soft w are is able to obtain the CPN solution for V esta,
the impro v ed camera orien tations and the p ole axis orien tation including the
sto c hastic mo del ap osteori in less than four hours on the reference machine.
This corresp onds to a sp eed-up factor of 42 with resp ect to the reference soft-
w are.
6.2 Outlo ok
F urther steps
It is planned to p erform a bundle blo c k adjustmen t for 18 000 high resolution
images mapping the equatorial region of Mercury . The fo cus hereb y will b e on
the CPN solution, but sim ultaneously p ole axis orien tation and spin rate can
b e determined. F urther geo detic pro ducts of Mercury will result from subse-
quen t analysis.
Comet CP67, the target of the Europ ean Rosetta mission, is a v ery in terest-
ing target with resp ect to its rotational mo del. The a v ailable images can b e
used to study the rotational b eha viour in greater detail with the inertial frame
analysis.
The soft w are should b e enlarged to a general rotational mo del. All p ossible
rotational parameters of an IA U mo del (Arc hinal et al., 2011) can b e formally
included. The initial rotational mo del as w ell as the parameters of in terest
could then b e sp ecified in a configuration file together with the w eigh t infor-
mation. Not only the p ole axis co ordinates α 0 , δ 0 , also the rate of precession,
the rotation rate as w ell as the acceleration term for ob jects lik e Phob os can
b e resolv ed.
The pap er of K onopliv et al. (2014) indicates that V esta has a precessing p ole,
but this is a result of a theoretical consideration and the momen t of precession
could not b e determined. It is planned to mo del the precession mo v emen t and
to determine its p erio d and momen tum with the inertial frame adjustmen t
metho d.
100
GPU implemen tation and further optimisation
The Op enCL implemen tation of the decomp osition algorithm in c hapter 4 is
fully applicable for GPU calculation. Since graphic engines can launc h thou-
sands of parallel threads, the m ultiplication on this computation device can
b e further optimised. On a CPU device, the n um b er of threads is less than
the n um b er of images and this case, there should b e only one thread assigned
to eac h w ork group ( l ocal w or k g r oupsiz e = 1 ). On a GPU the size of the
lo cal w ork group can b e increased and the w ork can b e distributed to sev eral
threads instead. In this case further care has to b e tak en to sync hronise the
w ork-items of eac h group. With the curren t configuration, the algorithm w as
tested on a NVIDIA Quadro 600 with one GB memory . Here, only data sets
of 1000 images could b e tested and they are to o small to pa y off in terms of
running time. F or the V esta data set, a card with 6 GB w ould b e sufficien t.
Ho w ev er, the memory limitation of GPUs needs to b e considered.
The algorithm computes the decomp osition ( S, D ) with S T D S = N − 1
11 , the
upp er left region of the in v erse normal matrix. Considering the V esta data
set, at the curren t lev el of optimisation this part tak es no w a quarter of total
running time on HLRN K onrad Berlin. The solution for the unkno wn v ector x
is obtained quic kly with the pro duct routines for compressed sparse matrices.
After that part, the diagonal of HA T matrix AN − 1 A T is computed sequen-
tially . This can b e done in parallel and should b e implemen ted in the future.
The building of the sparse system itself, in particular the preparation step for
the decomp osition algorithm can b e done in parallel as w ell. On the HLRN,
with 24 cores this w ould reduce the running time from curren tly 34 min utes
to ab out 10 min utes (p er iteration) and increase the curren t sp eed-up factor
of 3 to 9.
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App endix A
App endix
A.1 Pro of of the split tec hnique
F or a general matrix M ∈ M dim with | det ( M ) | > 0 and sub division sc heme
M = " A B
C D # the follo wing relation holds (Bronstein and Semendja jew, 2005):
M − 1 = " A 0 − A 0 B D − 1
− D − 1 C A 0 D − 1 + D − 1 C A 0 B D − 1 # (A.1)
where A 0 = ( A − B D − 1 C ) − 1 , A ∈ M dim 1 and D ∈ M dim 2 , where dim 1 is
arbitrary and dim 2 = dim − dim 1 . Pro of:
Using the same sub division sc heme for M − 1 = " A 0 B 0
C 0 D 0 # , the m ultiplication
M M − 1 = I leads to four systems of equations
1) AA 0 + B C 0 = I dim 1
2) AB 0 + B D 0 = 0
3) C A 0 + D C 0 = 0 ⇐ ⇒ C 0 = − D − 1 C A 0
4) C B 0 + D D 0 = I dim 2 ⇐ ⇒ D 0 = D − 1 − D − 1 C B 0
where the righ t hand side is the sub division of the unit matrix I dim .
Replacing C 0 in equation 1) giv es
1 0 ) AA 0 − B D − 1 C A 0 = I ⇐ ⇒ ( A − B D − 1 C ) A 0 = I
⇐ ⇒ ( A − B D − 1 C ) − 1 = A 0
105
106
Then D 0 is replaced in the second equation to obtain
2 0 ) AB 0 + B D − 1 − B D − 1 C B 0 = 0
( A − B D − 1 C ) B 0 + B D − 1 = 0 ⇐ ⇒ ( A 0 ) − 1 B 0 + B D − 1 = 0
B 0 = − A 0 B D − 1 .
If B 0 is set in to equation 4)
D 0 = D − 1 − D − 1 C B 0 = D − 1 + D − 1 C A 0 B D − 1
equation 4.3 follo ws.
A.2 Bac kw ard op eration in split mo de
The transformation of S 2 in the bac kw ard op eration on page 77 is accomplished
b y sequen tially replacing S 1 with its definition. Note S 0 = L − 1
0 , B 2 = b 20
b 21 ! .
S 1 = " S 0 , 0
0 , I # " I , B 1
0 , L − 1
1 # = " S 0 , S 0 B 1
0 , L − 1
1 #
S 2 = " S 1 , 0
0 , I # " I , B 2
0 , L − 1
2 # = " S 1 , S 1 B 2
0 , L − 1
2 #
=
S 1 , " S 0 , 0
0 , I # " I , B 1
0 , L − 1
1 # B 2
0 , L − 1
2
=
" S 0 , S 0 B 1
0 , L − 1
1 # , " S 0 , 0
0 , I # " b 20 + B 1 b 21
L − 1
1 b 21 #
0 , L − 1
2
=
L − 1
0 , L − 1
0 B 1 , L − 1
0 ( b 20 + B 1 b 21 )
L − 1
1 , L − 1
1 b 21
0 L − 1
2
The computation of B 1 = b 10 is the sync hronisation p oin t for the first bac kw ard
op eration step (transforming the final column-group C G = 2 ):
1 a ) b 20 + = b 10 ∗ b 21
107
1 b ) b 21 ∗ = L − 1
1 .
The final forw ard step is then the computation of L − 1
0 (and D 0 ); the final
bac kw ard op eration step is on the first ro w-group ( R G = 0 ):
2 a ) b 10 ∗ = L − 1
0
2 b ) b 20 ∗ = L − 1
0
2a) and 2b) can b e done in parallel straigh t a w a y . 1b) has do b e p erformed
after 1a): Since b oth are realised in the same w ork-group this is easy if there is
only one w ork-item p er group. If the n um b er will b e increased, care has to b e
tak en here. A lo cal w ork-group barrier or a separate call for all w ork-groups
can b e used, analogue to step 2.
The steps ab o v e are for n = 2 , for general n ∈ N they are as follo ws:
1 a ) b j i + = b k i ∗ b j k , i = 0 , . . . , k − 1 ∀ k < j < n
1 b ) b j k ∗ = L − 1
k ∀ k < j < n
and the final computation step is
2) b j 0 ∗ = L − 1
0 j = 1 , . . . , n − 1 .
There is a w ork-group for eac h j in ev ery step; k = n − 2 in the first iteration
decreases to k = 1 .
A.3 Phob os tables
The tables here are additional information to section 5.1.
108
initial SR C w eigh ts
Image No σ X σ Y σ Z σ φ σ ω σ κ
44470005 1 1 1 0.01 0.01 0.5
43810004 1 1 1 0.01 0.01 0.5
28050024 1 1 1 0.01 0.01 0.5
46030005 1 1 1 0.05 0.05 0.05
45540005 1 1 1 0.05 0.05 0.05
43400005 1 1 1 0.05 0.05 0.05
28130005 1 1 1 0.05 0.05 0.05
38430003 1 1 1 0.05 0.05 0.05
38430004 1 1 1 0.05 0.05 0.05
28460007 1 1 1 0.05 0.05 0.05
22330005 1 1 1 0.05 0.05 0.05
38680004 1 1 1 0.05 0.05 0.05
38680005 1 1 1 0.05 0.05 0.05
7150004 1 1 1 0.5 0.5 0.5
27390005 1 1 1 0.5 0.5 0.5
46360005 1 1 1 0.5 0.5 0.5
27560005 1 1 1 0.5 0.5 0.5
44140004 50 50 50 0.5 0.5 0.5
43810005 50 50 50 0.5 0.5 0.5
43730004 50 50 50 0.5 0.5 0.5
43730005 50 50 50 0.5 0.5 0.5
27800004 50 50 50 0.5 0.5 0.5
4130002 100 100 100 0.05 0.05 0.05
4130004 1000 1000 1000 0.5 0.5 0.5
26730006 1000 1000 1000 0.5 0.5 0.5
others 1 1 1 0.01 0.01 0.01
T able A.1: SR C w eigh ting mo del in metres for p ositions ( σ X , σ Y , σ Z ) and
gon for p oin ting data ( σ φ , σ ω , σ κ ).
109
pre-adjusted Viking w eigh ts
Image No σ X σ Y σ Z σ φ σ ω σ κ
30609008 466.65 1215.66 944.42 0.325 0.123 0.296
30682813 447.66 885.56 905.40 0.199 0.095 0.156
30756611 448.02 692.62 746.18 0.144 0.081 0.104
30756617 460.72 766.88 850.31 0.154 0.080 0.111
30793514 1313.21 1199.22 1477.15 0.225 0.196 0.156
30793516 733.70 836.98 868.49 0.137 0.113 0.094
31958652 551.44 1096.07 1223.99 0.265 0.113 0.240
30627448 552.17 1376.97 1269.21 0.476 0.172 0.443
30793518 609.99 746.15 789.33 0.125 0.092 0.087
30756618 863.02 919.09 1136.50 0.195 0.123 0.173
30738070 609.31 1093.13 1346.96 0.521 0.189 0.456
30719636 476.17 807.77 1041.87 0.379 0.212 0.286
30719635 585.06 1121.50 1332.78 0.464 0.277 0.343
30682738 566.19 761.58 830.95 0.372 0.225 0.295
30682739 619.37 846.42 1276.26 0.445 0.362 0.281
30719637 579.59 1131.27 1319.47 0.442 0.299 0.316
30719639 781.21 1198.14 1431.54 0.463 0.344 0.347
33379477 18055.54 12512.32 15113.71 0.654 0.623 0.195
47290097 2067.44 2289.28 1817.14 0.215 0.174 0.107
T able A.2: W eigh ting mo del for Viking images resulted from the pre-
adjustmen t in metres for p ositions ( σ X , σ Y , σ Z ) and gon for p oin ting data
( σ φ , σ ω , σ κ ). In the initial mo del p osition w eigh ts are set to 24 km for the final
t w o images and to 5 km for the rest; p oin ting w eigh ts w ere set to 0.4 gon.
110
pre-adjusted Viking orien tation
Image No X Y Z φ ω κ
30609008 -509701.44 233275.16 -278300.01 267.91 -24.74 203.47
30682813 -412993.96 329067.86 -280656.79 261.69 -36.25 198.47
30756611 -318507.28 358711.4 -280030.15 254.78 -43.92 190.33
30756617 -363553.65 376609.38 -315200.95 255.45 -42.20 191.47
30793514 -270506.44 416395.79 -277643.96 248.27 -51.31 183.18
30793516 -285298.13 420170.54 -289207.82 249.33 -50.74 184.67
31958652 445092.62 149187.44 -531627.23 155.33 -13.14 -55.83
30627448 -426187.25 201295.38 -232174.43 267.58 -25.45 203.29
30793518 -298170.19 426946.68 -301266.05 250.13 -50.47 186.35
30756618 -370371.87 382187.65 -319847.74 253.68 -42.43 193.25
30738070 333021.49 121682.06 243868.15 59.90 -17.68 205.10
30719636 198906.29 148837.57 156295.07 57.30 -32.43 205.71
30719635 203879.1 144407.99 163983.05 58.34 -30.99 205.17
30682738 142520.48 126899.79 146962.95 48.03 -33.86 208.77
30682739 132253.28 127256.51 141588.25 48.91 -34.95 208.09
30719637 189124.81 147705.46 152857.53 58.15 -33.10 205.12
30719639 172517.79 154358.52 140094.59 58.00 -36.17 204.50
33379477 393514.46 1168304.25 1395701.66 17.54 -42.69 135.39
47290097 -497585.57 432978.41 585023.8 -44.26 -33.50 186.32
T able A.3: Pre-adjusted Viking p ositions ( X , Y , Z ) in metres and p ointing
angles ( φ , ω , κ ) in gon.
111
A.4 Cen tre of figure
The origin of the b o dy-fixed co ordinates is the cen tre of mass (COM), the
ph ysical barycen tre. It agrees with the cen troid or cen tre of figure (COF), if
the b o dy has a uniform densit y . Hence, a difference of COM and COF w ould
indicate a non-uniform mass distribution. It is no w sho wn, ho w the first order
co efficien ts ( C 10 , C 11 , ¯
C 11 ) of the surface spherical harmonics are related to the
cen ter of figure and further, what assumptions ab out the measuremen ts need
to b e made in order to p erform an analysis that allo ws reliable conclusions.
Consider a cloud of n con trol p oin ts and let in th e follo wing θ i the latitude, φ i
the longitude and r i the distance to the barycen ter for p oin t i . The cen troid
is giv en b y
M = 1
n
n
X
i =1
X i , 1
n
n
X
i =1
Y i , 1
n
n
X
i =1
Z i ! =: ( M x , M y , M z ) .
F urther let
M S = 1
n
n
X
i =1
X i
r i
, 1
n
n
X
i =1
Y i
r i
, 1
n
n
X
i =1
Z i
r i ! =: ( m x , m y , m z ) ,
the cen troid of the pro jected p oin ts whic h la ys inside the unit sphere. It is
imp ortan t to see, that M S is only related to the distribution of con trol p oin ts
on the surface and it do es not dep end on th e top ograph y . F or example, the
V esta data set yields
M =
− 1704 . 82
3185 . 18
− 19604 . 1
, M S =
− 0 . 00618
0 . 0132
− 0 . 0831
.
F rom b oth p oin ts a strong shift to w ards the southern p ole can b e noticed. M S ,
in p olar co ordinates ( − 80 ◦ , 115 ◦ ), rev eals a high concen tration of measuremen ts
around the south p ole and that M is not represen tativ e for the real planetary
ob ject. Indeed the northern p ole is not sufficien tly co v ered.
Since the in tegral ov er the unit sphere v anishes, M S is alw a ys zero, if the whole
top ograph y f is considered. Hence, the first reasonable assumption to mak e is
M S = 0 , whic h is equiv alen t to a balanced distribution of con trol p oin ts. This
is ac hiev ed if to eac h ( θ , φ ) the diametric p oin t ( − θ , φ + 180 ◦ ) is included in
the p oin t cloud, but one relies on an in terp olation metho d or a DTM to obtain
112
the heigh t v alue for the diametric p oin ts.
No w consider the appro ximation (5.2) for degree d = 1 . The p oin t cloud then
is represen ted b y a system of n linear equations whic h con tain the spherical
harmonics of order j = 0 and j = 1 .
Y 0 is a constan t function and equal to one. Hence for j = 0 the n equations
are simply giv en b y C 00 = f 1 ( θ i , φ i ) = r i . F or j = 1 it is
C 00 + C 10 sin θ i + C 11 cos θ i cos φ i + ¯
C 11 cos θ i sin φ i = r i
for eac h p oin t i . P erforming an adjustmen t just for the case d = 0 giv es the
mean radius
C 00 = 1
n
n
X
i =1
r i , (A.2)
but for d = 1 the normal equation is
N ( C 00 , C 10 , C 11 , ¯
C 11 ) T = n
X
i =1
r i
n
X
i =1
Z i ,
n
X
i =1
X i ,
n
X
i =1
Y i ! T
(A.3)
with the symmetric matrix
N =
n P z i P x i P y i
∗ P z 2
i P z i x i P z i y i
∗ ∗ P x 2
i P x i y i
∗ ∗ ∗ P y 2
i
,
x i = cos θ i cos φ i
y i = cos θ i sin φ i
z i = sin θ i .
The ∗ act as place holders for a b etter readabilit y .
Scaling equation (A.3) with 1
n leads to
1
n N C 00 , C 10 , C 11 , ¯
C 11 T = 1
n
n
X
i =1
r i , M z , M x , M y ! T
(A.4)
Since the scaling factor is applied to b oth sides, it do es not effect the v alues of
the co efficien ts. Note, that the solution v ector on the righ thand side con tains
the co ordinates of M and the first ro w of N b ecomes (1 , m x , m z , m y ) whic h
holds the co ordinates of M S . With the demand M S = 0 the first ro w (resp.
column) c hanges to the unit v ector and equation (A.4) can b e written as
C 00 = 1
n
n
X
i =1
r i whic h is the same as (A.2)
113
and the remaining
P x 2
i P x i y i P x i z i
∗ P y 2
i P y i z i
∗ ∗ P z 2
i
C 11
¯
C 11
C 10
=
M x
M y
M z
. (A.5)
The matrix w as p erm uted to align the order of co efficien ts with the principle
axes, so that the equation ab o v e can b e shortened to QC = M . If M = 0
than also C = 0 b ecause of the linear indep endence. The matrix Q is the
co v ariance matrix of the unit sphere co ordinates. It is again indep enden t of
the top ograph y and relies only on the c hoice of measuremen ts. Hence, the
relationship b et w een the cen troid M and the first order co efficien ts is giv en b y
C = Q − 1 M .
Mo ving from a discrete p oin t cloud to the con tin uous case (the whole sphere)
sho ws, that Q is constan t for all top ographies. F ollo wing the argumen tation,
that M S is alw ays zero for the con tin uous case, the side en tries of Q also v anish
and it is
λ 1 0 0
0 λ 2 0
0 0 λ 3
C = M = ⇒ C T = M x
λ 1
, M y
λ 2
, M z
λ 3 .
One can sho w λ 1 = λ 2 < λ 3 , hence C is not just the scaled COF.
This is the analytical connection b et w een the first order co efficien ts and the
cen troid. The λ i will v ary for discrete p oin t clouds with the c hoice of con trol
p oin ts, also for the same ob ject if p oin ts are remo v ed or included. They do
not c hange if only the heigh t v alues are altered (scaling).
In shap e studies it is a common praxis to force the co efficien ts to b e zero since
one wishes to consider a cen tred figure. Note, that for a balanced measuremen t
distribution, i.e. M S = 0 the co efficien ts will b e zero, if the p oin ts are shifted to
COF (see equation A.5). F or || M S || > 0 this is not true, b ecause the matrices
Q and N are not diagonal and the v alues of M S affect the solution for C :
QC = M − k C M S , k C = 1
n X r i − C T M s .
Therefore, after C is computed, the transformation C T P i is used to shift the
p oin ts iterativ ely (within a new adjustmen t) un til C = 0 is reac hed.
114
Remark: If a sto chastic mo del is use d for the adjustment, the solution ve c-
tor changes to a weighte d me an ve ctor. F urther note, that any normalisation
N ( j, m ) of the L e gendr e p olynomials P j m changes the values of the c o efficients.
This change c an b e r everse d, if the norm is known.
App endix B
Co de examples for Compressed
Sparse Matrices
The follo wing pages con tain in addition to section 4.1 the C++ class definition
of a compressed sparse matrix (csm) and selected pro duct routines of the class.
115
116
117
118
119
120
121
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