Feature Matching and Heat Flow in Centro-Affine Geometry

Feature Matching and Heat Flow
in Centro-Affine Geometry

Peter J. OLVER †, Changzheng QU ‡ and Yun YANG §

P.J. Olver, C. Qu and Y. Yang

† School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
<EMAIL_ADDRESS>http://www.math.umn.edu/~olver/

‡ School of Mathematics and Statistics, Ningbo University, Ningbo 315211, P.R.
China<EMAIL_ADDRESS>
§ Department of Mathematics, Northeastern University, Shenyang, 110819, P.R.
China<EMAIL_ADDRESS>
Received April 02, 2020, in final form September 14, 2020; Published online
September 29, 2020

In this paper, we study the differential invariants and the invariant heat
flow in centro-affine geometry, proving that the latter is equivalent to the
inviscid Burgers’ equation. Furthermore, we apply the centro-affine invariants
to develop an invariant algorithm to match features of objects appearing in
images. We show that the resulting algorithm compares favorably with the
widely applied scale-invariant feature transform (SIFT), speeded up robust
features (SURF), and affine-SIFT (ASIFT) methods.

centro-affine geometry; equivariant moving frames; heat flow; inviscid
Burgers’ equation; differential invariant; edge matching

53A15; 53A55

##  1 Introduction

The main objective in this paper is to study differential invariants and
invariant curve flows – in particular the heat flow – in centro-affine
geometry. In addition, we will present some basic applications to feature
matching in camera images of three-dimensional objects, comparing our method
with other popular algorithms.

Affine differential geometry is based on the Lie group ${\rm
A}(n,{\mathbb{R}})={\rm GL}(n,{\mathbb{R}})\ltimes{\mathbb{R}}\mkern
1.0mu^{n}$ consisting of affine transformations $x\longmapsto Ax+b$, $A\in{\rm
GL}(n,{\mathbb{R}})$, $b\in{\mathbb{R}}\mkern 1.0mu^{n}$ acting on
$x\in{\mathbb{R}}\mkern 1.0mu^{n}$. Basic references include Nomizu and Sasaki
[29] and Simon [43]. Keep in mind that, in most of the literature, the term
“affine geometry” usually refers to “equi-affine geometry”, in which one
restricts to the subgroup ${\rm SA}(n,{\mathbb{R}})={\rm
SL}(n,{\mathbb{R}})\ltimes{\mathbb{R}}\mkern 1.0mu^{n}$ of volume-preserving
affine transformations. A key issue is to study the resulting invariants
associated with submanifolds $M\subset{\mathbb{R}}\mkern 1.0mu^{n}$. In
particular, the classical theory for equi-affine hypersurfaces was developed
by Blaschke and his collaborators, [4].

_Centro-affine differential geometry_ refers to the geometry induced by the
general linear group $x\longmapsto Ax$, $A\in{\rm GL}(n,{\mathbb{R}})$,
$x\in{\mathbb{R}}\mkern 1.0mu^{n}$, which is the subgroup of the affine
transformation group that keeps the origin fixed. Similarly, _centro-equi-
affine differential geometry_ refers to the subgroup ${\rm
SL}(n,{\mathbb{R}})$ of volume-preserving linear transformations. These cases
are usually discussed in passing in books that are devoted to (equi-)affine
geometry [29].

Several methods have been developed to construct differential invariants and
other invariant quantities in such Klein geometries, [14, 34, 35, 39, 45]. In
particular, invariants can be straightforwardly and algorithmically obtained
by the method of equivariant moving frames introduced in [12]. They play a
prominent role in the study of the geometric properties, including equivalence
and symmetry, of curves, surfaces, and more general submanifolds, as well as
invariant geometric flows, [32], with many applications to computer vision and
image processing.

The term “invariant submanifold flow” refers to the motion of a curve or
surface by a prescribed partial differential equation that admits an
underlying transformation group as a symmetry group, e.g. the Euclidean group
of rigid motions (translations and rotations). Invariant curve and surface
flows arise in an impressive range of applications, including geometric
optics, elastodynamics, computer vision, visual tracking and control,
interface dynamics, thermal grooving, and elsewhere. A celebrated example is
the curve shortening flow (CSF), in which a plane curve moves in its normal
direction in proportion to its curvature. The CSF was first introduced by
Mullins [28] as a model for the motion of grain boundaries. It was shown by
Gage and Hamilton [13] and Grayson [15, 16] that a simple closed Jordan curve
will remain simple when evolving under the CSF, first becoming convex and then
shrinking to a point in finite time while becoming asymptotically circular,
often referred to as a “circular point”. These results were an important
preliminary step on the road to Hamilton’s celebrated analysis of the higher
dimension counterparts such as mean curvature flow and Ricci flow; the latter
was extended by Perelman to in his famous solution to the Poincaré conjecture,
[9]. The corresponding affine curve shortening flow (ACSF) was introduced and
studied by Angenent, Sapiro, and Tannenbaum [2, 42], and it was proved that a
closed convex curve when evolves according to the ACSF will shrink to an
“elliptical point”. See also [1, 6] and the references therein for further
developments. In computer vision, [35, 37, 38], Euclidean curve shortening and
its equi-affine counterpart have been successfully applied to image denoising
and segmentation and are actively used in practical computer implementations,
both academic and commercial. Euclidean-invariant three-dimensional curve
flows include the integrable vortex filament flow appearing in three-
dimensional fluid dynamics, [19, 22], while mean curvature and Willmore flows
of surfaces have been the subject of extensive analysis and applications, [10,
21]. More recently, similar results were obtained for the heat flow in centro-
equi-affine geometry [46]. Heat flows in more general Klein geometries were
proposed [32, 37].

In this paper, we are interested in the heat flow in centro-affine geometry.
Interestingly, we find that the heat flow for the centro-affine curvature
$\kappa(t,s)$ is equivalent to the well-known _inviscid Burgers’ equation_
$\kappa_{t}=\kappa\kappa_{s}$. This result is in contrast to the behavior of
heat flows in Euclidean, equi-affine, and centro-equi-affine geometries, which
yield second order nonlinear parabolic equations for the associated curvature
invariant.

A challenging problem arising in computer vision and pattern recognition, is
feature matching under viewpoint changes between different images. Image and
feature matching has wide range of applications, including robotic vision,
medical image registration, 3D reconstruction, optical character recognition,
object classification, content-based image retrieval, and so on. Traditional
methods, such as scale invariant feature transform (SIFT) [24] and speeded up
robust features (SURF) [3], have excellent performance and high precision.
However, when the images have less texture complexity and color diversity, it
is not easy to extract and describe the feature points. Another drawback to
these detectors is that they are only invariant under the planar Euclidean
group consisting of rigid motions (rotation, translation, and, possibly,
reflection). In fact, the apparent deformations of three-dimensional objects
caused by changes of the camera position can be locally approximated by affine
maps, and hence, during the image matching process, the Euclidean
transformation group should be extended to the affine transformation group by
including stretching and skewing transformations. Applications of equi-affine
and affine invariance to image processing can be found, for instance, in [25,
26, 44]. An affine-invariant extension of the SIFT algorithm (ASIFT) has been
proposed in [40, 47], which detects feature points in two images that are so
related by simulating many affine transformations of each image and performing
the SIFT algorithm between all image pairs. Going beyond affine-invariant
detectors, projective invariance and moving frame-based signatures have also
been successfully applied in computer vision applications [17]. Recently,
Damelin, Ragozin and Werman investigated the best uniform approximation to a
continuous function under affine transformations, which has applications in
the rapid rendering of computer graphics [8].

In contrast to the above mentioned methods, a centro-affine invariant
detection method offers the following features:

  1. (1)

It depends on the centro-affine differential invariants of smooth closed
curves, in contrast to the ordinary discrete mode matching methods.

  2. (2)

Centro-affine differential invariants involve lower order derivatives of the
curve parametrization, and hence are more accurate and less error prone than
their fully affine counterparts.

  3. (3)

In some situations, centro-affine invariance is equivalent to fully affine
invariance if we can find a pair of exact corresponding points (a point-
correspondence) with respect to an affine transformation. More precisely, for
every closed curve, its barycenter can serve as that point-correspondence (or
local origin) for its local centro-affine invariants.

  4. (4)

The method relies solely on edge detection, and hence can be applied to
untextured images.

The remainder of this paper is organized as follows. In Section 2, we provide
a brief review discussion on the moving frame method and differential
invariants. In Section 3, a classification for the planar curves with constant
centro-affine curvatures is provided. In Section 4, we study the centro-affine
invariant heat flow. In Section 5, an application of the centro-affine
invariants in the matching of images obtained by cameras is discussed.
Finally, Section 6 contains some concluding remarks on this work.

##  2 Preliminaries

###  2.1 Moving frame

Let us first review basic facts on the method of equivariant moving frames
introduced by Fels and the first author [12, 35]. Assume $G$ is an
$r$-dimensional Lie group acting smoothly on an $m$-dimensional manifold $M$:

$G\times M\rightarrow M,\qquad h\cdot(g\cdot z)=(hg)\cdot z.$

A right equivariant moving frame is defined as a smooth map $\rho\colon
M\rightarrow G$, that is equivariant with respect to the action on $M$ and the
inverse right action of $G$ on itself; explicitly,

$\rho\colon\ M\rightarrow G\quad\hbox{satisfies}\quad\rho(g\cdot
z)=\rho(z)g^{-1}.$ (2.1)

The existence of a (local) moving frame requires that the group act freely and
regularly on $M$. The regularity is a global condition and does not play a
role in any of the applications to date. In many cases, one gets by with a
locally free action, in which case the resulting moving frame is locally
equivariant, meaning that (2.1) holds for group elements $g$ sufficiently
close to the identity.

Given local coordinates $z=(z_{1},\ldots,z_{m})$ on $M$, let $w(g,z)=g\cdot z$
be the formulae for the transformed coordinates under the group
transformation. The right moving frame $g=\rho(z)$ associated to the
coordinate cross-section $K=\\{z_{1}=c_{1},\ldots,z_{r}=c_{r}\\}$ is found by
solving the normalization equations

$w_{1}(g,z)=c_{1},\ \ldots,\ w_{r}(g,z)=c_{r},$ (2.2)

for the group parameters $g=(g_{1},\ldots,g_{r})$ in terms of the coordinates
$z=(z_{1},\ldots,z_{m})$. The equivariant moving frames can be obtained by
choosing a cross-section $K$ and solving for the group element $g=\rho(z)$
that takes $z$ to a point $\rho(z)\cdot z\in K$, known as the “canonical form”
of $z$. The coordinates of the canonical form provide a complete system of
non-constant invariants.

######  Theorem 2.1.

If $g=\rho(z)$ is the moving frame obtained by solving equation (2.2), then

$I_{1}(z)=w_{r+1}(\rho(z)\cdot z),\ \ldots,\ I_{z-r}(z)=w_{m}(\rho(z)\cdot
z),$

form a complete system of functionally independent invariants.

######  Definition 2.2.

The invariantization of a scalar function $F\colon M\rightarrow{\mathbb{R}}$
with respect to a right moving frame is the invariant function $I=\iota(F)$
defined by $I(z)=F(\rho(z)\cdot z)$. In particular, if $I$ is any invariant
function, then $I=\iota(I)$.

Thus, invariantization defines a canonical projection from the algebra of
(smooth) functions to the algebra of invariant functions that respects all
algebraic operations.

######  Definition 2.3.

Given a smooth manifold $M$ of dimension $m$ and an integer $1\leq p<m$, the
$k$-th order jet bundle $J^{k}=J^{k}(M,p)$ is a fiber bundle over $M$, such
that the fiber of a point $z\in M$ consists of the set of equivalence classes
of $p$-dimensional submanifolds of $M$ under the equivalence relation of
$k$-th order contact at the point $z$.

The _regular subset_ of the jet bundle is where the action is (locally) free
and regular, which is non-empty when the order $k$ is sufficiently large. A
(local) moving frame of order $k$ can then be constructed on the regular
subset. A $p$-dimensional submanifold is called _regular_ at order $k$ if its
jet belongs to the regular subset, and hence is in the domain of the moving
frame map. See [31] for an algebraic characterization of totally singular
submanifolds, meaning those whose jets are singular at all orders. In centro-
affine geometry, the straight lines are totally singular.

Assume the manifold $M$ has local coordinates
$z=\big{(}x^{1},\ldots,x^{p},u^{1},\ldots,u^{q}\big{)}$ in some neighborhood
where the regular submanifold $S$ can be represented as a graph $u=u(x)$. The
fundamental differential invariants are obtained by invariantization of the
individual jet coordinate functions,

$H^{i}=\iota\big{(}x^{i}\big{)},\qquad
I^{\alpha}_{J}=\iota\big{(}u^{\alpha}_{J}\big{)},\qquad
i=1,\ldots,p,\quad\alpha=1,\ldots,q,\quad\\#J\geq 0.$

One can further apply the invariantization process to differential forms by
the same procedure. First transform the differential form by acting on it by a
general group element $g$ and then invariantization by replacing the group
parameters by their expressions $g=\rho(z)$ in terms of the moving frame.
Details can be found in [12]. In particular the (horizontal components of) the
invariantized basis horizontal one-forms ${\rm d}x^{i}$ give the fundamental
(contact-)invariant one-forms, which we denote by111We ignore all contact
forms, [30], which is the meaning of the $H$ subscript.

$\omega^{i}=\iota\big{(}{\rm d}x^{i}\big{)}_{H},\qquad i=1,\ldots,p.$ (2.3)

In curve geometries, $\omega=\iota({\rm d}x)_{H}$ is the invariant arc length
element, usually denoted as ${\rm d}s$. The corresponding dual _invariant
differential operators_ are denoted by
$\mathcal{D}_{1},\ldots,\mathcal{D}_{p}$, and can be directly obtained by
substituting the moving frame formulas for the group parameters into the
corresponding implicit differentiation operators used to produce the prolonged
group actions. The invariant differential operators map differential
invariants to differential invariants, and hence can be iteratively applied to
generate the higher order differential invariants.

######  Theorem 2.4.

If the moving frame has order $n$, then the set of fundamental differential
invariants

$\mathcal{I}^{(n+1)}=\big{\\{}H_{i},I^{\alpha}_{J}\,|\,i=1,\ldots
p,\,\alpha=1,\ldots,q,\,\\#J\leq n+1\big{\\}}$

of order $\leq n+1$ forms a generating set, meaning that all other
differential invariants can be obtained by invariant differentiation.

In many cases, $\mathcal{I}^{(n+1)}$ does not form a minimal generating set,
owing to the existence of syzygies (algebraic relations) among the
differentiated invariants. Nevertheless, these syzygies and indeed the entire
structure of the differential invariant algebra can be completely determined
by the moving frame calculus, using the powerful recurrence formulae. See [12,
35] for details.

###  2.2 Centro-affine differential invariants for plane curves

Let us now implement the moving frame calculation, based on Section 2.1, for
the centro-affine group ${\rm A}(2,{\mathbb{R}})$ acting on plane curves. In
this case the group is acting on $J^{5}\big{(}{\mathbb{R}}^{2},1\big{)}$. We
represent the planar centro-affine action explicitly in a slightly more
convenient form:

$\left(\begin{matrix}u\\\
v\end{matrix}\right)=\lambda\left(\begin{matrix}\alpha&\beta\\\
\gamma&\delta\end{matrix}\right)\left(\begin{matrix}x\\\
y\end{matrix}\right),\qquad\hbox{where}\quad\det\left(\begin{matrix}\alpha&\beta\\\
\gamma&\delta\\\ \end{matrix}\right)=1,\quad\lambda\neq 0.$

By a direct computation, the prolonged centro-affine transformations up to
order $4$ are given by

$\displaystyle u=\lambda\alpha x+\lambda\beta y,\qquad v=\lambda\gamma
x+\lambda\delta y,$ $\displaystyle v_{u}=\frac{\gamma+\delta
y_{x}}{\alpha+\beta y_{x}},\qquad v_{uu}=\frac{y_{xx}}{\lambda(\alpha+\beta
y_{x})^{3}},$ $\displaystyle v_{uuu}=\frac{(\alpha+\beta y_{x})y_{xxx}-3\beta
y^{2}_{xx}}{\lambda^{2}(\alpha+\beta y_{x})^{5}},$ $\displaystyle
v_{uuuu}=\frac{(\alpha+\beta y_{x})^{2}y_{xxxx}-10\beta(\alpha+\beta
y_{x})y_{xx}y_{xxx}+15\beta^{2}y^{3}_{xx}}{\lambda^{3}(\alpha+\beta
y_{x})^{7}}.$

Let $\epsilon=\operatorname{sign}[(y_{xx}/(xy_{x}-y)]$. Further, after
possibly reparametrizing or applying a centro-affine transformation, we can
specify $\operatorname{sign}(y-xy_{x})=1$. To construct a moving frame, we use
the cross-section normalization

$u=0,\qquad v=1,\qquad v_{u}=0,\qquad v_{uu}=-\epsilon.$

Solving for the group parameters yields

$\displaystyle\lambda^{4}=\frac{\epsilon
y_{xx}}{(xy_{x}-y)^{3}},\qquad\alpha=\lambda y,\qquad\beta=-\lambda x,$
$\displaystyle\gamma=-\frac{y_{x}}{\lambda(y-xy_{x})},\qquad\delta=\frac{1}{\lambda(y-xy_{x})},$
(2.4)

which prescribes the right-equivariant moving frame. Invariantizing the
horizontal one-form

${\rm d}u_{H}=(\lambda\alpha+\lambda\beta y_{x}){\rm d}x$

by substituting the moving frame formulae (2.4) produces the centro-affine
arc-length element

${\rm d}s=\iota({\rm
d}x)_{H}={\operatorname{sign}}(y-xy_{x})\sqrt{\epsilon\frac{y_{xx}}{xy_{x}-y}}\
{\rm d}x=\sqrt{\epsilon\frac{y_{xx}}{xy_{x}-y}}{\rm d}x,$ (2.5)

By a direct calculation, we produce the fundamental differential invariant

$\displaystyle\kappa=\iota(y_{xxx})={\operatorname{sign}}(y-xy_{x})\frac{3xy^{2}_{xx}+(y-xy_{x})y_{xxx}}{y^{2}_{xx}}\sqrt{\epsilon\,\frac{y_{xx}}{xy_{x}-y}}$
$\displaystyle\hphantom{\kappa=\iota(y_{xxx})}{}=\frac{3xy^{2}_{xx}+(y-xy_{x})y_{xxx}}{y^{2}_{xx}}\sqrt{\epsilon\,\frac{y_{xx}}{xy_{x}-y}},$
(2.6)

which we identify as the _centro-affine curvature_. Higher order invariants
are all obtained by invariant differentiation of $\kappa$ with respect to the
centro-affine arc length (2.5), and so a complete list of differential
invariants is given by $\kappa,\kappa_{s},\kappa_{ss},\ldots$. In particular,
one can show, either by direct calculation or by using the recurrence formulae
[12], that

$\iota(y_{xxxx})=\kappa_{s}+\frac{3}{2}\kappa^{2}-3,\qquad\iota(y_{xxxxx})=\kappa_{ss}+5\kappa\kappa_{s}+3\kappa^{3}-16\kappa,$

and so on.

It is also of interest to obtain the formulas for the centro-affine curvature
and arc length for a general parametrized curve. Consider a smooth curve
parameterized by

$\mathbf{x}(p)=(x(p),y(p))^{\mathrm{T}},$

where $x(p)$, $y(p)$ are smooth functions of the parameter $p$ defined over a
certain interval $I$, and the superscript “T” represents the transpose of a
vector or matrix. We use dots to denote derivatives with respect to the
parameter $p$. In particular $\dot{\mathbf{x}}={\rm d}\mathbf{x}/{\rm d}p$ is
the tangent vector.

To write out the formulas, we will use the bracket notation

$[\mathbf{a},\mathbf{b}]=\det(\mathbf{a},\mathbf{b}),\qquad\mathbf{a},\mathbf{b}\in{\mathbb{R}}^{2},$

to denote the cross product in the plane. Let $s$ be the centro-affine arc-
length parameter, where $\mathbf{x}^{\prime}={\rm d}\mathbf{x}/{\rm d}s$ is
used to distinguish
$\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{\mathbf{x}}={\rm
d}\mathbf{x}/{\rm d}p$. We first note that a parametrized curve
$\mathbf{x}(p)$ is regular if and only if it satisfies

$\bigl{[}\mathbf{x},\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{\mathbf{x}}\bigr{]}\neq
0,\qquad\bigl{[}\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{\mathbf{x}},\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{\mathbf{x}}\bigr{]}\neq
0.$ (2.7)

The required formulas are obtained by replacing the jet derivative coordinates
$y_{x},y_{xx},\ldots$ by using the chain rule to express $x$-derivatives in
terms of $p$-derivatives, as given by

$y_{nx}\longmapsto D_{x}^{n}y,$

where

$D_{x}=\frac{1}{\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{x}}\frac{{\rm
d}}{{\rm d}p}$

is the differentiation operator dual to the horizontal one-form ${\rm
d}x=\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{x}\,{\rm
d}p$. Thus,

$y_{x}\longmapsto\frac{\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{y}}{\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{x}},\qquad
y_{xx}\longmapsto\frac{\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{x}\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{y}-\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{x}\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{y}}{\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{x}{}^{3}},$

and so on. Substituting into (2.5) produces the general formula for the
centro-affine arc-length element of a parametrized curve

${\rm
d}s=\sqrt{\epsilon\frac{\bigl{[}\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{\mathbf{x}},\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{\mathbf{x}}\bigr{]}}{\bigl{[}\mathbf{x},\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{\mathbf{x}}\bigr{]}}}{\rm
d}p,\qquad\textrm{where}\quad\epsilon=\operatorname{sign}\left(\frac{\bigl{[}\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{\mathbf{x}},\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{\mathbf{x}}\bigr{]}}{\bigl{[}\mathbf{x},\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{\mathbf{x}}\bigr{]}}\right).$
(2.8)

One easily verifies that $\epsilon$ is invariant under centro-affine
transformations and reparametrizations, including those that are orientation
reversing. Similarly, its centro-affine curvature (2.6) is given by the
general formula

$\displaystyle\sqrt{\epsilon\frac{\bigl{[}\mathbf{x},\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{\mathbf{x}}\bigr{]}}{\bigl{[}\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{\mathbf{x}},\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{\mathbf{x}}\bigr{]}}}\left(\frac{\bigl{[}\mathbf{x},\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{\mathbf{x}}\bigr{]}}{\bigl{[}\mathbf{x},\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{\mathbf{x}}\bigr{]}}+\frac{\bigl{[}\mathbf{x},\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{\mathbf{x}}\bigr{]}\bigl{[}\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{\mathbf{x}},\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{\mathbf{x}}\bigr{]}-\bigl{[}\mathbf{x},\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{\mathbf{x}}\bigr{]}\bigl{[}\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{\mathbf{x}},\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{\mathbf{x}}\bigr{]}}{2\bigl{[}\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{\mathbf{x}},\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{\mathbf{x}}\bigr{]}\bigl{[}\mathbf{x},\overset{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}{\mathbf{x}}\bigr{]}}\right).$
(2.9)

It is easy to check that (2.9) is equivalent to (2.6), up to a constant
factor. For the sake of convenience, we choose (2.9) as the centro-affine
curvature $\kappa$. In particular, parametrizing the curve by centro-affine
arc-length $s$, that is,

$\frac{\bigl{[}\mathbf{x}^{\prime},\mathbf{x}^{\prime\prime}\bigr{]}}{[\mathbf{x},\mathbf{x}^{\prime}\bigr{]}}=\epsilon,\qquad\hbox{which
implies}\quad\frac{\bigl{[}\mathbf{x}^{\prime},\mathbf{x}^{\prime\prime\prime}\bigr{]}\bigl{[}\mathbf{x},\mathbf{x}^{\prime}\bigr{]}-\bigl{[}\mathbf{x}^{\prime},\mathbf{x}^{\prime\prime}\bigr{]}\bigl{[}\mathbf{x},\mathbf{x}^{\prime\prime}\bigr{]}}{\bigl{[}\mathbf{x},\mathbf{x}^{\prime}\bigr{]}^{2}}=0,$

one obtains the formula [14, 34, 39, 45]:

$\kappa=\frac{\bigl{[}\mathbf{x}^{\prime\prime},\mathbf{x}\bigr{]}}{\bigl{[}\mathbf{x}^{\prime},\mathbf{x}\bigr{]}},$
(2.10)

and

$\mathbf{x}^{\prime\prime}=\kappa\mathbf{x}^{\prime}-\epsilon\mathbf{x}.$
(2.11)

Furthermore, from equation (2.11), we have

$\mathbf{x}^{\prime\prime\prime}=\big{(}\kappa_{s}+\kappa^{2}-\epsilon\big{)}\mathbf{x}^{\prime}-\epsilon\kappa\mathbf{x},$
(2.12)

and

$\mathbf{x}^{\prime\prime\prime\prime}=\big{(}\kappa^{\prime\prime}+3\kappa\kappa_{s}-2\epsilon\kappa+\kappa^{3}\big{)}\mathbf{x}^{\prime}-\epsilon\big{(}2\kappa_{s}+\kappa^{2}-\epsilon\big{)}\mathbf{x}.$

######  Remark 2.5.

For a regular curve $\mathbf{x}(p)$, $\epsilon\equiv 1$ or $\epsilon\equiv-1$.
The value of $\epsilon$ indicates that whether the vectors $\mathbf{x}$,
$\mathbf{x}_{ss}$ lie on the same or opposite sides of the tangent vector
$\mathbf{x}_{s}$. For example, in Fig. 1, $\epsilon=1$ on the red parts and
$\epsilon=-1$ on the blue parts of the curve. The points in between the red
and blue parts are irregular points.

|
---|---
Figure 1: The geometrical properties of the centro-affine invariant
$\epsilon$.

######  Remark 2.6.

If $\kappa(s)<0$, $s\in(a,b)$, let $\tilde{\kappa}(\tilde{s})=-\kappa(s)>0$,
$\tilde{s}=-s\in(-b,-a)$. Then, by solving the differential equation

$\frac{{\rm d}^{2}\mathbf{x}(\tilde{s})}{{\rm
d}\tilde{s}^{2}}-\tilde{\kappa}(\tilde{s})\frac{{\rm
d}\mathbf{x}(\tilde{s})}{{\rm
d}\tilde{s}}+\epsilon\mathbf{x}(\tilde{s})=0,\qquad\tilde{s}\in(-b,-a),$

modeled on (2.11), we obtain a curve $\tilde{\mathbf{C}}(\tilde{s})$ with the
centro-affine curvature $\tilde{\kappa}(\tilde{s})>0$, $\tilde{s}\in(-b,-a)$.
If we perform the orientation-reversing reparametrization
$s=-\tilde{s}\in(a,b)$, then the resulting curve $\mathbf{C}(s)$ is equivalent
to the curve $\tilde{\mathbf{C}}(\tilde{s}),\tilde{s}\in(-b,-a)$, and its
centro-affine curvature satisfies $\kappa(s)<0$.

##  3 Centro-affine planar curves with constant curvature

In any Klein geometry, submanifolds that have the property that all their
differential invariants are constant play a particularly important role. The
regular ones can all be algebraically characterized by the following theorem,
originally due to Cartan [5, 12]. See also [31, 36] for further details.

######  Theorem 3.1.

Let $G$ be a Lie group acting on an $m$-dimensional manifold $M$. Then, for
$1\leq p<m$, a regular $p$-dimensional submanifold $S\subset M$ has all
constant differential invariants if and only if it is a subset of an orbit of
a $($suitable$)$ $p$-dimensional subgroup $H\subset G$. In this case, $H$ can
be identified with the local symmetry group of $S$.

The totally singular $p$-dimensional submanifolds, as defined above, are
characterized by their admitting a (local) symmetry group of dimension $>p$.
They can be characterized differentially by a Lie determinant condition [31].
The caveat “suitable” in the above result means that the subgroup $H$ has
$p$-dimensional orbits and that such orbits are not totally singular. See [31]
for an algebraic characterization.

Specializing to plane curves, we assume the transformation group is _ordinary_
, meaning that it acts transitively and does not pseudo-stabilize [30], which
is the case for all fundamental geometric transformation groups including the
centro-affine and centro-equi-affine groups. In this case, the differential
invariant algebra is generated by invariantly differentiating a single
differential invariant, which we identify as the $G$-invariant curvature.
Thus, by the above results, a regular curve $C\subset X$ has constant
curvature if and only if it forms part of the orbit of a suitable one-
parameter subgroup $H\subset G$, and admits $H$ as its (local) symmetry group.
Two such curves are _equivalent_ , meaning the $\widetilde{C}=g\cdot C$ for
some $g\in G$ is their corresponding symmetry groups are related by the
adjoint map $\widetilde{H}=\operatorname{Ad}g\cdot H=g\cdot H\cdot g^{-1}$.
Thus one can use the methods of classification of optimal subalgebras of the
Lie algebra of $G$ to classify constant curvature curves up to equivalence. On
the other hand, totally singular curves are characterized by their admitting
local symmetry groups of dimension $\geq 2$. All other curves have at most a
discrete group of local symmetries.

Now let us apply these considerations to the centro-affine group. The first
remark is that a curve is totally singular if and only if it is a straight
line segment. This follows easily from the prolonged action, or by calculation
of the Lie determinant. For the group ${\rm GL}(2,{\mathbb{R}})$, there are
three inequivalent families of one-parameter subgroups, respectively generated
by one of the Lie algebra elements

$\left(\begin{matrix}1&0\\\
0&\alpha\end{matrix}\right),\qquad\left(\begin{matrix}1&1\\\
0&1\end{matrix}\right),\qquad\left(\begin{matrix}\cos\varphi&-\sin\varphi\\\
\sin\varphi&\cos\varphi\end{matrix}\right),$ (3.1)

where $\alpha\neq 0$ and $0<\varphi<\pi$. In particular, when $\alpha=1$, the
corresponding orbit is a straight line and hence totally singular.
Exponentiating the infinitesimal generators (3.1) to determine the
corresponding one-parameter subgroups and orbits, we deduce the following
result.

######  Theorem 3.2.

Let $\mathbf{C}$ be a constant centro-affine curvature curve with
$\epsilon=1$.

  * $(i)$

If $\kappa>2$, then $\mathbf{C}$ is centro-affine equivalent to a curve

$y=x^{\alpha},\qquad x>0,\quad
0<\alpha<1,\qquad\textrm{where}\quad\kappa=\frac{1}{\sqrt{\alpha}}+\sqrt{\alpha}.$

  * $(ii)$

If $\kappa=2$, then $\mathbf{C}$ is centro-affine equivalent to the curve
$y=x\log x$.

  * $(iii)$

If $\kappa=0$, then $\mathbf{C}$ is centro-affine equivalent to the unit
circle.

  * $(iv)$

If $0<\kappa<2$, then $\mathbf{C}$ is centro-affine equivalent to the
logarithmic spiral with polar coordinates

$\rho=\exp(\theta\cot\varphi),\qquad
0<\varphi<\pi/2,\qquad\textrm{where}\quad\kappa=2\cos\varphi.$

Note that the curves in Theorem 3.2 are generated from the subgroups in (3.1),
with $\varphi=\pi/2$ corresponding to the circle. The above values of the
centro-affine curvature can be found either by direct computation or by
applying the intrinsic algebraic method of [33].

######  Proposition 3.3.

A non-degenerate centro-affine curve has centro-affine curvature $\kappa=0$ if
and only if it is locally centro-affine equivalent to the unit hyperbola
$y=x^{-1}$ or unit circle $x^{2}+y^{2}=1$.

##  4 Centro-affine invariant geometric heat flow

Let us next investigate centro-affine invariant evolutionary processes.
Consider a family of embedded smooth Jordan curves parametrized by
$\mathbf{C}\colon S^{1}\times I\rightarrow{\mathbb{R}}^{2}$, where $t\in
I\subset{\mathbb{R}}$ can be viewed as the time parameter and $p\in S^{1}$ is
a free parameter of each individual curve in the family. We assume that the
curve family $\mathbf{C}(p,t)$ evolves according to the centro-affine
invariant evolution equation

$\frac{\partial\mathbf{C}}{\partial t}=\beta(\kappa(s,t))\mathbf{C}_{ss}$
(4.1)

with the initial condition

$\mathbf{C}(s,0)=\mathbf{C}_{0}(s),$

where $s$ is the corresponding centro-affine arc-length, $\kappa(s,t)$ is the
centro-affine curvature, $\beta(\kappa)$ is a prescribed function of $\kappa$.

Geometrically, equation (4.1) means that any point of the curve moves with a
velocity in the direction of the “normal” vector $\mathbf{C}_{ss}$, with speed
proportional to a function of the centro-affine curvature of the curve at this
point. These kinds of equations arise in differential geometry and a variety
of applications due to their inherent invariance. On the other hand, because
the arc length parameter $s$ changes with time, (4.1) is a non-linear
evolutionary equation.

In view of (2.8), let

$g(p):=\sqrt{\epsilon\frac{[\mathbf{C}_{p},\mathbf{C}_{pp}]}{[\mathbf{C},\mathbf{C}_{p}]}}$
(4.2)

be the invariant centro-affine metric for the curve $C(p,t)$. The arc length
parameter $s$ is obtained by integration:

$s(p)=\int^{p}_{p_{0}}g(\xi){\rm d}\xi.$

In view of the elementary commutator relation
$\displaystyle\frac{\partial}{\partial t}\frac{\partial}{\partial
p}=\frac{\partial}{\partial p}\frac{\partial}{\partial t}$, we have

$\frac{\partial}{\partial t}\frac{\partial}{\partial
s}=\frac{\partial}{\partial t}\left(\frac{1}{g}\frac{\partial}{\partial
p}\right)=-\frac{g_{t}}{g}\frac{\partial}{\partial s}+\frac{\partial}{\partial
s}\frac{\partial}{\partial t}.$

Next, let us compute the centro-affine metric evolution. Firstly, using
equation (4.2),

$\frac{\partial\big{(}g^{2}\big{)}}{\partial
t}=\epsilon\frac{([\mathbf{C}_{pt},\mathbf{C}_{pp}]+[\mathbf{C}_{p},\mathbf{C}_{ppt}])[\mathbf{C},\mathbf{C}_{p}]-[\mathbf{C}_{p},\mathbf{C}_{pp}]([\mathbf{C}_{t},\mathbf{C}_{p}]+[\mathbf{C},\mathbf{C}_{pt}])}{[\mathbf{C},\mathbf{C}_{p}]^{2}}.$

Note that since the tangent $\mathbf{C}_{s}$ is not parallel to $\mathbf{C}$,
then the right hand side of (4.1) can be expressed as the linear combination
of $\mathbf{C}_{s}$ and $\mathbf{C}$, which means

$\beta\,\mathbf{C}_{ss}=W\mathbf{C}_{s}+U\mathbf{C},\qquad{\rm where}\quad
W=\beta\kappa,\quad U=-\epsilon\beta.$

By a direct computation, we obtain

$\displaystyle\mathbf{C}_{p}=g\mathbf{C}_{s},$
$\displaystyle\mathbf{C}_{pp}=gg_{s}\mathbf{C}_{s}+g^{2}\mathbf{C}_{ss}=\big{(}gg_{s}+g^{2}{\kappa}\big{)}\mathbf{C}_{s}-\epsilon
g^{2}\mathbf{C},$
$\displaystyle\mathbf{C}_{pt}=(W\mathbf{C}_{s}+U\mathbf{C})_{p}$
$\displaystyle\hphantom{\mathbf{C}_{pt}}{}=g(W_{s}\mathbf{C}_{s}+W\mathbf{C}_{ss}+U_{s}\mathbf{C}+U\mathbf{C}_{s})=g(W_{s}+W{\kappa}+U)\mathbf{C}_{s}+g(U_{s}-\epsilon
W)\mathbf{C},$ $\displaystyle\mathbf{C}_{ppt}=\big{[}gg_{s}(W_{s}+\kappa
W+U)+g^{2}(W_{ss}+2\kappa W_{s}+(\kappa_{s}+\kappa^{2}-\epsilon)W+U_{s}+\kappa
U)\big{]}\mathbf{C_{s}}$
$\displaystyle\hphantom{\mathbf{C}_{ppt}=}{}+\big{[}gg_{s}(U_{s}-\epsilon
W)-\epsilon g^{2}(2W_{s}+\kappa W-\epsilon U_{ss}+U)\big{]}\mathbf{C}.$

Then

$\displaystyle[\mathbf{C}_{pt},\mathbf{C}_{pp}]=\epsilon
g^{2}(g(W_{s}+U+\epsilon\kappa U_{s})-g_{s}(W-\epsilon
U_{s}))[\mathbf{C},\mathbf{C}_{s}],$
$\displaystyle[\mathbf{C}_{p},\mathbf{C}_{ppt}]=\epsilon
g^{2}(g(2W_{s}+{\kappa}W-\epsilon U_{ss}+U)-\epsilon g_{s}(U_{s}-\epsilon
W))[\mathbf{C},\mathbf{C}_{s}],$
$\displaystyle[\mathbf{C}_{p},\mathbf{C}_{pp}]=\epsilon
g^{3}[\mathbf{C},\mathbf{C}_{s}],$
$\displaystyle[\mathbf{C}_{t},\mathbf{C}_{p}]=gU\left[\mathbf{C},\mathbf{C}_{s}\right],$
$\displaystyle[\mathbf{C},\mathbf{C}_{pt}]=g(W_{s}+W{\kappa}+U)[\mathbf{C},\mathbf{C}_{s}].$

It follows that

$\displaystyle\frac{1}{g}\frac{\partial g}{\partial
t}=\;\frac{1}{2}(2W_{s}+\epsilon{\kappa}U_{s}-\epsilon
U_{ss})=\frac{1}{2}(\beta_{ss}+\kappa\beta_{s})+\kappa_{s}\beta.$ (4.3)

We now come to a crucial computation, namely the centro-affine curvature
evolution

$\displaystyle\left.\frac{\partial{\kappa}}{\partial
t}\right|_{p}=\frac{\partial}{\partial
t}\frac{[\mathbf{C},\mathbf{C}_{ss}]}{[\mathbf{C},\mathbf{C}_{s}]}=\frac{([\mathbf{C}_{t},\mathbf{C}_{ss}]+[\mathbf{C},\mathbf{C}_{sst}])[\mathbf{C},\mathbf{C}_{s}]-[\mathbf{C},\mathbf{C}_{ss}]([\mathbf{C}_{t},\mathbf{C}_{s}]+[\mathbf{C},\mathbf{C}_{st}])}{[\mathbf{C},\mathbf{C}_{s}]^{2}}.$

A direct computation gives

$\displaystyle\mathbf{C}_{st}=\mathbf{C}_{ts}-\frac{g_{t}}{g}\mathbf{C}_{s}=(W\mathbf{C}_{s}+U\mathbf{C})_{s}-\frac{g_{t}}{g}\mathbf{C}_{s}=\left(W_{s}+{\kappa}W+U-\frac{g_{t}}{g}\right)\mathbf{C}_{s}+(U_{s}-\epsilon
W)\mathbf{C},$
$\displaystyle\mathbf{C}_{sst}=\mathbf{C}_{sts}-\frac{g_{t}}{g}\mathbf{C}_{ss}=\left[\left(W_{s}+{\kappa}W+U-\frac{g_{t}}{g}\right)\mathbf{C}_{s}+(U_{s}-\epsilon
W)\mathbf{C}\right]_{s}-{\kappa}\frac{g_{t}}{g}\mathbf{C}_{s}+\epsilon\frac{g_{t}}{g}\mathbf{C}$
$\displaystyle\hphantom{\mathbf{C}_{sst}}{}=\left[W_{ss}+\big{(}\kappa_{s}+\kappa^{2}-\epsilon\big{)}W+2U_{s}+\kappa\left(2W_{s}+U-2\frac{g_{t}}{g}\right)-\left(\frac{g_{t}}{g}\right)_{s}\right]\mathbf{C}_{s}$
$\displaystyle\hphantom{\mathbf{C}_{sst}=}{}-\epsilon\left[-2W_{s}+{\kappa}W-\epsilon
U_{ss}+U-2\frac{g_{t}}{g}\right]\mathbf{C}.$

Thus we arrive at

$\displaystyle\frac{\partial{\kappa}}{\partial
t}={\kappa}_{s}W+2U_{s}-\frac{\epsilon}{2}\big{(}{\kappa}_{s}U_{s}+{\kappa}^{2}U_{s}-U_{sss}\big{)}=\beta{\kappa}{\kappa}_{s}-2\epsilon\beta_{s}+\frac{1}{2}\big{(}\big{(}\kappa_{s}+{\kappa}^{2}\big{)}\beta_{s}-\beta_{sss}\big{)}.\\!\\!\\!$
(4.4)

We now focus on the case of $\beta\equiv 1$, namely the _heat flow_ in centro-
affine geometry

$\frac{\partial\mathbf{C}}{\partial t}=\mathbf{C}_{ss}.$ (4.5)

Equation (4.3) implies that

$g_{t}=g\kappa_{s}.$ (4.6)

Consequently, in view of equation (4.4), we see that the centro-affine
curvature satisfies the first order inviscid Burgers’ equation

$\frac{\partial\kappa}{\partial t}=\kappa\kappa_{s}.$ (4.7)

To sum up, we arrive at the following results.

######  Theorem 4.1.

The centro-affine curve evolution process

$\frac{\partial\mathbf{C}}{\partial
t}=\mathbf{C}_{ss},\qquad\mathbf{C}(s,0)=\mathbf{C}_{0},$

is equivalent to the initial problem of the inviscid Burgers’ equation:

$\frac{\partial\kappa}{\partial t}=\kappa\frac{\partial\kappa}{\partial
s},\qquad\kappa(s,0)=\kappa_{0}(s),$

where $\kappa_{0}(s)$ is the signed centro-affine curvature of the initial
curve $\mathbf{C}$.

Comparing equation (4.6) with equation (4.7), one has

$\frac{\partial\kappa}{\partial t}-\frac{\kappa}{g}\frac{\partial g}{\partial
t}=0.$

Hence

$\frac{\partial(\kappa/g)}{\partial
t}=\frac{1}{g}\left(\frac{\partial\kappa}{\partial
t}-\frac{\kappa}{g}\frac{\partial g}{\partial t}\right)=0.$

Thus we conclude:

######  Corollary 4.2.

$\displaystyle\frac{\kappa}{g}(p,t)=\frac{\kappa}{g}(p,0)$ remains invariant
for any $t>0$.

Therefore, equation (4.7) can be written as

$\frac{\partial\kappa}{\partial
t}=\frac{\kappa}{g}(p,0)\frac{\partial\kappa}{\partial p}.$ (4.8)

In fact, if $\kappa_{0}(s)=\kappa(s,0)\equiv 0$, by (4.8), the solution for
(4.5) is $\mathbf{C}(\cdot,t)=\exp(-\epsilon t)\mathbf{C}_{0}$. We immediately
deduce the following result.

######  Corollary 4.3.

If the initial centro-affine curvature $\kappa_{0}(s)=\kappa(s,0)\equiv 0$,
then at any time $t\geq 0$, the curve $\mathbf{C}(t)$ is centro-affine
equivalent to the initial curve $\mathbf{C}_{0}$.

In the following, we assume $\kappa(p,0)\neq 0$. Solve the above partial
differential equation (4.8) by the method of characteristics yields

$\kappa(p,t)=\Phi(t+h(p)),$

where $\Phi$ is any differentiable function of one variable and $\displaystyle
h(p)=\int^{p}\frac{g}{\kappa}(\tilde{p},0)\,{\rm d}\tilde{p}$. Since
$\kappa(p,0)\neq 0$, the ratio $g(p,0)/\kappa(p,0)$ will be of one sign, and
hence $h(p)$ defines a one-to-one map. Thus we conclude:

######  Corollary 4.4.

The curvature $\kappa(p,t)$ remains invariant on the curve $t+h(p)=C$, where
$C$ is constant. At the same time,
$\kappa(p,t)=\kappa\big{(}h^{-1}(C),0\big{)}$, i.e., at any given time $t$, we
have $\kappa(p,t)=\kappa\big{(}h^{-1}(t+h(p)),0\big{)}$.

The flow (4.1) can be written as

$\frac{\partial\mathbf{C}}{\partial
t}=\mathbf{C}_{ss}=\kappa\mathbf{C}_{s}-\epsilon\mathbf{C}.$ (4.9)

In view of Corollary 4.2, the heat flow (4.9) is equivalent to

$\epsilon\frac{\kappa}{g}(p,0)\mathbf{C}_{p}-\epsilon\mathbf{C}_{t}=\mathbf{C},$

which can be solved easily, to get

$\mathbf{C}(p,t)=\exp(-\epsilon
t)\mathbf{\Psi}\left(t+\int^{p}\frac{g}{\kappa}\,{\rm d}p\right),$

where the vector function
$\displaystyle\mathbf{\Psi}\left(\int^{p}\frac{g}{\kappa}\,{\rm
d}p\right)=\mathbf{C}_{0}$, $\mathbf{C}_{0}=\mathbf{C}(p,0)$,
$p\in(p_{1},p_{2})$, is the initial value of $\mathbf{C}(p,t)$, which implies

$\mathbf{\Psi}(\tilde{p})=\mathbf{C}_{0}\big{(}h^{-1}(\tilde{p})\big{)},\qquad\tilde{p}\in(h(p_{1}),h(p_{2})).$
(4.10)

At any given time $\tau$,
$\mathbf{\Psi}(\tilde{p}+\tau)=\mathbf{C}_{0}\big{(}h^{-1}(\tilde{p}+\tau)\big{)}$,
where $\tilde{p}\in(h(p_{1})-\tau,h(p_{2})-\tau)$.

On the other hand, by (4.3), the evolution of centro-affine arc-length $L$ for
$\mathbf{C}(p,0)$ at $(p_{1},p_{2})$ is

$\frac{\partial L}{\partial t}=\frac{\partial}{\partial
t}\int^{p_{2}}_{p_{1}}g\,{\rm d}p=\kappa(p_{2})-\kappa(p_{1}).$

Hence, we have

######  Corollary 4.5.

The solution to the initial problem of the heat flow (4.9) with initial curve
$\mathbf{C}_{0}$, $p\in(p_{1},p_{2})$ and $\kappa\neq 0$ is given by
$\displaystyle\mathbf{C}(p,t)=\exp(-\epsilon
t)\mathbf{\Psi}\left(t+\int^{p}\frac{g}{\kappa}\,{\rm d}p\right)$ with
$\mathbf{\Psi}$ satisfying (4.10).

A basic fact on the theory of curve evolution [6, 11] states that the
geometric shape of the curve, sometimes referred to as the trace or the image
of the planar curve, is affected only by the normal component of the flow
field. The tangential component affects only the parametrization, and not the
the curves’ overall geometric shape. Thus, equation (4.9) is equivalent to

$\frac{\partial\mathbf{C}}{\partial t}=-\epsilon\mathbf{C}.$ (4.11)

Solving the above equation, we have

$\mathbf{C}(p,t)=\exp(-\epsilon\,t)\mathbf{C}(p,0).$

In this manner, we arrive at the long time behavior of the curves governed by
the flow (4.9):

######  Proposition 4.6.

The curve family $\mathbf{C}(p,t)$ with $\epsilon=1$ will converge smoothly to
the origin as $t\to\infty$.

##  5 Edge matching of curve profiles in digital images

For many vision tasks, including 3D reconstruction, image alignment, and
tracking, a key issue is finding correspondences between common objects in
images. The SIFT and SURF algorithms are among the most widely applied to the
identification of corresponding feature points. In fact, when the images have
less texture complexity and color diversity, it is not so easy to accurately
extract and describe the feature points through application of the SIFT and
SURF algorithms. Another drawback to these detectors is that they are only
Euclidean-invariant.

In general, the camera is often modeled as a projective transformation from
scene coordinates to image coordinates. Especially, if a physical object has a
smooth or piecewise smooth boundary, its images obtained by cameras in varying
positions undergo apparent deformations, which are locally well approximated
by affine transforms of the image plane. In consequence, the solid object
recognition problem will lead back to the computation of affine invariant
local image features. That is, during image matching, the Euclidean group
(rotation, translation, reflection) should be extended to the equi-affine or
full affine transformation group by including stretching and skewing
transformations.

In comparison with the above-mentioned methods, differential centro-affine-
invariant detection has the following advantageous features:

  1. (1)

It depends on the differential invariants of smooth curves for reducing
errors.

  2. (2)

Centro-affine differential invariants involve lower order derivatives of the
curve parametrization, and hence are more accurate and less error prone than
fully affine differential invariants.

  3. (3)

In some situations, centro-affine invariance is equivalent to fully affine
invariance if we can find a pair of exact corresponding points (a point-
correspondence) with respect to an affine transformation.

  4. (4)

More precisely, for every closed curve, its barycenter can be temporarily
served as that point-correspondence (or the local origin) only for its local
centro-affine invariants.

  5. (5)

The method relies solely on edge detection, which may well be adequate for
untextured images.

|  |  |
---|---|---|---
Figure 2: The original images.

In this section, we apply the centro-affine differential invariants defined in
the previous sections to identify objects in different images. In Fig. 2,
photos of a sign with the word “Standards” were taken by the camera at
different angles and directions. We seek correspondences between the letters
according to the centro-affine invariants given in equations (2.8) and (2.9).

---
Figure 3: The smooth edges generated by the B-spline curve fitting method and
their ordinals, which will be used in Table 1.

In order to describe more clearly and accurately this process, we now focus on
the initial letter “S” as a main example; the others can be dealt with
similarly. First of all, it is necessary to obtain the boundaries of the
objects in the images. Accurate boundaries of the individual letters are
segmented using level set method [23] or Canny edge detection [27]. We shall
apply the centro-affine arc-length, the integral invariant of centro-affine
curvature, the area, the barycenter and corner points of every boundary as its
descriptor, which can be employed to recognize the corresponding boundaries in
different digital images. Corner point in a boundary represents critical
information in describing object features, which is the local extreme point
related to the Euclidean curvature. Then after the boundary extraction
(segmentation), the letter “S” is located inside of its discrete boundary
points.

Notice that, for a given closed curve, its local centro-affine invariants may
be obtained by placing the temporary origin at the barycenter of its closed
boundary. To be precise, the smoothing algorithm – curve shortening flow [10,
13] or Gaussian convolution for a curve [27] – can play an auxiliary role in
the pretreatment process for decreasing the jaggedness in the curve. For
simplicity, we use Gaussian convolution for smoothing. The curve $\Gamma$ is
first parameterized by the parameter $u$, so $\Gamma(u)=(x(u),y(u))$. An
evolved version $\Gamma_{\sigma}$ of $\Gamma$ can then be computed, and is
defined by $\Gamma_{\sigma}=(x(u,\sigma),y(u,\sigma))$, where
$x(u,\sigma)=x(u)*g(u,\sigma)$, $y(u,\sigma)=y(u)*g(u,\sigma)$, with $*$
denoting the convolution operator, while $g(u,\sigma)$ denotes a Gaussian of
width $\sigma$. For all boundaries, we apply k-means clustering [18] to ensure
that they are represented by the same number of points, which are used for
further fitting purposes. By experimentation, we find that it is adequate to
select $85$ points for every boundary. To obtain the centro-affine invariants
by (2.8), (2.9), the discrete boundary points should generate a curve smooth
enough for computing derivatives. B-spline curves are suitable for this role
because of their affine invariance and smoothness [7, 20]. Thus, by a direct
operation, we obtain the smooth boundaries as shown in Fig. 3.

If the boundary curve of the letter “S” is sufficiently smooth, it is easy to
calculate its centro-affine invariants $\epsilon$ and centro-affine curvature
$\kappa$ appearing in (2.8) and (2.9), respectively. The final results are
shown in Fig. 4. To reduce the disturbance caused by irregular points, we set
threshold value of centro-affine curvatures $|\kappa|$ to $100$, deleting the
points whose centro-affine curvatures $|\kappa|>100$. In view of the pictures
in Fig. 4, we observe that they look similar modulo an overall translation in
$s$. Now, it is crucial to find the corresponding points between them.

|
---|---
|
Figure 4: The differential centro-affine invariants $\epsilon$ and $|\kappa|$
for the first letter “S”. The horizontal axis represents the centro-affine
arc-length up to a factor.

We employ the ${\rm L}^{2}$ inner product

$\displaystyle\langle\,{f(x)}\,,{g(x)}\,\rangle=\int^{b}_{a}f(x)g(x)\,{\rm
d}x$

between functions $f(x)$ and $g(x)$ over the interval $[a,b]$. The associated
norm

$\operatorname{dist}(f(x),g(x))=\|\,{f(x)-g(x)}\,\|=\sqrt{\int^{b}_{a}(f(x)-g(x))^{2}\,{\rm
d}x},$

will be used as a measure of their distance.

Now we can apply this concept to compare two discrete curves. Let us assume
that, the curve $A$ consists of $N$ points, denoted by

$(x_{a_{1}},y_{a_{1}})^{\rm T},\ (x_{a_{2}},y_{a_{2}})^{\rm T},\ \ldots,\
(x_{a_{N}},y_{a_{N}})^{\rm T}.$

In this way, we denote the curve $B$ consists of $M$ points by

$(x_{b_{1}},y_{b_{1}})^{\rm T},\ (x_{b_{2}},y_{b_{2}})^{\rm T},\ \ldots,\
(x_{b_{M}},y_{b_{M}})^{\rm T}.$

Locally, we can always assume that $x_{a_{1}}<x_{a_{2}}<\cdots<x_{a_{N}}$ and
$x_{b_{1}}<x_{b_{2}}<\cdots<x_{b_{M}}$. For the sake of convenience, we use
the following notations to indicate the maps of curve $A$ and curve $B$, that
is,

$y_{a_{i}}=f(x_{a_{i}}),\qquad y_{b_{j}}=g(x_{b_{j}}),\qquad
i=1,2,\ldots,N,\quad j=1,2,\ldots,M.$

Two new sets of $x$-coordinates are generated by trimming the original
$\\{x_{a_{i}}\\}$ and $\\{x_{b_{i}}\\}$ to only those values such that every
$x^{\prime}_{a_{i}}$ in the new set has two neighbours from $\\{x_{b_{i}}\\}$
and every $x^{\prime}_{b_{i}}$ in the new set has two neighbours from
$\\{x_{a_{i}}\\}$. At the same time, we have
$x^{\prime}_{a_{i}}>x^{\prime}_{b_{i-1}}$ and
$x^{\prime}_{b_{i}}>x^{\prime}_{a_{i-1}}$. Then, we need to generate a common
set of points for both curves with $x$-coordinates from the following set.

$\\{x_{1},x_{2},\ldots,x_{K}\\}=\\{x^{\prime}_{a_{1}},x^{\prime}_{a_{2}},\ldots,x^{\prime}_{a_{N^{\prime}}}\\}\cup\\{x^{\prime}_{b_{1}},x^{\prime}_{b_{2}},\ldots,x^{\prime}_{b_{M^{\prime}}}\\},$

where $x_{1}<x_{2}<\cdots<x_{K}$ and $\max\\{M^{\prime},N^{\prime}\\}\leq
K\leq M^{\prime}+N^{\prime}$. The next step is to define the maps for the set
$\\{x_{1},x_{2},\ldots,x_{K}\\}$, and here we take

$f(x_{i})=\begin{cases}\displaystyle\frac{(x_{i}-x^{\prime}_{a_{l-1}})f(x^{\prime}_{a_{l}})+(x^{\prime}_{a_{l}}-x_{i})f(x^{\prime}_{a_{l-1}})}{x^{\prime}_{a_{l}}-x^{\prime}_{a_{l-1}}},&x^{\prime}_{a_{l-1}}\leq
x_{i}\leq x^{\prime}_{a_{l}},\quad 1<l\leq N^{\prime},\\\
g(x_{i}),&x_{i}<x^{\prime}_{a_{1}},\\\
g(x_{i}),&x_{i}>x^{\prime}_{a_{N^{\prime}}},\end{cases}$

and

$g(x_{i})=\begin{cases}\displaystyle\frac{(x_{i}-x^{\prime}_{b_{l-1}})g(x^{\prime}_{b_{l}})+(x^{\prime}_{b_{l}}-x_{i})g(x^{\prime}_{b_{l-1}})}{x^{\prime}_{b_{l}}-x^{\prime}_{b_{l-1}}},&x^{\prime}_{b_{l-1}}\leq
x_{i}\leq x^{\prime}_{b_{l}},\quad 1<l\leq M^{\prime},\\\
f(x_{i}),&x_{i}<x^{\prime}_{b_{1}},\\\
f(x_{i}),&x_{i}>x^{\prime}_{b_{M^{\prime}}}.\end{cases}$

The missing $y$-coordinates (if any) for each curve are obtained via
interpolating neighboring points. Thus, we can calculate the difference
between curve $A$ and curve $B$ by the normalized ${\rm L}^{1}$ or ${\rm
L}^{2}$ distance:

$\mathrm{error}_{1}=\frac{\mathrm{dist}_{1}}{\displaystyle\frac{1}{2N}\sum^{N}_{i=1}|y_{a_{i}}|+\frac{1}{2M}\sum^{M}_{i=1}|y_{b_{i}}|}\qquad\mathrm{or}\qquad\mathrm{error}_{2}=\frac{\mathrm{dist}_{2}}{\displaystyle\frac{1}{2N}\sqrt{\sum^{N}_{i=1}y^{2}_{a_{i}}}+\frac{1}{2M}\sqrt{\sum^{M}_{i=1}y^{2}_{b_{i}}}},$

where

$\displaystyle\mathrm{dist}_{1}=\max\\{|f(x_{i})-g(x_{i})|,i=1,2,\ldots,K\\}\qquad\mathrm{and}\qquad\mathrm{dist}_{2}=\frac{1}{K}\sqrt{\sum^{K}_{i=1}|f(x_{i})-g(x_{i})|^{2}}.$

|  |
---|---|---
|  |
Figure 5: Finding the corresponding relations by using the data in Fig. 4. Table 1: The centro-affine arc-lengths for the smooth boundaries in Fig. 2, where the row labels and column labels correspond to the ordinals in Figs. 2 and 3. | 1 | 2 | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---|---
Image A | 21.5764 | 17.3975 | 16.0039 | 6.0827 | 13.1816 | 10.2993 | 6.1379
Image B | 21.4343 | 18.1060 | 16.2536 | 6.0291 | 13.4718 | 9.9786 | 6.1225
Image C | 21.3836 | 16.3080 | 15.0370 | 5.6557 | 12.2398 | 9.9733 | 6.1182
Image D | 21.4965 | 17.0939 | 15.8373 | 5.9567 | 13.2712 | 11.4628 | 6.1752
(continued) | 8 | 9 | 10 | 11 | 12 | 13 |
Image A | 16.1042 | 6.0655 | 17.9754 | 10.5182 | 6.1844 | 20.9430 |
Image B | 16.3201 | 5.8780 | 12.8081 | 9.4719 | 6.0826 | 20.6943 |
Image C | 15.7962 | 5.8537 | 15.7287 | 11.1307 | 6.1087 | 20.9512 |
Image D | 16.7117 | 6.0081 | 18.1944 | 10.3007 | 6.1984 | 20.7571 |
Table 2: The correlation coefficents for Table 1. | Image A | Image B | Image C | Image D
---|---|---|---|---
Image A | 1 | 0.9672 | 0.9930 | 0.9976
Image B | 0.9672 | 1 | 0.9767 | 0.9624
Image C | 0.9930 | 0.9767 | 1 | 0.9897
Image D | 0.9976 | 0.9624 | 0.9897 | 1

By using the above method, under translations or orientation reversals of data
sets, we can find the minimum errors about $\epsilon$ and centro-affine
curvatures between the first graph in the Fig. 4 and the remaining three. In
the first row of Fig. 5, the comparison results are shown together. We apply
the dynamic time warping (DTW) algorithm [41] to find the optimal alignment
between the two sets of data points, which we view as time series, as shown in
the second row of Fig. 5. In general, DTW is often used to determine
similarity, classification, and corresponding regions between two time series.

Finally, applying the same algorithm to the other letters in the label, the
corresponding points can be found. The results are shown in Fig. 6, where,
between two images of every column, the correspondences are indicated by the
different colors. For comparative purposes, we also apply the SIFT, SURF and
ASIFT methods to obtain their corresponding points, which are shown in Fig. 7,
respectively.

|  |
---|---|---
|  |
Figure 6: The corresponding points between the two images of every column obtained by using the centro-affine invariants method. |  |
---|---|---
|  |
|  |
Figure 7: The corresponding points obtained by the SURF method (the first row), the SIFT method (the second row) and the ASIFT method (the third row). Centro-Affine | SURF | SIFT | ASIFT
---|---|---|---
|  |  |
|  |  |
|  |  |
Figure 8: The homography transformation results by using the corresponding
points obtained in Figs. 6 and 7.

Further, by using equation (2.8), we can also obtain the centro-affine arc-
lengths for the boundaries of the letters in the label “Standards”, which are
shown as in Table 1. The centro-affine arc-lengths for the letters in the
$i$th image of Fig. 3 are listed in the $i$th row of Table 1. It is easy to
see that there is a strong correspondence between the centro-affine arc-
lengths of the corresponding objects. Furthermore, we calculate the
correlation coefficients between them, as shown in Table 2, which indicate
that they are almost same.

In projective geometry, a homography is an isomorphism of projective spaces,
induced by an isomorphism of the vector spaces from which the projective
spaces derive. Estimating the 2D homography (or projective transformation)
from a pair of images is a fundamental task in computer vision. Now let us
find the homography transformations between these images and compare the
results with the corresponding points obtained by the SURF, SIFT, ASIFT, and
our centro-affine method. Those corresponding points data sets in Figs. 6 and
7 are used to transform the other three images to the second one in Fig. 2,
the final results are shown in Fig. 8. The transformation results derived by
the corresponding points of the centro-affine invariant method are shown at in
the first column; the second column shows the results by the SURF method; the
third column shows the results by the SIFT method; the last column shows the
results by the ASIFT method. (The second figure in the SIFT column is blank
due to incorrect matching points; see the second image of the SIFT row in Fig.
7.)

######  Remark 5.1.

When the images have less texture complexity and color diversity, it is more
challenging to extract and describe the feature points through the SIFT, SURF,
and ASIFT methods. The centro-affine invariant method can match the
corresponding points by the closed boundary, and hence is less affected by
texture and color. In view of Figs. 6, 7 and 8, we find that between the
images under the large scale transformation of the camera, the centro-affine
invariant method for finding the corresponding points, using the corresponding
boundaries, offers certain advantages over SURF, SIFT and ASIFT. However, if
the object itself admits affine symmetries, for example, the “o” inside the
letter “d”, it is not so easy to find the proper corresponding points; this
defect can be observed in Fig. 6.

##  6 Concluding remarks

In this paper, we have investigated the heat flow in centro-affine geometry
and applications of differential centro-affine invariants in edge matching.
Differential invariants and their algebraic relations for curves in centro-
affine geometry are easily obtained by applying the equivariant moving frame
method [12]. A classification of curves with constant centro-affine curvature
has been provided. More interestingly, we have shown that the heat flow in
centro-affine geometry is equivalent to the first-order inviscid Burgers’
equation, in contrast to the equations governed by the heat flows in
Euclidean, affine, and centro-equi-affine geometries, which are nonlinear
second-order parabolic equations. Thus, the evolution process for the centro-
affine heat flow is described by solving the inviscid Burgers equation through
the method of characteristics. An interesting question, to be explored later,
is what the presence of shock waves in the solution might mean for the
corresponding curve evolution. In addition, an application of centro-affine
invariants to edge matching is presented. It turns out that the resulting
method offers certain advantages over other well-used methods.

In conclusion, we would like to mention further possible issues relating to
this work.

  * •

The Gaussian kernel, while being one of the most used in image analysis, has
several undesirable properties, principally when applied to planar curves. One
of these is that the filter is not intrinsic to the curve. This can be
remedied by replacing the linear heat equation by a geometric heat equation.
In particular, if the Euclidean geometric heat flow is used, a scale space
invariant to rotations and translations is obtained, while the (equi-)affine
version leads to an(equi-)affine invariant multi-scale representation; see
[37, 44] for general results and recent developments. It would thus be of
interest to apply the centro-affine heat flow analyzed here to construct a
corresponding scale space.

  * •

As shown here, the differential invariants for curves in centro-affine
geometry can be used to study edge matching in images. A natural question
arises: whether we can use differential invariants for surfaces in centro-
affine geometry to study the edge matching of three-dimensional images. In
fact, the differential invariants for the equi-affine group acting on image
volumes have been constructed in [44].

### Acknowledgements

The authors thank the referees for their valuable comments and suggestions.
The work of C.Z. Qu is supported by the NSF-China grant-11631007 and
grant-11971251.

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