Möbius Invariants of Shapes and Images

Möbius Invariants of Shapes and Images

Stephen MARSLAND † and Robert I. MCLACHLAN ‡

S. Marsland and R.I. McLachlan

† School of Engineering and Advanced Technology, Massey University,
† Palmerston North, New Zealand<EMAIL_ADDRESS>http://www.stephenmonika.net

‡ Institute of Fundamental Sciences, Massey University, Palmerston North, New
Zealand<EMAIL_ADDRESS>http://www.massey.ac.nz/~rmclachl/

Received April 01, 2016, in final form August 08, 2016; Published online
August 11, 2016

Identifying when different images are of the same object despite changes
caused by imaging technologies, or processes such as growth, has many
applications in fields such as computer vision and biological image analysis.
One approach to this problem is to identify the group of possible
transformations of the object and to find invariants to the action of that
group, meaning that the object has the same values of the invariants despite
the action of the group. In this paper we study the invariants of planar
shapes and images under the Möbius group $\mathrm{PSL}(2,\mathbb{C})$, which
arises in the conformal camera model of vision and may also correspond to
neurological aspects of vision, such as grouping of lines and circles. We
survey properties of invariants that are important in applications, and the
known Möbius invariants, and then develop an algorithm by which shapes can be
recognised that is Möbius- and reparametrization-invariant, numerically
stable, and robust to noise. We demonstrate the efficacy of this new invariant
approach on sets of curves, and then develop a Möbius-invariant signature of
grey-scale images.

invariant; invariant signature; Möbius group; shape; image

68T45; 68U10

##  1 Introduction

Lie group methods play a fundamental role in many aspects of computer vision
and image processing, including object recognition, pattern matching, feature
detection, tracking, shape analysis, tomography, and geometric smoothing. We
consider the setting in which a Lie group $G$ acts on a space $M$ of objects
such as points, curves, or images, and convenient methods of working in $M/G$
are sought. One such method is based on the theory of invariants, i.e., on the
theory of $G$-invariant functions on $M$, which has been extensively developed
from mathematical, computer science, and engineering points of view.

When $M$ is a set of planar objects the most-studied groups are the Euclidean,
affine, similarity, and projective groups. In this paper we make a first study
of invariants of planar objects under the Möbius group
$\mathrm{PSL}(2,\mathbb{C})$, which acts on the Riemann sphere
$\overline{\mathbb{C}}=\mathbb{C}\cup\infty$ by

$\displaystyle\phi\colon\
\overline{\mathbb{C}}\to\overline{\mathbb{C}},\qquad\phi(z)=\frac{az+b}{cz+d},\qquad
a,b,c,d\in\mathbb{C},\qquad ad-bc\neq 0.$

We work principally in the school of invariant signatures, developed by
(amongst others) Olver and Shakiban [4, 9, 18, 30, 35, 36], and widely used
for Euclidean object recognition (see, e.g., [2]).

$\mathrm{PSL}(2,\mathbb{C})$ is a 6-dimensional real Lie group. It forms the
identity component of the inversive group, which is the group generated by the
Möbius transformations and a reflection. In addition to its importance on
fundamental grounds – it is one of the very few classes of Lie groups that act
on the plane, it crops up in numerous branches of geometry and analysis, it is
the smallest nonlinear planar group that contains the direct similarities, and
it is the set of biholomorphic maps of the Riemann sphere – it also has direct
applications in image processing since it arises in the conformal camera model
of vision, in which scenes are projected radially onto a sphere [20, 39, 40].
It may also correspond to neurological aspects of vision, such as grouping of
lines and circles (which are equivalent under Möbius transformations) [41].

From an applications point of view, different objects may be related by Möbius
transformations, as is explored by Petukhov [34] for biological objects in
fascinating detail in the context of Klein’s Erlangen program111D’Arcy
Wentworth Thompson [38] famously deformed images of one species to match those
of another; his theory of transformations is reviewed and interpreted in light
of modern biology in [43]. In particular, it has given rise to image
processing techniques such as Large Deformation Diffeomorphic Metric Mapping
$($LDDMM$)$ [15], in which images are compared modulo infinite-dimensional
groups such as the diffeomorphism group. Petukhov writes that Thompson “did
not use the Erlanger program as the basis in this comparative analysis.”
However, the totality of Thompson’s examples and the explanations in his text
do indicate that in all cases he selected his transformations from the
simplest group that would do an (in his view) acceptable job. Eight different
groups are identified in [22, Table 1]; four are finite-dimensional. Thus,
whether consciously or not, Thompson’s work was fully consistent with the
Erlangen program. Of relevance to the present paper is that many of his
examples use conformal mappings, and thus may be approximated (or even
determined by) Möbius transformations., giving examples of many body parts
that are loxodromic to high accuracy (loxodromes have constant Möbius
curvature and play the role in Möbius geometry that circles do in Euclidean
geometry), that change shape by Möbius and other Lie group actions, and that
grow via Möbius transformations (including 2D representations of the human
skull). Other examples of 1-dimensional growth patterns, such as antenatal and
postnatal human growth, appear to be well modelled by 1-dimensional linear-
fractional transformations $x\mapsto(ax+b)/(cx+d)$. Discussing this work,
Milnor [25] argued that “The geometrically simplest way to change the relative
size of different body parts would be by a conformal transformation. It seems
plausible that this simplest solution will often be the most efficient, so
that natural selection tend to choose it”. (Milnor was thinking of 3D
conformal transformations, whose restriction to 2D is the Möbius
transformations.)

In this paper we present an integral invariant for the 2D Möbius group that is
reparameterization independent, and demonstrate its use to identify curves
that are related by Möbius transformations. We then consider the case of
images, and describe an invariant signature by which Möbius transformed images
can be recognised. We begin in Section 2 by discussing the desirable
properties of invariants for object classification, and introduce the property
of bounded distortion, which subsumes most of the requirements identified in
the literature. This is followed by an overview of the methods that have been
used to give object classification invariants for curves and images (most
often with respect to the Euclidean, similarity, and affine groups).

In Section 3 we present the classical Möbius invariants and discuss their
utility with respect to the properties identified in Section 2, before using a
numerical example based on an ellipse to demonstrate the difference between
them. This is followed by the introduction of the invariant that we have
identified as the best behaved for curves with respect to the requirements
previously discussed. In order to demonstrate its utility, we present an
experiment where a set of smooth Jordan curves are created, and then the
invariant distance is computed, and compared with direct registration of each
pair of curves in the Möbius group using the $H^{1}$ similarity metric. The
results show that the invariant is well-behaved with respect to noise, and can
be used to separate all but the most similar shapes.

In Section 4 we move on to images and demonstrate the use of a 3D Möbius
signature that is very sensitive and relatively cheap to compute, while still
being more robust than the analogous signature for curves. As far as we know,
such differential invariant signatures (in the sense of Olver [30]) for images
have not been considered previously.

##  2 Invariants of shapes and images

###  2.1 Computational requirements of invariants for object classification

The mathematical definition of an invariant, namely, a $G$-invariant function
on $M$, is not sufficiently strong for many computational applications. For
example, object classification via invariants involves comparing the values of
the invariant on different $G$-orbits, and the definition says nothing about
this. Recognising this, Ghorbel [14] has given the following partial list of
qualities needed for object recognition:

  * (i)

fast computation;

  * (ii)

good numerical approximation;

  * (iii)

powerful discrimination (if two objects are far apart modulo $G$, their
invariants should be far apart);

  * (iv)

completeness (two objects should have the same invariants iff they are the
same modulo $G$);

  * (v)

provide a $G$-invariant distance on $M$; and

  * (vi)

stability (if two objects have nearby invariants, they should be nearby modulo
$G$).

Calabi et al. [9] include in (ii) the requirement that the numerical
approximation itself should be $G$-invariant, while Abu-Mostafa et al. [1] add
the further desirable qualities of:

  * (vii)

robustness (if two objects are nearby modulo $G$, especially when one is a
noisy version of the other, their invariants should be nearby);

  * (viii)

lack of redundancy (i.e., all invariants are independent); and

  * (ix)

lack of suppression (in which the invariants are insensitive to some features
of the objects).

Manay [21] includes, in addition:

  * (x)

locality (which allows matching subparts and matching under occlusion).

To this already rather demanding list we add one more, that the set of
invariants should be:

  * (xi)

small.

Figure 1: Sketch of the approximate relationships between $d_{G}(x,y)$, which
measures the distance between two objects $x$ and $y$ modulo a symmetry group
$G$, and $\|I(x)-I(y)\|$, which measures the distance between their
invariants, illustrating the role of the desired properties of completeness,
robustness, stability, and discrimination. In object recognition, ‘good
discrimination’ is sometimes taken to mean the avoidance of type I errors
(relative to the null hypothesis that two objects are in the same group orbit,
so that in a type I error dissimilar objects are classified as similar), the
avoidance of type II errors (similar objects being classified as dissimilar),
or is not specified; we take it to mean that both type I and type II errors
are avoided.

The motivation for this last is purely a parsimony argument; it is easy to
produce large sets of invariants without adding much utility, and smaller sets
of easily computable invariants are to be preferred. This is particularly the
case since some of these criteria are in conflict, so there will usually not
be one invariant that is preferred for all applications; instead, the best
choice will depend on the dataset and the particular application. Others are
closely related, especially discrimination, distance, stability, robustness,
and suppression; these all depend on which features of the objects are deemed
to be signal and which are deemed to be noise.

These criteria can be unified and quantified in the following way: suppose
that a distance on objects has been chosen that measures the features that we
are interested in (the signal), and that is small for differences that we are
not interested in (the noise). This distance induces a distance on objects
modulo $G$ by (where $\|\cdot\|_{d}$ is some appropriately chosen norm):

$\displaystyle d_{G}(x,y):=\max\big{(}\inf_{g\in G}\|g\cdot
x-y\|_{d},\,\inf_{g\in G}\|g\cdot y-x\|_{d}\big{)}.$

(In principle one can compute $d_{G}(x,y)$ by optimizing over $G$, and indeed
this is done in many applications. However, optimization is relatively
difficult and unreliable, and becomes impractical on large sets of objects;
this is the step that invariants are intended to avoid.) Suppose that a norm
on invariants $I\colon M\to\mathbb{R}^{k}$ has been chosen. Then we can
express the criteria as follows:

  * (iii)

discrimination:
$d_{G}(x,y)\gg\varepsilon\Leftrightarrow\|I(x)-I(y)\|\gg\varepsilon$,

  * (iv)

completeness: $\|I(x)-I(y)\|=0\Rightarrow d_{G}(x,y)=0$,

  * (vi)

stability: $\|I(x)-I(y)\|\lesssim\varepsilon\Rightarrow
d_{G}(x,y)\lesssim\varepsilon$,

  * (vii)

robustness:
$d_{G}(x,y)\lesssim\varepsilon\Rightarrow\|I(x)-I(y)\|\lesssim\varepsilon$.

These may be subsumed and strengthened by the property of bounded distortion:
an invariant has bounded distortion with respect to $d_{G}$ if there exist
positive constants $c_{1}$, $c_{2}$ such that for all $x,y\in M$ we have

$\displaystyle c_{1}d_{G}(x,y)\leq\|I(x)-I(y)\|\leq c_{2}d_{G}(x,y).$ (2.1)

Suppose the invariant is to be used to test the hypothesis that two objects
are the same modulo symmetry group $G$. The situation is illustrated in Fig.
1, which plots the distance between the invariants of two objects against the
distance between the objects modulo $G$. The smaller the value of $c_{2}$, the
more robust the invariant is, and the fewer false positives will be reported.
The larger the value of $c_{1}$, the more stable the invariant is, the better
its discrimination, and the fewer false negatives will be reported. If (2.1)
holds only with $c_{1}=0$, the invariant is neither stable nor complete; if
there is no $c_{2}$ such that (2.1) holds, the invariant is not robust. In
addition, in some applications one may be more interested in some parts of the
space shown in Fig. 1 than others: for example, in discriminating very similar
or very dissimilar objects.

In practice, bounded distortion may not be possible. Even completeness, the
centrepiece of the mathematical theory of invariants, can be very demanding.
It turns out that many invariants that are roughly described as ‘complete’ are
not fully complete, that is, they do not distinguish all group orbits, but
only distinguish almost all group orbits. For example, Ghorbel’s Euclidean
invariants of grey-level images [14] only distinguish images in which two
particular Fourier coefficients are non-zero. Thus, it will be stable only on
images in which these two Fourier coefficients are bounded away from zero. The
most commonly-used Euclidean signature of curves, $(\kappa,\kappa_{s})$, is
complete only on nondegenerate curves [17]. The fundamental issue is that even
for very simple group actions, the space of orbits can be very complicated
topologically, and thus very difficult (or even impossible) to coordinatize
via invariants. We illustrate these ideas via an example:

######  Example 2.1.

Consider $n$ points $z_{1},\dots,z_{n}$ in the complex plane. Let $G=S^{1}$
act by rotating the points, i.e.,
$e^{i\theta}\cdot(z_{1},\dots,z_{n})=(e^{i\theta}z_{1},\dots,e^{i\theta}z_{n})$.
Then $\\{\bar{z}_{i}z_{j}\colon 1\leq i\leq j\leq n\\}$ forms a complete set
of invariants. It is very large (there are $n^{2}$ real components) compared
to the dimension $2n-1$ of the space, $\mathbb{C}^{n}/S^{1}$, in which we are
trying to work. However, if any one real component of the set is omitted, the
resulting set is not complete. For example, if $|z_{1}|^{2}$ is omitted, then
the points $(1,0,\dots,0)$ and $(2,0,\dots,0)$ have the same invariants, but
do not lie on the same orbit. This is the situation considered in the problem
of phase retrieval [6]. By choosing combinations of $\\{\bar{z}_{i}z_{j}\\}$
it is possible to create smaller sets of complete invariants, but the
computational complexity of calculating the description of the $n$ points is
still $\mathcal{O}(n^{2})$ [6].

In cases where one settles for an incomplete set of invariants, or a set that
is not of bounded distortion – so that the worst-case behaviour of the
invariants is arbitrarily poor – it can make sense instead to study the
average-case behaviour of the invariants over some distribution of the
objects. This is analogous to the role that ill-posed and ill-conditioned
problems play in numerical linear algebra, although in that case the ill-
conditioning is intrinsic rather than imposed by considerations of
computational complexity. Indeed, one can sometimes very usefully describe
objects based on an extremely small number of invariants, for example,
describing planar curves by their length and enclosed area, or by the first
few Fourier moments of their curvature. What is wanted is to optimize the
behaviour of the invariants, over some distribution of the objects, with
respect to their time and/or space complexity. However, we know of no genuine
cases in which such a program has been carried out.

###  2.2 Invariants of curves

There is a large literature concerning invariants to the Euclidean and
similarity groups for both curves and images. The purpose of this section is
to provide an overview of the dominant themes within that work that are
relevant to our goal of identifying invariants for the Möbius group that
satisfy at least some of the criteria listed in the previous section.

We define a (closed) shape as the image of a function $\phi\colon
S^{1}\to\mathbb{R}^{2}$. For different restrictions on $\phi$ this defines
different spaces: if $\phi$ is a continuous injective mapping then this
defines simple closed curves, while if $\phi$ is differentiable and
$\phi^{\prime}(t)\neq 0$ for all $t$ this defines the ‘shape space’
$\operatorname{Imm}(S^{1},\mathbb{R}^{2})/\operatorname{Dif\/f}(S^{1})$ [26],
while if $\phi$ is an immersion and $\phi(S^{1})$ is diffeomorphic to $S^{1}$,
we obtain the shape space
$\operatorname{Emb}(S^{1},\mathbb{R}^{2})/\operatorname{Dif\/f}(S^{1})$
(roughly, the curve has no self-intersections). There are also smooth
($C^{\infty}$) versions of these, piecewise smooth versions, shapes of bounded
variation, and so on. The geometry of these shape spaces is now studied
intensively in its own right [23, 24] as well as as a setting for computer
vision; see [8] for a recent review.

The quotient by $\operatorname{Dif\/f}(S^{1})$ in the shape spaces has the
effect of factoring out the dependence on the parameterization. Various
methods have been used to achieve this, including:

  1. 1.

Using a standard parameterization, such as Euclidean arclength. This leaves a
dependence on the choice of starting point, i.e., the subgroup of
$\operatorname{Dif\/f}(S^{1})$ consisting of translations is not removed.

  2. 2.

Moment invariants, $\int_{0}^{L}f(\phi(s))\,\mathrm{d}s$, where $L$ is the
length of the curve, $s$ is Euclidean arclength and $f$ ranges over a set of
basis functions, such as monomials or Fourier modes [1].

  3. 3.

Currents $\int_{S^{1}}\phi^{*}\alpha$, where $\alpha$ ranges over a set of
basis 1-forms on $\mathbb{R}^{2}$, such as monomials times $\mathrm{d}x$ and
monomials times $\mathrm{d}y$ [15].

The next step is to consider a Lie group $G$ acting on $\mathbb{R}^{2}$ that
induces an action on the shape space. Here, many methods and types of
invariants have been investigated (see, e.g., [42]).

The moving frame method

is a general approach to constructing invariants. Objects are put into a
reference configuration and their resulting coordinates are then invariant.
The moving frame or reference configuration method was developed as a way of
finding differential invariants and variants of them (see [31] for an
overview). A simple application of the method is the situation considered in
Example 2.1, of $n$ points in the plane under rotations. Consider
configurations with $z_{1}\neq 0$, $|\arg z_{1}|<\pi$. Rotate the
configuration so that $z_{1}$ lies on the positive real axis. The coordinates
of the resulting reference configuration, $\bar{z}_{1}z_{j}/|z_{1}|^{2}$, are
invariant. In this case, the invariants are well behaved as $|\arg
z_{1}|\to\pi$, but not as $z_{1}\to 0$.

Joint invariants

are functions of several points; for example, pairwise distances for a
complete set of joint invariants for the action of the Euclidean group on sets
of points in the plane.

Differential invariants

are functions of the derivatives of a curve at a point. There exist algorithms
to generate all differential invariants [31]. For the action of the Euclidean
group on planar curves, the Euclidean curvature
$\kappa=\phi^{\prime}\times\phi^{\prime\prime}/\|\phi^{\prime}\|^{3}$ is
$E(n)$ invariant (and parameterization invariant), and its derivatives
$d^{n}\kappa/ds^{n}$ with respect to arclength form a complete set of
differential invariants.

Semi-differential invariants

(also known as joint differential invariants [29]) of a curve are functions of
several points and derivatives.

Integral invariants

are formed from the moments or the partial moments
$\int_{s_{0}}^{s}f(\phi(t))\,\mathrm{d}t$. With some care they can be made
parameterization- and basepoint-independent [11, 21]. Initially, these
invariants appear to have some advantages, being relatively robust and often
including some locality. However, they are not always applicable; for example
[21], which used regional integral invariants, still requires a point
correspondence optimization in order to get a distance between shapes. In
addition, groups such as the projective group do not act on any finite subset
of the moments [42]. Astrom [5] shows that there are no stable projective
invariants for closed planar curves. While local integral invariants are
promising, as reported by [16], there may be analytic difficulties in deriving
them.

Another example of the moving frame method, which is common in image
processing and shape analysis, is centre of mass reduction. The centre of mass
of a shape may be moved to the origin in order to remove the translations.
This may be calculated by, for example,
$\int_{\phi(S^{1})}(x^{2}/2,xy)\mathrm{d}y=\iint_{\mathrm{int}\phi(S^{1})}(x,y)\mathrm{d}x\mathrm{d}y$.
However, the shapes $x=a+\sin t$, $y=0$ have centre of mass equal to 0 for all
values of $a$, even though they are related by translations. Thus, this method
would not be robust on any dataset that contained shapes approaching such
degenerate shapes. For rotations, the reference configuration method will not
be robust if there are objects close to having a discrete rotational symmetry.
The underlying problem with the reference configuration approach is that it is
attempting to use a set of invariants equal to the dimension of the desired
quotient space $M/G$. This space is almost always non-Euclidean, so such a set
can only be found on some subset of $M/G$ of Euclidean topology. The set is
then robust only on datasets that are bounded away from the boundary of this
subset.

Calabi et al. [9] propose the use of differential invariant signatures for
shape analysis, and further argue that these should be approximated in a
group-invariant way. For example, for the Euclidean group, the signature is
the shape $(\kappa,\kappa_{s})$ regarded as a subset of $\mathbb{R}^{2}$. The
claimed advantages of the approach are that the signature determines the
shape; that it does not depend on the choice of initial point on the curve or
on parameterization by arclength, and that its $G$-invariance makes it robust;
and that it is based on a general procedure for arbitrary objects and groups.
See [2, 4, 9, 18, 30, 35, 36] for further developments and applications of the
invariant signature.

Although the signature at first sight appears to be complete (e.g., Theorem
5.2 in [9]), a more detailed treatment (e.g., [17, 28]) highlights the fact
that it is not complete on shapes that contain singular parts – straight and
circular segments in the Euclidean case. For these parts, the signature
reduces to a point. Thus, for shapes that have nearly straight or nearly
circular segments, the signature cannot be robust. In addition, while the
signature does not depend on the starting point or the parameterization, it
takes values in a very complicated set, namely, the planar shapes. To compare
the invariants of two shapes requires comparing two shapes. Essentially, the
parameterization-dependence has only been deferred to a later stage of the
analysis (unless one is content to compare shapes visually). One approach to
this is to weight the signature, see [32] for more details.

The claim in [9] that the method’s $G$-invariance makes it robust should be
assessed through further analysis and experiment. It would appear to be most
relevant for datasets in which the errors due to the presentations of the
shapes are comparable to those resulting from the errors in the shapes
themselves (i.e., noise) and from the distribution of the objects in a
classification problem. Their final point, that the method is extremely
general, is a powerful one. Calabi [9] carry out the procedure for the
Euclidean and for the 2D affine group. However in Section 6.7 they report that
“the interpolation equations in general are transcendentally nonlinear and do
not admit a readily explicit solution”, indicating that the method may not
succeed for all group actions.

It is also possible to represent the shape as a binary image and apply image-
based invariants (which are described next). This has the advantage of working
directly in the space $\mathbb{R}^{2}$ on which the group acts, and avoiding
all questions of parameterization, etc., but it does sacrifice a lot of
information about the shape. A related approach, which is popular in PDE-based
method for curves, is to represent the shape as the level set of a smooth
function $\phi\colon\mathbb{R}^{2}\to\mathbb{R}$. This is also
parameterization-independent, and retains smoothness, but we have never seen
it used in for constructing invariants.

###  2.3 Invariants of images

Most studies of image invariants has been based on grey-level images
$f\colon\Omega\to[0,1]$, where $\Omega\subset\mathbb{R}^{2}$. The methods are
primarily based on moments [1] or Fourier transforms [14, 19] of the images,
and have been highly developed for the translation, Euclidean, and similarity
groups, where the linearity of the action and the special structure of these
Lie groups means that the approach is particularly fast and robust. Attempts
have been made to extend the method of Fourier invariants to other groups.
There is an harmonic analysis for many non-Abelian Lie groups, including, in
fact, the Möbius group [37], as well as a general theory for compact non-
commutative groups [13]. There are some applications of this theory to image
processing [19, 41] and to other problems in computational science. Fridman
[12] discusses a Fourier transform for the hyperbolic group, the 3-dimensional
subgroup of the Möbius group that fixes the unit disc. However, the theory
appears not to have been developed to the point where it can be used as
effectively as the standard Fourier invariants. Therefore, in this paper, in
Section 4, we develop a Möbius invariant of images, based on a differential
invariant signature of the image.

##  3 Möbius invariants for curves

In this section we consider invariants of curves for the Möbius group, and
derive one suitable for practical computation, demonstrating its application
for a set of simple closed curves.

###  3.1 Known Möbius invariants

The most well-known invariant of the Möbius group is the cross-ratio, also
known as the wurf, which is based on the ratio of the distances between a set
of 4 points:

$\displaystyle\mathrm{CR}(z_{1},z_{2},z_{3},z_{4}):=\frac{(z_{1}-z_{3})(z_{2}-z_{4})}{(z_{2}-z_{3})(z_{1}-z_{4})},$

where the invariance means that
$\mathrm{CR}(Tz_{1},Tz_{2},Tz_{3},Tz_{4})=\mathrm{CR}(z_{1},z_{2},z_{3},z_{4})$
for Möbius transformation $T$.

Since the Möbius group can send any triple of distinct points in
$\overline{\mathbb{C}}$ into any other such triple, there are no joint
invariants of 2 or 3 points; the orbits are the configurations consisting of
1, 2, and 3 distinct points. When $n>4$, the set of cross-ratios of any four
of the points forms a complete invariant.

For large numbers of points, this set of all cross-ratios has cardinality
$\mathcal{O}(n^{4})$ which is impractically large (although some may be
eliminated using functional relations amongst the invariants, known as
syzygies [29]). However, if the dataset of shapes or images is tagged with a
small number of clearly-defined landmarks, then some subset of the set of
cross-ratios may form a useful invariant. This is the method by which Petukhov
[34] was able to identify linear-fractional, Möbius, and projective
relationships in biological shapes. For untagged objects, automatic tagging
may be possible using critical points (e.g., maxima and minima) of images, and
their values; these are homomorphism- and hence Möbius-invariant, and can be
identified, even in the presence of noise, by the method of persistent
homology [10]. However, such invariants are clearly highly incomplete for
shapes and images, and we do not study them further here.

In order to derive differential invariants for the Möbius group, the most
useful starting point is the Schwarzian derivative

$\displaystyle(Sz)(t)=\left(\frac{z^{\prime\prime}}{z^{\prime}}\right)^{\prime}-\frac{1}{2}\left(\frac{z^{\prime\prime}}{z^{\prime}}\right)^{2}=\frac{z^{\prime\prime\prime}}{z^{\prime}}-\frac{3}{2}\left(\frac{z^{\prime\prime}}{z^{\prime}}\right)^{2}.$
(3.1)

By an abuse of notation, which is standard in the literature (see, for
example, the very readable [3]), the same formula (3.1) is used in three
different situations: when $z\colon\mathbb{R}\to\mathbb{R}$ (used in studying
linear-fractional mappings in real projective geometry); when
$z\colon\overline{\mathbb{C}}\to\overline{\mathbb{C}}$ (used in studying
complex analytic mappings); and when $z\colon M\to\overline{\mathbb{C}}$, $M$
a real 1-dimensional manifold (used in studying the Möbius geometry of
curves). We adopt the latter setting so that $z^{\prime}$ is the tangent to
the curve. The Schwarzian derivative is then invariant under Möbius
transformations $\phi$:

$\displaystyle S(\phi\circ z)(t)=S(\phi)(t)$

and under reparameterizations $\psi\colon M\to M$ transforms as

$\displaystyle S(z\circ\psi)=(S(z)\circ\psi)\cdot(\psi^{\prime})^{2}+S(\psi),$

where the last term is the real Schwarzian derivative. Therefore (where
$\operatorname{Im}$ represents the imaginary part of a complex number)

$\displaystyle\operatorname{Im}S(z\circ\psi)=(\operatorname{Im}S(z)\circ\psi)\cdot(\psi^{\prime})^{2}.$

This can be used to construct a distinguished Möbius-invariant
parameterization of the curve. Let $\tilde{z}(\lambda)=z(t)$ where
$\lambda=\psi(t)$. The parameter $\lambda$ will be chosen so that
$\operatorname{Im}S(\tilde{z})\equiv 1$. This gives

$\displaystyle\operatorname{Im}S(z)=\operatorname{Im}S(\tilde{z}\circ\psi)=(\operatorname{Im}S(\tilde{z})\circ\psi)\cdot(\psi^{\prime})^{2}=(\psi^{\prime})^{2}.$

The choice $\psi^{\prime}(t)=\sqrt{|\operatorname{Im}S(z)(t)|}$ achieves this
while preserving the sense of the curve, while the choice
$\psi^{\prime}(t)=-\sqrt{|\operatorname{Im}S(z)(t)|}$ achieves it while
reversing the sense. Put another way, the parameter

$\displaystyle\lambda=\int_{t_{0}}^{t}\sqrt{|\operatorname{Im}S(z)(t)|}$

is invariant under Möbius transformations and almost invariant under sense-
preserving reparameterizations of $t$, which act as translations in $\lambda$
(because of the freedom to choose $t_{0}$). Equivalently, the 1-form

$\displaystyle\mathrm{d}\lambda=\sqrt{|\operatorname{Im}S(z)(t)|}\mathrm{d}t,$

known as the Möbius or inversive arclength, is Möbius- and sense-preserving
parameterization-invariant.

This is often stated in a form (originally due, according to Ahlfors [3], to
Georg Pick) using the Euclidean curvature $\kappa$. If the curve is
parameterized by Euclidean arclength $s$, then its tangent
$\theta(s)=z^{\prime}(s)/\|z^{\prime}(s)\|$ and
$\kappa(s)=\theta^{\prime}(s)$. Differentiating again leads to
$\operatorname{Im}S(z)(s)=\kappa^{\prime}(s)$, or

$\displaystyle\mathrm{d}\lambda=\sqrt{|\kappa^{\prime}(s)|}{\rm d}s.$

The 1-form $\mathrm{d}\lambda$ provides a useful discrete invariant, the
Möbius length $L$ of the curve

$\displaystyle L=\int_{M}\mathrm{d}\lambda.$

The real part of $S(\tilde{z})(\lambda)$ is now a parameterization-invariant
Möbius invariant known as the inversive or Möbius curvature [33]

$\displaystyle\kappa_{{\text{\tiny
M\"{o}b}}}=\frac{4\kappa^{\prime}(\kappa^{\prime\prime\prime}-\kappa^{2}\kappa^{\prime})-5(\kappa^{\prime\prime})^{2}}{8(\kappa^{\prime})^{3}},$

where ′ denotes differentiation with respect to arclength $s$.

The set of all differential Möbius shape invariants is then
$\mathrm{d}^{n}\kappa_{{\text{\tiny M\"{o}b}}}/d\lambda^{n}$, $n\geq 0$.
Following Calabi et al. [9], two possible candidate invariants that could be
used to recognize Möbius shapes are the function $\kappa_{\text{\tiny
M\"{o}b}}(\lambda)$ modulo translations and the signature
$(\kappa_{\text{\tiny M\"{o}b}},\mathrm{d}\kappa_{\text{\tiny
M\"{o}b}}/\mathrm{d}\lambda)(S^{1})\subset\mathbb{R}^{2}$. These are complete
on sections of shapes with no vertices (points where $\kappa^{\prime}(s)=0$).
However, since $\kappa_{\text{\tiny M\"{o}b}}$ requires the 5th derivative of
the curve (third derivative of the curvature), it is not robust in the
presence of noise, and we do not explore it further.

There are also invariants based on forms of higher degree. We give just one
example, the Möbius energy

$\displaystyle-\frac{1}{2}\iint_{M\times
M}\frac{\sin\theta_{u}\sin\theta_{v}}{|v-u|^{2}}\mathrm{d}u\,\mathrm{d}v,$
(3.2)

introduced by O’Hara and Solanes [27], see also the related Kerzman–Stein
distance [7]. Here $\mathrm{d}u$, $\mathrm{d}v$ are Euclidean arclength and
$\theta_{u}$ (resp. $\theta_{v}$) is the angle between $v-u$ and
$z^{\prime}(u)$ (resp. $v$). It represents a renormalization of the energy of
particles on the curve, distributed evenly with respect to Euclidean arclength
and interacting under an $r^{-4}$ potential. (The authors of [27] comment that
“due to divergence problems, almost nothing is known about integral geometry
[i.e., about invariant differential forms] under the Möbius group”.) The
Möbius energy has one distinguishing feature compared to the other invariants:
its definition depends only on the first derivative of the curve. The
singularity at $u=v$ is removable, since the integrand obeys

$\displaystyle\frac{\sin\theta_{u}\sin\theta_{v}}{|v-u|^{2}}=\frac{1}{4}\kappa(u)\kappa(v)+\mathcal{O}(|v-u|).$

Thus, the energy is defined for $C^{2}$ curves, and even near the singularity
it depends only on the curvature of the curve.

However, in numerical experiments we found the integrand of (3.2) difficult to
evaluate accurately and invariantly, particularly near the diagonal, while the
double integral generated little improvement in robustness. The most negative
feature of the energy is its cost: its evaluation apparently requires
considering all pairs of points on a discrete curve. There is another reason
why invariant 2-forms are less useful than invariant 1-forms like
$\mathrm{d}\lambda$: invariant $k$-forms distinguish coordinates on $M^{k}$
only up to $k$-form- (i.e., volume-) preserving maps. These are infinite-
dimensional for $k>1$, but 1-dimensional for $k=1$: the coordinates are
determined up to translations. For these reasons, we do not consider the
energy, or related invariant 2-forms, any further.

Finally, Möbius transformations map circles to circles, and thus critical
points of Euclidean curvature (the ‘vertices’ of the shape) are Möbius
invariants. The four vertex theorem states that all smooth curves have at
least four vertices. The number of vertices is a discrete Möbius invariant.
Consequently, sections of shapes on which the Euclidean curvature is
monotonic, and their Möbius lengths, are Möbius invariant.

###  3.2 Example: Evaluating the Möbius length of an ellipse

Having described a number of possible invariants, and ruled many of them out
based on the criteria discussed in Section 2.1, we now provide a concrete
example, illustrating and testing some of these constructions on the ellipse
$z(t)=\cos 2\pi t+2{\rm i}\sin 2\pi t$, $0\leq t\leq 1$. The ellipse is
discretized at $t_{i}=(i+1/4)h$, $i=0,\dots,n$, giving points
$z_{i}=z(t_{i})$, and the Möbius arclength $\mathrm{d}\lambda$ is calculated
in two ways:

  * •

From the curvature method, in which the Euclidean curvature $\kappa$ is
calculated in a Euclidean-invariant way222Following Calabi et al. [9], let
$A$, $B$, $C$ be three points on a curve, and let $a=d(A,B)$, $b=d(B,C)$, and
$c=d(C,A)$ be the Euclidean distances between them. Then the curvature of the
circle interpolating $A$, $B$, and $C$ is  $\displaystyle\kappa(A,B,C)=\pm
4\frac{\sqrt{s(s-a)(s-b)(s-c)}}{abc}.$ by interpolating a circle through 3
adjacent points, and $\mathrm{d}\kappa/\mathrm{d}s$ by a finite difference,
giving

$\displaystyle\mathrm{d}\lambda(t_{i+3/2})\approx|\kappa(z_{i+1},z_{i+2},z_{i+3})-\kappa(z_{i},z_{i+1},z_{i+2})|(|z_{i+2}-z_{i+1}|).$
(3.3)

  * •

From the cross-ratio method, using

$\displaystyle\mathrm{d}\lambda(t_{i+3/2})\approx\sqrt{6|\mathrm{Im}(\log(\mathrm{CR}(z_{i},z_{i+1},z_{i+2},z_{i+3})))|}.$
(3.4)

Figure 2: The ellipse used as a test case, showing 50 points equally spaced
with respect to Möbius arclength. Figure 3: The Möbius arclength density,
$\mathrm{d}\lambda/\mathrm{d}t$, for the ellipse, and its errors when
calculated by the two approximations (3.3) and (3.4).

Away from vertices (points where $\mathrm{d}\lambda=0$), both of these are
second-order finite difference approximations to the Möbius length
$\int_{t_{i+1}}^{t_{i+2}}\mathrm{d}\lambda$ of the arc $z([t_{i+1},t_{i+2}])$,
or, dividing by $h$, to its arclength density $\mathrm{d}\lambda/\mathrm{d}t$.
This can be established by expanding the approximations in Taylor series333A
related approximation is
$\mathrm{d}\lambda(t_{i+3/2})\approx\sqrt{\frac{9}{2}|\mathrm{Im}(\mathrm{CR}(z_{i},z_{i+1},z_{i+2},z_{i+3}))|}$.
This is given in [7] but with an apparent error ($\frac{9}{2}$ replaced by
6).. We test their accuracy as a function of the step size $h$. The total
Möbius length of the curve is

$\displaystyle L:=\int_{0}^{1}d\lambda=\int_{0}^{1}\frac{12\pi\sqrt{|\sin 4\pi
t|}}{5+3\cos 4\pi t}\,\mathrm{d}t=6.86\quad(\mbox{to }2\mbox{ d.p.}).$

The arclength density $\mathrm{d}\lambda/\mathrm{d}t$ is shown in Fig. 3,
showing its 4 zeros at the vertices of the ellipse, where the ellipse is
approximately circular, and its square-root singularities at the vertices. The
Möbius length is approximated by the trapezoidal rule, i.e., by
$L_{h}:=\sum\limits_{i=1}^{n}\mathrm{d}\lambda(t_{i+1/2})$. The error
$L-L_{h}$ is expected to have dominant contributions of order $h^{3/2}$, due
to the singularities at the vertices, and of order $h^{2}$, due to the finite
differences used to approximate $\mathrm{d}\lambda$.

In this example, the cross-ratio approximation has errors approximately 0.176
times those of the curvature approximation (see Fig. 3). However, due to some
cancellations in this particular example, the curvature approximation actually
gives a slightly more accurate approximation to the Möbius length (see Fig.
4). The dominant sources of error can be eliminated by two steps of Richardson
extrapolation, first to remove the $\mathcal{O}(h^{3/2})$ error, and then to
remove the $\mathcal{O}(h^{2})$ error. This is highly successful and allows
the calculation of the Möbius length with an error of less than $10^{-10}$,
even though it is singular and involves a 3rd derivative.

Next, we subject the ellipse to a variety of Möbius transformations. The
resulting shapes and errors are shown in Fig. 5. The errors increase markedly
for the curvature method, which is not Möbius invariant, but are unchanged for
the cross-ratio method, which is Möbius invariant. Thus, this experiment
supports the argument of [9] that numerical approximations of invariants
should themselves be invariant.

Figure 4: The error in the two approximations of the total Möbius length of
the ellipse. Here there are $n:=25\cdot 2^{j}$ points on the ellipse. Figure
5: The ellipse subjected to 16 different Möbius transformations
$z\mapsto\frac{z}{1+dz}$. 10 landmarks, equally spaced in $\lambda$, are
shown. Next to each shape is given the error in its Möbius length as
calculated by the curvature method. The error in the cross-ratio method is
$0.005$ for all shapes, because it is Möbius invariant. (The other Möbius
transformations are the Euclidean similarities, which are easy to visualize.
Figures not to scale.)

###  3.3 The Möbius invariant $\boldsymbol{\mathrm{CR}(\lambda;z,\delta)}$

The method proposed in this paper for closed curves is to parameterize the
curve by Möbius arclength, giving $z(\lambda)$, and to use as an invariant the
cross-ratio of all sets of 4 points a distance $\delta$ apart. We call this
the Shape Cross-Ratio, or SCR.

######  Definition 3.1.

Let $z\colon S^{1}\to\mathbb{C}$ be a smooth curve. Let $L$ be its Möbius
length, and let $\tilde{z}\colon\mathbb{R}\to\mathbb{C}$ be an $L$-periodic
function representing the curve parameterized by Möbius arclength. The shape
cross-ratio of $z$ is the $L$-periodic function
$\mathrm{SCR}\colon\mathbb{R}\to\overline{\mathbb{C}}$ defined by

$\displaystyle\mathrm{SCR}(\lambda;z,\delta)=\mathrm{CR}(\tilde{z}(\lambda),\tilde{z}(\lambda+\delta),\tilde{z}(\lambda+2\delta),\tilde{z}(\lambda+3\delta)).$

The shape cross-ratio signature of $z$ is the shape
$\mathrm{SCR}(\mathbb{R};z,\delta)\subset\overline{\mathbb{C}}$.

The shape cross-ratio is invariant under the Möbius group, and sense-
preserving reparameterizations of the curve act as translations in $\lambda$.
(Reversals of the curve will be considered in Section 3.5.) The shape cross-
ratio signature is invariant under the Möbius group and under
reparameterizations of the curve.

For an $L$-periodic function $f\colon\mathbb{R}\to\mathbb{C}$ we will denote
its Fourier coefficients by

$\displaystyle\mathcal{F}(f)_{n}=\frac{1}{L}\int_{0}^{L}f(t)e^{-2\pi{\rm
i}nt/L}\mathrm{d}t.$

The translation $t\mapsto t+c$ acts on the Fourier coefficients as
$\mathcal{F}(f)_{n}\mapsto e^{-2\pi{\rm i}cn/L}\mathcal{F}(f)_{n}$. The
Fourier amplitudes $|\mathcal{F}(f)_{n}|^{2}$ are invariant under
translations, and can be used to recognize functions up to translations, but
are clearly not a complete invariant: for a function discretized at $N$
equally-spaced points, and using the DFT, the space of orbits has dimension
$2N-1$ and we have only $N$ invariants. The bispectrum [19]
$\mathcal{F}(f)_{m}\mathcal{F}(f)_{n}\mathcal{F}(f)_{-m-n}$ is better, being
complete on functions all of whose Fourier coefficients are non-zero, but it
is a very large set of invariants. Other invariants are
$\mathcal{F}(\phi_{1}\circ f)_{n}\mathcal{F}(\phi_{2}\circ f)_{-n}$ for any
functions $\phi_{1,2}$. Each such choice provides $2N$ invariants. The choice
of $\phi_{1}$ and $\phi_{2}$ determines which aspects of $f$ are measured by
the invariant. If necessary, several such pairs may be used.

######  Definition 3.2.

The Fourier cross-ratio of the shape $z$ is

$\displaystyle\mathrm{FCR}(\cdot;z,\delta)\colon\
{\mathbb{Z}}\to\mathbb{C},\qquad\\!\\!\mathrm{FCR}(n;z,\delta)=\mathcal{F}(\phi_{1}\circ\mathrm{SCR}(\cdot;z,\delta))_{n}\mathcal{F}(\phi_{2}\circ\mathrm{SCR}(\cdot;z,\delta))_{-n},\\!\\!$
(3.5)

where the Fourier transforms are based on the Möbius length $L$ of $z$.

In the numerical illustrations we use

$\displaystyle\phi_{1}(w)=\frac{w}{\sqrt{1+|w|^{2}}},\qquad\phi_{2}(w)=\phi_{1}(w)^{2}$
(3.6)

and the distance between the invariants of two shapes $z_{1}$ and $z_{2}$
given by

$\displaystyle\|\mathrm{FCR}(\cdot;z_{1},\delta)-\mathrm{FCR}(\cdot;z_{2},\delta)\|_{2}.$
(3.7)

The motivation here is that the cross-ratio becomes arbitrarily large when two
different parts of the curve approach one another. If left untouched (i.e., if
we use just $\phi_{1}(w)=w$), then these large spikes in
$\mathrm{SCR}(\lambda;z,\delta)$ will dominate all other contributions to the
shape measurement. By scaling them using (3.6), they will still contribute to
the description of the shape, but in a way that is balanced with respect to
other parts of the shape. $\phi_{1}$ and $\phi_{2}$ take values in the unit
disk and $\phi_{2}$ is sensitive to the main range of features of shapes; only
values of $\mathrm{SCR}$ near 0 are suppressed, and these are rare.

The invariant $\mathrm{SCR}(\lambda;z,\delta)$ is smooth on simple closed
curves, and also on most curves with self-intersections (blow-up requires the
close approach of two points Möbius distance $n\delta$ apart). It is locally
complete, as given $z([a,a+3\delta])$, the invariant $\mathrm{SCR}(\lambda)$
determines $z$. We do not know if it is globally complete, i.e., if, given
$\mathrm{SCR}(\lambda)$ which is the invariant of some shape, the shape can be
determined up to Möbius transformations, because this requires solving a
nonlinear functional boundary value problem. Subject to this restriction, the
invariant $\mathrm{FCR}(n;z,\delta)$ is complete except on a residual set of
shapes (those for which enough Fourier coefficients in (3.5) are zero).

We will first study the numerical approximation of $\mathrm{SCR}$ and then
study its use in recognizing shapes modulo Möbius transformations.

The numerical experiments in Section 3.2 convinced us to approximate the
Möbius arclength using the cross-ratio. However, when combined with piecewise
linear interpolation to locate points on the curve the required distance
$\delta$ apart, we found that the resulting values of
$\mathrm{SCR}(\lambda;z,\delta)$ did not converge as $h\to 0$. This is due to
accumulation of errors along the curve, which arise particularly at the
vertices due to the singularities there. This prompted us to develop a more
refined interpolation method that takes into account the singularities of
$\mathrm{d}\lambda/\mathrm{d}t$ at the vertices. We call it the modified
cross-ratio method:

  1. 1.

Calculate the square of the Möbius arclength density at the centre of each
cell, as

$\displaystyle\left(\frac{\mathrm{d}\lambda}{\mathrm{d}t}\right)^{2}_{i+3/2}=6\operatorname{Im}(\log(\mathrm{CR}(z_{i},z_{i+1},z_{i+2},z_{i+3}))).$

If the curve is smooth, this is a smooth function.

  2. 2.

Let $f(t)$, $0\leq t<1$, be the piecewise linear interpolant of
$\left(\frac{\mathrm{d}\lambda}{\mathrm{d}t}\right)^{2}_{i+3/2}$.

  3. 3.

Calculate $\lambda(t)=\int_{0}^{t}\sqrt{|f(\tau)|}\,\mathrm{d}\tau$ and its
inverse, $t(\lambda)$, used in locating the parameter values at which points a
desired length apart are located, exactly (we omit the formulas).

  4. 4.

The desired points $z(t(\lambda+n\delta))$ are calculated using linear
interpolation from the known values $z(ih)$.

  5. 5.

The cross-ratio is evaluated at $N$ points equally spaced in $\lambda$, giving
$\mathrm{SCR}(iL/N;z,\delta)$ for $i=1,\dots,N$, and the Fourier invariant
$\mathrm{FCR}$ evaluated using two FFTs.

The resulting cross-ratio is globally second-order accurate in $h$. Its
accuracy could be increased for smooth curves using higher order
interpolation, but the calculation of the inverse $t(\lambda)$ would be much
more complicated and the method would be less robust.

The error in the length $L$ of the ellipse used in Section 3.2 as calculated
by this method is shown in Fig. 4. It behaves extremely reliably over a wide
range of scales of $h$; its error after one Richardson extrapolation is
observed to be $\mathcal{O}(h^{4})$, which indicates that the singularities at
the vertices have been completely removed.

The parameter $\delta$ is the length scale on which
$\mathrm{SCR}(\lambda;z,\delta)$ describes the shape. However, if $\delta\to
0$ then $\mathrm{SCR}(\lambda;z,\delta)\to\kappa_{\text{\tiny
M\"{o}b}}\pm\mathrm{i}$: the real part becomes extremely nonrobust and the
imaginary part yields no information [33]. In the experiments in this paper we
have used $\delta=L/8$. Choosing $L/\delta\in\mathbb{Z}$ seems to yield
somewhat improved accuracy, as the same values of $z$ are used repeatedly.

Figure 6: Average error in $L$ (left) and $\mathrm{FCR}$ (right) for the
ellipse shown in Fig. 2 as a function of the number of points $N$ and noise
level $\varepsilon$.

As the method relies on parameterization by Möbius arclength, it is
unavoidably sensitive to noise in the data. We test its sensitivity for a
noise model in which each point on the discrete curve is subject to normally
distributed noise of standard deviation $\varepsilon$. The dependence of the
error on $\epsilon$ and $N$ is shown for length and cross-ratio invariants in
Fig. 6. Clearly, both are sensitive to relatively small amounts of noise.
However, some positive features can also be seen:

  * (i)

The errors in $L$ and $\mathrm{FCR}(n;z,\delta)$ are both $\mathcal{O}(h^{2})$
in the absence of noise.

  * (ii)

The errors in $\mathrm{FCR}(n;z,\delta)$ are much smaller than those in $L$ –
their relative errors are about 8 times smaller. (For the ellipse, $L\approx
6.86$ and $\|\mathrm{SCR}(\cdot;z,\delta)\|_{2}\approx 0.65$.)

  * (iii)

The error in $\mathrm{FCR}(n;z,\delta)$ appears to saturate at about 13% as
$\varepsilon$ increases and as $N$ increases.

Point (iii) is particularly striking. It appears to hold because (a) $\delta$
is chosen to be proportional to $L$, and thus when the noise is high, the
chosen points stay roughly in their correct places; and (b) noise in the
chosen points is averaged out by the Fourier transform, which remains
dominated by its first few terms. This effect is illustrated in Fig. 7 in
which a single noise realization is illustrated for value of $\varepsilon$
from 0 to $10^{-2}$. Even though $L$ is overestimated by a factor of 100, the
signature cross-ratio $\mathrm{SCR}(\lambda;z,\delta)$ is still recognizable.

Figure 7: The cross-ratio signature $\mathrm{SCR}\colon S^{1}\to\mathbb{C}$
(blue), and the error in its Fourier invariant $\mathrm{FCR}$, is shown for
the ellipse discretized with $N=128$ points and various levels of noise. The
exact signature is shown in red. Figure 8: The set of 16 shapes used in the
numerical experiments. The shapes are shown to scale with the shape number
marking the origin, along with the shape’s Möbius length $L$ and number of
vertices $V$. Figure 9: Scatter plot of distance between all 120 pairs of 16
shapes with respect to (i) the $H^{1}$ distance between the shapes, as defined
in (3.8)–(3.10), and (ii) the distance between their Möbius invariants,
defined in (3.5)–(3.7). The correlation coefficient is 0.85. The registration
between the 4 pairs marked with circles is illustrated in Figs. 10–13.

###  3.4 Comparison with shape registration

We now compare the results of the Möbius invariant $\mathrm{FCR}(n;z,\delta)$
with direct registration of shapes. Given two shapes $z$ and $w$ we define the
$G$-registration of $z$ onto $w$ as

$\displaystyle r_{G}(z,w)=\min_{\varphi\in
G\atop\psi\in\operatorname{Dif\/f}^{+}(S^{1})}\|\varphi\circ z\circ\psi-w\|.$
(3.8)

Different choices of norm in (3.8) will give different registrations; we have
used the $H^{1}$ norm

$\displaystyle\|z\|_{H^{1}}^{2}=\int_{0}^{1}|z(t))|^{2}+\alpha|z^{\prime}(t))|^{2}\,\mathrm{d}t,$
(3.9)

where the constant $\alpha$ was chosen as $0.1$, a value which made both
contributions to the norm roughly equal. One of the peculiarities of the
Möbius group is that $z$ may register very well onto $w$ while $w$ registers
poorly onto $z$. This happens when $z$ has a distinguished feature which can
be squashed, thus minimizing its contribution to $r_{G}(z,w)$. Therefore in
our experiments we use the ‘distance’

$\displaystyle d_{G}(z,w)=\max(r_{G}(z,w),r_{G}(w,z)).$ (3.10)

Figure 10: Registration of shapes 12 (blue) & 13 (red). This pair has the
closest Möbius invariants (distance 0.0426, see (3.7)), is 9th closest after
Möbius registration, and 96th closest after similarity registration. (a)
Similarities act on blue shape; (b) Similarities act on red shape; (c) Möbius
acts on blue shape; (d) Möbius acts on red shape. Here $d=r_{G}(x,y)$ where
$x$ and $y$ are the two shapes, see equation (3.8). Figure 11: Registration of
shapes 2 (blue) & 16 (red). This pair has the 2nd closest Möbius invariants
(distance 0.0697), see (3.7)), is 3rd closest after Möbius registration, and
42rd closest after similarity registration. (a) Similarities act on blue
shape; (b) Similarities act on red shape; (c) Möbius acts on blue shape; (d)
Möbius acts on red shape. Here $d=r_{G}(x,y)$ where $x$ and $y$ are the two
shapes, see equation (3.8). Figure 12: Registration of shapes 8 (blue) & 10
(red). This pair is closest after Möbius registration. It has the 11st closest
Möbius invariants (distance 0.1165) and is the closest pair after similarity
registration. (a) Similarities act on blue shape; (b) Similarities act on red
shape; (c) Möbius acts on blue shape; (d) Möbius acts on red shape. Here
$d=r_{G}(x,y)$ where $x$ and $y$ are the two shapes, see equation (3.8).
Figure 13: Registration of shapes 9 (blue) & 11 (red). This pair has the 3rd
closest Möbius invariants (distance 0.0798), is 9th closest after Möbius
registration, and 96th closest after similarity registration. (a) Similarities
act on blue shape; (b) Similarities act on red shape; (c) Möbius acts on blue
shape; (d) Möbius acts on red shape. Here $d=r_{G}(x,y)$ where $x$ and $y$ are
the two shapes, see equation (3.8).

Note that this should not be regarded as any kind of ‘ground truth’ for the
$G$-similarity of $z$ and $w$. It is not $G$-invariant. However, as we shall
see, it does correspond remarkably well to the Möbius invariants described
earlier.

To calculate $d_{G}(z,w)$ numerically, we discretize
$\operatorname{Dif\/f}^{+}(S^{1})$ by piecewise linear increasing functions
with 16 control points and perform the optimization using Matlab’s lsqnonlin,
with initial guesses for $\varphi$ chosen to be each of 4 rotations and 3
scale factors for $\varphi$, and initial $\psi$ chosen to be the identity.

We generated 16 random shapes from a 14-dimensional distribution that favours
smooth Jordan curves of similar sizes (where $U(0,2\pi)$ denotes uniform
random numbers in the range $0$ to $2\pi$):

$\displaystyle
z(t)=\sum_{n=-4}^{4}a_{n}\mathrm{e}^{2\pi\mathrm{i}nt},\qquad\operatorname{Re}(a_{n}),\,\operatorname{Im}(a_{n})\in
N\big{(}0,1/\big{(}1+|n|^{3}\big{)}\big{)},\qquad n\neq\pm 1,$
$\displaystyle\arg a_{1}\in U(0,2\pi),\qquad|a_{1}|=1,\qquad a_{-1}/a_{1}\in
U(0,0.6).$

The shapes are shown in Fig. 8. They are all simple closed curves, which we
take to be positively oriented. The 120 pairs of distinct shapes are
registered in both directions, and the scatter plot between this distance and
the 2-norm of the distance between their invariants is shown in Fig. 9. The
correlation (0.85) is extremely striking, and suggests that this pair of
measures may be related by bounded distortion (2.1), implying that they meet
many of the requirements that we identified in Section 2.1; they are also
quick to compute and provide a good numerical approximation.

Some examples of the registrations in the similarity and Möbius groups are
shown for four close pairs in Figs. 10–13. Including the invariants $L$ and
$V$ in the list of invariants did not improve the correlation. Note that the
errors in $\|\mathrm{FCR}(\cdot;z,\delta)\|$ observed in Fig. 4 are small
enough to allow the separation of all but the closest pairs of shapes,
regardless of the level of noise $\varepsilon$.

###  3.5 Reversals and reflections

Orientation reversals and reflections are examples of actions of discrete
groups and can, in theory, be handled by any of the approaches in Section 2.2.
First, consider the sense-reversing reparameterizations, $z(t)\mapsto z(-t)$.
These map the SCR invariant as
$\mathrm{SCR}(\lambda;z,\delta)\mapsto\mathrm{SCR}(-\lambda;z,\delta)$ and
hence $\mathrm{FCR}(n)\mapsto\mathrm{FCR}(-n)$. It is convenient to pass to
the equivalent invariants:

$\displaystyle x_{n}=\begin{cases}\mathrm{FCR}(n)-\mathrm{FCR}(-n),&n>0,\\\
\mathrm{FCR}(n)+\mathrm{FCR}(-n),&n\leq 0,\end{cases}$

for which $x_{n}$ is invariant for $n\leq 0$ and $x_{n}\mapsto-x_{-n}$ for
$n>0$. Suppose that we wish to identity curves with their reversals, that is,
to work with unoriented shapes. Some options are the following:

  1. 1.

The moving frame method: the shapes are put into a reference orientation
first. This is only possible if the problem domain is restricted suitably; in
this case, for example, to simple closed curves, which can be taken to be
positively oriented. This is the approach that we have taken in Sections 3.3
and 3.4. If the problem domain includes non-simple curves this approach may
not be possible. For example, in the space of plane curves with the topology
of a figure 8, each such shape can be continuously deformed into its reversal;
thus we cannot assign them orientations, since they vary continuously with the
shape.

  2. 2.

Finding a complete set of invariants: this is $x_{n}$ for $n\leq 0$ and
$x_{i}x_{j}$ for $i,j>0$. Again we see that the quotient by a relatively
simple group action is expensive to describe completely using invariants.

  3. 3.

Use an incomplete set of invariants that is “good enough”: here, using $x_{n}$
for $n\leq 0$ and $x_{n}x_{n+1}$ for $n>0$ is a possibility. This creates a
complicated effect on the metric used to compare invariants.

  4. 4.

As the group action is of a standard type, one can work in unreduced
coordinates ($x_{n}$) together with a natural metric induced by the quotient,
such as some function of $\|x\|-\|y\|$ and the Fubini–Study metric on
projective space, $\cos^{-1}(|\bar{x}^{T}y|/\|x\|\|y\|)$.

  5. 5.

Finally, and most easily in this case, for finite groups one can represent
points in the quotient as entire group orbits. A suitable metric is then

$\displaystyle d(x,y)=\min_{g\in G}\|x-g\cdot y\|,$

where $G$ is the group. Although this is impractical for large finite groups,
here $G$ is $\mathbb{Z}_{2}$.

Similar considerations apply to recognizing shapes modulo the full inversive
group, generated by the Möbius group together with a reflection, say
$z\mapsto\bar{z}$. The reflection maps
$\mathrm{FCR}(n)\mapsto\overline{\mathrm{FCR}(n)}$ and hence is an action of
the same type as reversal (a sign change in some components). The reflection
symmetry of the ellipse in Fig. 2, for example, can be detected by the
reflection symmetry of the cross-ratio signature $\mathrm{SCR}(\lambda)$ in
Fig. 7. (Its second discrete symmetry, a rotation by $\pi$, is manifested in
Fig. 7 by the signature curve retracting itself twice.)

The invariants developed here are for closed shapes. They can be adapted for
other types of shapes (for example, shapes with the topology of two disjoint
circles), but as the topology gets more complicated (for example, shapes
consisting of many curve segments) the problem becomes significantly more
difficult.

##  4 Möbius invariants of images

Let $f\colon\overline{\mathbb{C}}\to[0,1]$ be a smooth grey-scale image.
Diffeomorphisms act on images by $\varphi\cdot f:=f\circ\varphi^{-1}$. It is
easy, in principle, to adapt the Möbius shape invariant $\mathrm{SCR}$ to
images by computing level sets of $f$, each of which is an invariant shape for
which $\mathrm{SCR}$ can be calculated. In addition, if $\varphi$ is
conformal, the orthogonal trajectories of the level sets, i.e., the shapes
tangent to $\nabla f$, are also invariant shapes. In the neighbourhood of a
simple closed level set, coordinates $(\lambda,\mu)$ can be introduced, where
$\lambda$ is Möbius arclength along the level set and $\mu$ is Möbius
arclength along the orthogonal trajectories. The quantity

$\displaystyle\mathrm{CR}(z(\lambda,\mu),z(\lambda,\mu+\delta),z(\lambda+\delta,\mu+\delta),z(\lambda+\delta,\mu)),$
(4.1)

calculated from the cross-ratio of 4 points in a square, is then invariant
under the Möbius group, and reparameterizations of the level set act as
translations in $\lambda$.

In practice, however, the domain of this invariant is quite restricted. The
topology of level sets is typically very complicated and the domain of $f$ may
be restricted, so that level sets can stop at the edge of the image.
Restricting to level sets of grey-scales near the maximum and minimum of $f$
helps, but this is a severe restriction. Instead, we shall show that the extra
information provided by an image, as opposed to that provided by a shape,
determines a differential invariant signature using only 3rd derivatives,
compared to the 5th derivatives needed for differential invariants of shapes.
Because of this, we do not develop the cross-ratio invariant (4.1) any further
here.

######  Proposition 4.1.

Let $f\colon\overline{\mathbb{C}}\to\mathbb{R}$ be a smooth grey-scale image.
Let $R\subset\overline{\mathbb{C}}$ be the regular points of $f$. Identify
$x_{1}+{\rm i}x_{2}\in\mathbb{C}$ with $(x_{1},x_{2})\in\mathbb{R}^{2}$ so
that $\nabla f$ is the standard Euclidean gradient. On $R$, define

$\displaystyle n:=\frac{\nabla f}{\|\nabla
f\|},\qquad\lambda_{n}:=\frac{n\cdot\nabla(\nabla\times n)}{\|\nabla
f\|^{2}},\qquad\lambda_{t}:=\frac{n\times\nabla(\nabla\cdot n)}{\|\nabla
f\|^{2}}.$

Then

$\displaystyle(f,\lambda_{n},\lambda_{t})(R)$ (4.2)

is a subset of $\mathbb{R}^{3}$ that is invariant under the action of the
Möbius group on images.

###### Proof.

As defined above, $n$ is the unit vector field normal to the level sets of
$f$. Let $n^{\perp}$ be the unit vector tangent to the level sets given by
$n^{\perp}_{i}=\varepsilon_{ij}n_{j}$. (Here $i,j=1,2$ and $\varepsilon_{ij}$
is the Levi-Civita symbol defined by $\varepsilon_{11}=\varepsilon_{22}=0$,
$\varepsilon_{12}=1$, $\varepsilon_{21}=-1$; we sum over repeated indices and
write $n_{i,k}=\partial n_{i}/\partial x_{j}$.) From the Frenet–Serret
relation $n_{s}=\kappa n^{\perp}$, where $s$ is arclength along the level
sets, we have that the curvature of the level sets is

$\displaystyle\kappa$ $\displaystyle=n^{\perp}\cdot
n_{s}=n^{\perp}\cdot((n^{\perp}\cdot\nabla)n)=\varepsilon_{ij}n_{j}\varepsilon_{kl}n_{l}n_{i,k}=(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk})n_{j}n_{l}n_{i,k}$
$\displaystyle=n_{j}n_{j}n_{k,k}-n_{k}n_{i}n_{i,k}=n_{k,k}\quad\hbox{\rm(because\
}n_{j}n_{j}=1\Rightarrow n_{i}n_{i,k}=0\hbox{\rm\ for all\ }k)$
$\displaystyle=\nabla\cdot n.$

Recall that the Möbius arclength of the level sets of $f$ is
$\mathrm{d}\lambda:=\sqrt{|\kappa_{s}|}\mathrm{d}s$. Under any conformal map,
the scaling along and normal to the level sets is the same, and thus
$\mathrm{d}s$ and $1/\|\nabla f\|$ both scale by the same factor. Therefore
$\sqrt{|\kappa_{s}|}/\|\nabla f\|$ is invariant, as is its square
$|\kappa_{s}|/\|\nabla f\|^{2}$. The sign of $\kappa_{s}$ is also invariant
under Möbius transformations, resulting in the given invariant
$\lambda_{t}=\kappa_{s}/\|\nabla f\|^{2}$.

The invariant $\lambda_{n}$ arises in the same way from the orthogonal
trajectories, whose curvature is $\nabla\cdot n^{\perp}=\nabla\times n$. ∎

######  Example 4.2.

As a test image we take the smooth function

$\displaystyle f(x,y)=e^{-4x^{2}-8\big{(}y-0.2x-0.8x^{2}\big{)}^{2}}$ (4.3)

and calculate its invariant signature before and after the Möbius
transformation with parameters

$\displaystyle a=0.9+0.1\mathrm{i},\qquad b=0.1,\qquad
c=0.1+0.4\mathrm{i},\qquad d=1.$ (4.4)

on the domain $[-1,1]^{2}$. The invariants are approximated by finite
differences with mesh spacing $1/80$, corresponding to $161\times 161$ pixel
images. The invariants are shown as functions of $(x,y)$ in Fig. 14 for
$\lambda_{n}$ and Fig. 15 for $\lambda_{t}$. The resulting signature surfaces,
shown for $f$ in Fig. 16 in $\mathbb{R}^{3}$, are quite complicated. A useful
way to visualize and compare them is shown in Fig. 17. For example, one can
plot the contours of $f$ in the $(\lambda_{n},\lambda_{t})$ plane, and
similarly for other projections. This enables a sensitive comparison of the
signatures of the image and its Möbius transformed version and reveals that
they are extremely close.

Figure 14: Contours 0.1,0.2,…,0.9 of a function are shown in blue, together
with its invariant $\lambda_{n}$: contour 0 in green, contours $-0.25$, $-1$,
and $-100$ in red, and contours 0.25, 1, and 100 in black. Top: function $f$
from (4.3). Bottom: Möbius related function $f\circ\varphi^{-1}$, parameters
in (4.4). The invariance can be seen, along with the way that $\lambda_{n}$
typically blows up as $\nabla f\to 0$. A small discretization error is visible
in the top figure: the saddle point near $(-0.5,-0.5)$ has $\lambda_{n}\approx
1.07$, whereas the exact value is 0.94. This results in the wrong topology of
the $+1$ contour (cf. bottom figure near $(-0.8,-0.2)$).

Figure 15: Contours 0.1,0.2,…,0.9 of a function are shown in blue, together
with its invariant $\lambda_{t}$: contour 0 (which locates vertices (points of
stationary curvature) of the level sets) in green, contours $-0.25$, $-1$, and
$-100$ in red, and contours 0.25, 1, and 100 in black. Top: function $f$ from
(4.3). Bottom: Möbius related function $f\circ\varphi^{-1}$, parameters in
(4.4).

\begin{overpic}[width=411.93767pt]{blobhill} \put(75.0,-5.0){\small$-1$}
\put(90.0,11.0){\small$x$} \put(102.0,25.0){\small$1$}
\put(0.0,5.0){\small$1$} \put(30.0,0.0){\small$y$} \end{overpic}

Figure 16: The sample image defined in equation (4.3) is shown in grayscale
(top left) and as a graph $(x,y,f(x,y))$ (bottom left). Its Möbius signature
surface (4.2) is shown at right. Figure 17: The invariant signature
$(f,\lambda_{n},\lambda_{t})$ shown for $f$ in the left column and for
$f\circ\varphi^{-1}$ in the right column. Top: contours 0.2, 0.4, 0.6, and 0.8
of $f$; middle and bottom: contours $-1$, $-0.25$, 0, 0.25, and 1 of
$\lambda_{n}$ (resp. $\lambda_{t}$). The two invariants are almost identical
in appearance (see, e.g., the $-1$ (dark blue) contour of $\lambda_{t}$ near
$(\lambda_{n},f)=(1,0.6)$, which is slightly different in the left and right
columns).

######  Example 4.3.

As a more numerical example, we take 9 similar blob-like functions,
constructed as the sum of four random 2D Gaussian functions, and their Möbius
images under a random Mobius transform, and compare their invariant
signatures. The functions and their Mobius-transformed variants
$f\circ\varphi^{-1}$ are shown in Fig. 18 as level set contours, while the
invariant signatures are shown in Fig. 19. Because the whole invariant
signature surfaces are very complicated, we show just the signature curve
corresponding to the level set $f^{-1}(0.5)$. This depends only on the first 3
derivatives of $f$ on the level set. Because $\lambda_{n}$ and $\lambda_{t}$
take values in $[-\infty,\infty]$, we use coordinates
$(\arctan(\lambda_{t}/4),\arctan(\lambda_{n}/4))$. Clearly, even this very
limited portion of the signature serves to distinguish the Möbius-related
pairs extremely sensitively. In some cases, the invariants change extremely
rapidly along the level set, so that even though they are evaluated
accurately, the resulting contours of the Möbius-related pairs do not overlap.
This would need to be taken into account in the development of a distance
measure on the invariant signatures.

Figure 18: Nine random blob-like functions are shown on the left. Each is
given by the sum of 4 random Gaussians, with the range of the resulting
function scaled to $[0,1]$. The domain is $[-1,1]^{2}$ and the functions are
discretized with $h=1/80$ giving $161\times 161$ grey-scale images. For each
of the 9 functions $f$, a random Möbius transformation $\varphi$ is chosen and
the composition $f\circ\varphi^{-1}$ shown on the right, evaluated on the
domain $[-1,1]^{2}$. The transformations have parameters $b=0$, $a$ uniform in
an annulus with inner radius 0.7 and outer radius 1.3, $d=1$, and $c$ with
uniform argument and normal random modulus with standard deviation 0.6. The
contours 0.1, 0.2,…,0.9 of the functions are shown. Figure 19: The invariant
signature $(\arctan(\lambda_{t}/4),\arctan(\lambda_{n}/4))$ evaluated on the
level set $f^{-1}(0.5)$ is calculated by central differences for each of the
images in Fig. 18 left (shown in blue) and for the corresponding images in
Fig. 18 right (shown in red). The domains are $[-\pi/2,\pi/2]^{2}$. The
signature curves distinguish the Möbius-related pairs very sensitively; only
tiny finite difference errors are visible. However, some errors related to
insufficient resolution of the signature curves are clearly visible.

######  Example 4.4.

In this example we illustrate the extreme sensitivity of the invariant
signature by evaluating it on 9 very similar images, together with their
Möbius transformations. Each original image is a blob function generated as in
Example 4.3, but with parameters varying only by $\pm 5\%$. The Möbius
transformations have the form $1/(1+cz)$ where $c$ is normally distributed
with standard deviation 0.1. The 0.5-level contours of the original and
transformed images are shown in Fig. 20, and their signatures in Fig. 21. The
signature is extremely sensitive to tiny changes in the image, but not to
Möbius transformations.

Figure 20: The 0.5-level contour of 9 very similar blob-like images are shown
in blue, and of their Möbius transformations in red. Only the central
$80\times 80$ portion of the $161\times 161$ images are shown. Figure 21: The
invariant signature $(\arctan(\lambda_{t}/4),\arctan(\lambda_{n}/4))$
evaluated on the level set $f^{-1}(0.5)$ is shown for each of the images in
Fig. 20 (blue) and for their Möbius transformations (red).

We do not have a full understanding of the properties of this invariant
signature with respect to the criteria listed in Section 2. It is certainly
fast, small, local, and lacks redundancy and suppression. It has a good
numerical approximation on smooth (or smoothed) images. Is it complete? That
is, given an image, does its signature surface determine the image up to a
Möbius transformation? Suppose we are given a small piece of signature
surface, parameterized by $(u,v)$, say. We are given three functions
$\tilde{f}(u,v)$, $\tilde{\lambda}_{n}(u,v)$, and $\tilde{\lambda}_{t}(u,v)$,
and need to determine (by solving three PDEs) three functions $f(x,y)$ (the
image), $x(u,v)$, and $y(u,v)$ (the coordinates). Typically, the solution of
these PDEs will be determined by some boundary data. This suggests that
distinct images with the same signature are parameterized by functions of 1
variable; a kind of near completeness that may be good enough in practice.

Although very sensitive, the fact that it is not continuous at critical points
means that it does not have good discrimination in the sense of Section 2. (It
falls into the ‘more false negatives’ region of Fig. 1.) Near nondegenerate
critical points, the signature blows up in a well-defined way, so it is
possible that there exists a metric on signatures that leads to robustness and
good discrimination.

##  5 Conclusion

In this paper we have developed Möbius invariants of both curves and images,
and proposed computational methods to evaluate both, demonstrating them on a
variety of examples. In Section 2 we identified a set of properties that are
important for invariants, principally that there was a small set of invariants
that were quick to compute, numerically stable, robust (so that noisy versions
of the same curve have similar invariants) and yet sufficiently discriminatory
(so that different objects have different invariants).

While differential invariants are not generally robust when dealing with
noise, they offer good discrimination and are cheap to compute; this leads us
to the Möbius arc-length. The cross-ratio is more robust, but requires a large
set of points to be evaluated, and blows up as the pairs of points approach
each other. In order to make this reparameterization-invariant, we used a
Fourier transform. This lead to a method of computing Möbius invariants that
satisfies the properties that we have outlined, as is demonstrated in the
numerical experiments, see particularly Fig. 9.

For images, the extra information means that it is possible to compute a
relatively simple three-dimensional signature based on the function value at
each point together with two functions of the Möbius arclength, along and
perpendicular to level sets of the image intensity. It is computationally
cheap, extremely sensitive to non-Möbius changes in the image, but insensitive
to Möbius transformations of the image.

### Acknowledgements

This research was supported by the Marsden Fund, and RM by a James Cook
Research Fellowship, both administered by the Royal Society of New Zealand. SM
would like to thank the Erwin Schrödinger International Institute for
Mathematical Physics, Vienna, where some of this research was performed.

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