Foundations of Computational Mathematics (2023) 23:1273–1333
https://doi.org/10.1007/s10208-022-09569-5
The Moving-Frame Method for the Iterated-Integrals
Signature: Orthogonal Invariants
Joscha Diehl1·Rosa Preiß2·Michael Ruddy3·Nikolas Tapia2,4
Received: 7 April 2021 / Revised: 31 January 2022 / Accepted: 28 February 2022 /
Published online: 1 June 2022
© The Author(s) 2022
Abstract
Geometric, robust-to-noise features of curves in Euclidean space are of great inter-
est for various applications such as machine learning and image analysis. We apply
Fels–Olver’s moving-frame method (for geometric features) paired with the log-
signature transform (for robust features) to construct a set of integral invariants under
rigid motions for curves in Rdfrom the iterated-integrals signature. In particular, we
show that one can algorithmically construct a set of invariants that characterize the
equivalence class of the truncated iterated-integrals signature under orthogonal trans-
formations, which yields a characterization of a curve in Rdunder rigid motions (and
tree-like extensions) and an explicit method to compare curves up to these transfor-
mations.
Keywords Geometric invariants ·Orthogonal group ·Shuffle product ·
Log-signature ·Coordinates of the first kind ·Polynomial invariants ·Integral
invariants ·Signed volume ·Signed area ·Almost-polynomial moving-frame
Mathematics Subject Classification 60L10 ·14L24 ·53A04 ·16T05 ·22E66
Communicated by Teresa Krick.
BNikolas Tapia
1Universität Greifswald, Institut für Mathematik und Informatik, Walther-Rathenau-Str. 47,
17489 Greifswald, Germany
2Institut für Mathematik, Technische Universität Berlin, Str. des 17. Juni 136, 10623 Berlin,
Germany
3University of San Francisco, 101 Howard St., San Francisco, CA 94501, USA
4Weierstraß-Institut Berlin, Mohrenstr. 39, 10117 Berlin, Germany
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1274 Foundations of Computational Mathematics (2023) 23:1273–1333
Nomenclature
chThe coordinate of c≤n∈g≤n(( Rd)) corresponding to the Hall
basis element bh
c≤nAn element of g≤n(( Rd)) with coordinates given by ci1i2···imfor
m≤n
G(R)A real variety with associated complex variety G, of which it is
also a subgroup
g≤n(( Rd)) The free step-nnilpotent Lie algebra over Rd
GzThe stabilizer of a point z, the largest subgroup of Gthat keeps z
invariant
IdThe set of rational invariants defining Ud(C)
IIS(Z)The iterated-integrals signature of the curve Z
IMThe set of polynomial invariants generating C(Cd⊕sod(C))Od(C)
given by φk·φk,1≤k<d
IWd(C)The generating set for C(L(d−1)
d)Wd(C)given by σd−1(Id)
k(X)The field of rational functions on the variety Xwith coefficients
in k
k(X)GThe subfield of k(X)of rational invariants for the action of Gon
X
K2,≤2The cross-section for the action of O2(R)on U2;≤2
Kd;≤nThe cross-section for the action of Od(R)on Ud;≤n
k[X]The ring of polynomial functions on the variety Xwith coefficients
in k
k[X]GThesubringofk[X]of polynomial invariants for the action of G
on X
L(d−1)
dThe relative Wd(C)-section for the action of Od(C)on Cd⊕
sod(C)
LdThe Lyndon words over the alphabet {1,...,d}
L(d−1);R
dThe intersection of L(d−1)
dand Rd⊕sod(R)
L(i)
dThe relative Nd−i
d(C)-section for the action of Od(C)on Cd⊕
sod(C)
log(IIS(Z)) The log-signature of the curve Z
Ni
d(C)The product of the groups Oi
d(C)and Wd(C); the normalizer of
L(d−i)
d
Oi
d(C)The subgroup of Od(C)isomorphic to Oi(C)which leaves the
last d−icomponents of a Cdvector invariant
φkThe map φk:Cd⊕sod(C)→Cd,(v,M)→ Mkv
proj≤nThe canonical projection proj≤n:T(( Rd)) →T≤n(( Rd))
proj≤n→≤2The canonical projection proj≤n→≤2:g≤n(( Rd)) →g≤2(( Rd))
˜ρ2The moving-frame map ˜ρ2:U2;≤2→O2(R)for the action of
O2(R)on U2;≤2
ρ2The moving-frame map for the action of O2(R)on U2;≤n, where
ρ2(c≤n)=˜ρ2(proj≤2c≤n)
˜ρdThe moving-frame map for the action of Od(R)on Ud;≤2
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Foundations of Computational Mathematics (2023) 23:1273–1333 1275
ρdThe moving-frame map for the action of Od(R)on Ud;≤n, where
ρd(c≤n)=˜ρd(proj≤2c≤n)
σiThe field isomorphism σi:C(Cd⊕sod(C))Od(C)→C(L(i)
d)Nd−i
d(C)
so(d,R)The space of skew-symmetric Rd×dmatrices
U2;≤2The domain of the moving-frame ˜ρ2, a Zariski-open subset of
g≤2(( R2))
Ud;≤nThe domain of the moving frame ρd, a Zariski-open subset of
g≤n(( Rd))
Ud(C)The Zariski-open subset of Cd⊕sod(C)where none of the invari-
ants in Idvanishes
Ud(R)The intersection of Ud(C)and Rd⊕sod(R), a Zariski open subset
of Rd⊕sod(R)
Wd(C)The group of diagonal matrices with diagonal entries in {−1,1};
the normalizer of L(d−1)
d
X(R)A real variety with associated complex variety X
Contents
1 Introduction ............................................ 1275
2 Preliminaries ............................................ 1278
2.1 The Tensor Algebra ...................................... 1278
2.2 The Iterated-Integrals Signature ................................ 1283
2.3 Invariants ........................................... 1285
2.4 Moving-Frame Method .................................... 1289
2.5 Algebraic Groups and Invariants ............................... 1292
3 Rigid-Motion Invariant Iterated-Integrals Signature in Low Dimensions ............. 1295
3.1 Planar Curves ......................................... 1296
3.2 Spatial Curves ......................................... 1300
4 Orthogonal Invariants on g≤2(( Rd)) ................................ 1307
5O
d(R)-Invariant Iterated-Integrals Signature ........................... 1319
5.1 Moving Frame on g≤n(( Rd)) ................................. 1319
5.2 Toward a Fundamental Set of Polynomial Invariants ..................... 1325
6 Discussion and Open Problems .................................. 1330
References ............................................... 1331
1 Introduction
A central problem in image science is constructing geometrically relevant features
of curves that are robust to noise. In this sense, rigid motions of space make up a
natural group of “nuisance” transformations of the data. For this reason, rotation- and
translation-invariant features are often desired, for instance, in human activity recog-
nition [39, Section 6] or in matching contours [52]. Classically, differential invariants
such as curvature have been used for this purpose [25], and more recently, integral
invariants of curves have been of interest [13,16]. In this work, we construct a rigid
motion-invariant representation of a curve through its iterated-integrals signature by
applying the Fels–Olver moving-frame method. We show that this yields sets of inte-
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1276 Foundations of Computational Mathematics (2023) 23:1273–1333
gral invariants that characterize the truncated iterated integral signature up to rigid
motions.
Iterated integrals, a subject of study introduced by Chen in the 50s [6,8], will be
properly reviewed in Sect. 2.2. In a nutshell, they are descriptive features of continuous
curves that moreover possess desirable stability properties. Regarding their use for
invariant theory, we consider two concrete examples, reproduced from [13]. Given a
smooth curve X=(X(1),X(2)):[0,1]→R2, starting at X0=0, the norm squared
of total displacement is clearly invariant to the orthogonal group O2(R)actingonthe
ambient space. Using the fundamental theorem of analysis, we can write this invariant
as
||X1||2=(X(1)
1)2+(X(2)
1)2
=21
0
X(1)
r˙
X(1)
rdr+21
0
X(2)
r˙
X(2)
rdr
=21
0r
0˙
X(1)
udu˙
X(1)
rdr+21
0r
0˙
X(2)
udu˙
X(2)
rdr
=: 21
0r
0
dX(1)
udX(1)
r+21
0r
0
dX(2)
udX(2)
r,
where we introduced the shorthand dX(i)
t:= ˙
X(i)
tdt. We have expressed this invariant
as the linear combination of iterated integrals. A less trivial invariant is given by the
square1of the signed area2enclosed by the curve (for simplicity assume that the
curve is closed, i.e., X1=0). By Green’s theorem (see [48, Theorem 10.33] and [36,
Proposition 1]), the signed area can be expressed in terms of iterated integrals, namely
as
1
21
0r
0
dX(1)
udX(2)
r−1
0r
0
dX(2)
udX(1)
r.
These examples illustrate that simple, and geometrically relevant, invariants can be
found in the collection of iterated integrals.
The Fels–Olver moving-frame method, introduced in [15], is a modern generaliza-
tion of the classical moving-frame method formulated by Cartan [3]. In the general
setting of a Lie group Gacting on a manifold M, a moving frame is defined as a
G-equivariant map from Mto G. A moving frame is determined by a choice of cross-
section to the orbits of Gand hence a unique “canonical form” for elements of M
under G. Thus, the moving-frame method provides a framework for algorithmically
constructing G-invariants on Mthat characterize orbits and for determining equiva-
lence of submanifolds of Munder G.
The moving-frame method has been used to construct differential invariants of
smooth planar and spatial curves under Euclidean, affine, and projective transforma-
1The signed area is an SO2(R)invariant; however, only its square (resp. its absolute value) is an O2(R)
invariant.
2For more on the specific relevance of signed area in the study of the iterated-integrals signature, see [12].
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Foundations of Computational Mathematics (2023) 23:1273–1333 1277
tions, and, in certain cases, these differential invariants lead to a differential signature,
which can be used to classify curves under these transformation groups [2]. The dif-
ferential signature has been applied in a variety of image science applications from
automatic jigsaw puzzle assembly [26] to medical imaging [22]. Also in the realm
of image science, the moving-frame method has been used to construct invariants of
grayscale images [1,51].
We consider the induced action of the orthogonal group of rotations on the log-
signature of a curve, which provides a compressed representation of a curve obtained
by applying the log transform to the iterated-integrals signature, and provide an explicit
cross-section for this action. We show that for most curves and any truncation of the
curve’s log-signature, the orbit is characterized by the value on this cross-section. As
a consequence, a curve is completely determined up to rigid motions and tree-like
extensions by the invariantization of its iterated-integrals signature induced by this
cross-section.
This yields a constructive method to compare curves up to rigid motions and to
evaluate invariants that characterize the iterated-integrals signature under rotations.
These invariants are constructed from integrals on the curve and hence are likely to
be more noise-resistant than their differential counterparts such as curvature. One can
easily set up an artificial example where this is visible. Consider, for instance, the
circle of radius n−3/2given by the parameterization γ:[0,1]→R2where
γ(t)=(x(t), y(t)) =cos(2πnt)
n3/2,sin(2πnt)
n3/2,
which as n→∞converges to the constant curve (at the origin). Now the curvature of
this curve does not converge (in fact, it blows up). In contrast, the iterated integrals do
all converge (to zero) since γconverges in variation norm. Then, the invariants built
out of the iterated integrals (Sect. 5.1) also converge to their value on the zero curve.
On this toy example, these integral invariants are hence more “stable”. More precisely,
iterated integrals are continuous in p-variation norm, for p<2, [20, Proposition 6.11],
thus even covering paths that are not even differentiable. Curvature is continuous only
in the (much) stronger C2-norm.
Additionally, in contrast to the methods in [13], the resulting set of integral invariants
is shown to uniquely characterize the curve under rotations, and moreover, does so in
a minimal fashion. Since the iterated-integrals signature of a curve is automatically
invariant to translations, this provides rigid motion-invariant features of a curve, which
can be used for applications such as machine learning or shape analysis.
This work is structured as follows: In Sect. 2, we provide background on the iterated-
integrals signature and the moving-frame method, as well as some facts about algebraic
groups and invariants. In Sect. 3, we construct the moving-frame map for paths in
R2and R3motivating the construction of the moving-frame map for Rd.Wealso
provide explicit sets of invariants at these lower dimensions, which might be useful
for applications. In Sect. 4, we consider the orthogonal action on the second-order
truncation of the log-signature over the complex numbers. Using tools from algebraic
invariant theory, we construct the linear space, which will form the basis for the
cross-section in the following section. We also provide an explicit set of polynomial
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