%gmChap05.tex.

\graphicspath{{./figures/}}

\chapter{The Main Curvature  Equations\label{Chap5}}

The purpose of this chapter is to introduce the curvature motion
 PDE's for Jordan curves and images.
  Our main task is to establish a formal link between curve evolution
 and image evolution. This link will be established through the PDE formulation.
  The basic differential geometry used in this chapter was thoroughly developed in Chapter \ref{Chapter4}, which must
therefore be read first.

\section{The definition of a shape and how it is recognized}
Relevant information in images has been reduced to the image level sets in Chapter \ref{Chap3}. By Corollary
\ref{corollary4.6}, if the image is $C^1$,  the boundary of its  level sets is a finite set Jordan curves at almost
every level. Thus, shape analysis can be led back to the study of these curves which we shall call ``elementary
shapes".

\begin{definition} We call
elementary shape any $C^1$ planar  Jordan curve.
\end{definition}
 The many experiments where we display  level lines of digital
 images make clear enough why a smoothing is necessary to restore
 their structure. These experiments also show that we can in no
 way assimilate these level lines with our common notion of shape
 as the silhouette of a physical object in full view.
Indeed, in images of a natural environment, most  observed objects are partially hidden (occluded) by other objects and
often deformed by perspective. When we observe a level line we cannot be sure that it belongs to a single object; it
may be composed of pieces of the boundaries of several objects that are occluding each other. Shape recognition
technology has therefore focused on local methods, that is, methods that work even if a shape is not in full view or if
the visible part is distorted. As a consequence, image analysis adopts the following principle: \textit{Shape
recognition must be based on local features of the shape's boundary, in this case local features of the Jordan curve,
and not on its global features. If the boundary has some degree of smoothness, then these local features are based on
the derivatives of the curve, namely the tangent vector, the curvature, and so on.} Many local recognition methods
involve the ``salient'' points of a shape, which are the points where the curvature is zero (inflection points) and
points where the curvature has a maximum or minimum (the ``corners'' of the shape).

\section{Multiscale features and scale space}
Computational shape recognition methods often make the following two basic assumptions, neither of which is true in
practice for the rough  shape data:
\begin{enumerate}[$\bullet$]\setlength\itemsep{2pt}
\item The shape is a smooth Jordan curve. \item The boundary has a
small number of inflexion points  and curvature extrema. This number can be made as  small as desired by smoothing.
\end{enumerate}

The fact that these conditions can be obtained by properly smoothing a $C^1$ Jordan curve was proven in 1986-87 by Gage
and Hamilton \cite{Gage:1986:HESa} and Grayson \cite{Grayson:1987:HESb}. They showed that it is possible to transform a
$C^1$ Jordan curve into a $C^\infty$ Jordan curve by using the so-called \textit{intrinsic heat equation}.

For convenience, and unless it would cause ambiguity, we will not make a distinction between a Jordan curve $\Gamma$ as
a subset of the plane and a function $s\mapsto\fxx(s)$ such that $\Gamma=\{\fxx(s)\}$. As we have already done, we will
speak of the Jordan curve $\fxx$. Since we will be speaking of families of Jordan curves dependent on a parameter
$t>0$, we will most often denote these families by $\fxx(t,s)$, where the second variable is a parameterization of the
Jordan curve. Thus, $\fxx(t,s)$ has three meanings: a family of Jordan curves, a family of functions that represent
these curves, and a particular point on one of these curves. The notation $s$ will be usually reserved to an arc-length
parameter.

\begin{definition}Let $\fxx(t)$, $t>0$, be a family of $C^2$ Jordan curves. We say
that $\fxx(t)$ satisfies the intrinsic heat equation if
\begin {equation}\label{eq61x}
\frac{\partial\fxx}{\partial t}=\bm{\kappa}(\fxx(t)).
\end{equation}
\end{definition}

\begin{theorem}[Grayson] \label{Graysontheorem}Let $\fxx_0$ be a $C^1$ Jordan  curve. By using the
intrinsic heat equation, it is possible to evolve $\fxx_0$ into a family of Jordan curves $\fxx(t,s)$ such that
$\fxx(0,s)=\fxx_0(s)$ and such that for every $t>0$, $\fxx(t,s)$ is $C^\infty$ (actually analytical) and satisfies the
equation \eqref{eq61x}. Furthermore, for every $t>0$, $\fxx(t,s)$ has only a finite number of inflection points and
curvature extrema, and the number of these points does not increase with $t$. For every initial curve, there is a scale
$t_0$ such that the curve $\fxx(t,s)$ is convex for $t\geq t_0$ and there is a scale $t_1$ such that the curve
$\fxx(t,s)$ is a single point for $t\geq t_1$.
\end{theorem}
 It is time to say
what we mean by ``curve scale space", or ``shape scale space.'' We will refer to any process that smooths a Jordan
curve and that depends on a real parameter $t$. Thus a shape scale space associates with an initial Jordan curve
$\fxx(0,s)=\fxx_0(s)$ a family of smooth curves $\fxx(t,s)$. For example, the intrinsic heat equation eliminates
spurious details of the initial shape and retains simpler, more reliable versions of the shape, and these smoothed
shapes have finite codes. A scale space is {\it causal} in the terminology of vision theory if it does not introduce
new features. Grayson's theorem therefore defines a causal scale space.



\section{From image motion to curve motion}

The intrinsic heat equation is only one example from a  large family of nonlinear equations that move curves with a
curvature-dependent speed, that is, $\partial\fxx/\partial t$ is a function of the curvature of the curve $\fxx$. There
are two conditions on the curvature dependence. The velocity vector must always
 point towards the concavity of the curve.  Its norm
  must be a nondecreasing function of the magnitude of the curvature $|\bm{\kappa}(\fxx)|$. The first
   condition ensures that the equation is a smoothing which reduces asperities. The second condition
  intuitively preserves the inclusion between curves. This can be appreciated by considering two circles $C$ and $C'$
  such that $C'$ surround $C$ and $C'$ and $C$ are tangent at some point $\fxx$. Then the first condition implies  that
  both circles shrink, but the  second condition implies that the smaller circle $C$ shrinks faster, so that the inclusion
  between circles is preserved by the evolution.


\begin{definition}We say that a $C^2$ function $u:\RR^+\times\RR^2\to\RR$
 satisfies a curvature equation if for some real-valued function $g(\kappa, t)$,
  which is nondecreasing in $\kappa$ and satisfies
  $g(0,t)=0$,
\begin{equation}\label{eq65x}
\frac{\partial u}{\partial t}(t,\fxx)=g({\rm curv}(u)(t,\fxx), t)|Du|(t,\fxx).
\end{equation}
\end{definition}

\begin{definition}Let $\fxx(t)$ be a family of
 $C^2$ Jordan curves. We say that the functions $\fxx(t)$
 satisfy a curvature equation if for some real-valued function $g(\kappa, t)$
nondecreasing in $\kappa$ with $g(0,t)=0$, they satisfy
\begin{equation}\label{eq64x}
\frac{\partial\fxx}{\partial t}= g(|\bm{\kappa}(\fxx)|,t)\fxn(t),
\end{equation}
where $\fxn$ is a unit vector in the direction of $\bm{\kappa}(\fxx)$.
\end{definition}
In the preceding definition, the equation makes sense if $\bm{\kappa}(\fxx)=0$  since then the second member is zero.
As we shall see, these equations are the only candidates to be curve or image scale spaces, and one of the objectives
of this book is to identify  which forms for $g$ are particularly relevant for image analysis. The above definitions
are quite restrictive because they require the curves or images to be $C^2$. A more generally applicable definition of
solutions for these equations will be given in Chapter \ref{Chapviscositysolutions} with the introduction of viscosity
solutions. Our immediate objective is to establish the link between the motion of an image and the motion of its level
lines. This will establish the relation between equations \eqref{eq65x} and \eqref{eq64x}.

\subsection{A link between image and curve evolution}

\begin{lemma} \label{defnormalflow}{\bf (Definition of the ``normal flow")}.
Suppose that
 $(t,\fxx)\mapsto u(t,\fxx)$ is $C^2$ in a neighborhood
 $T\times U$ of the point $(t_0,\fxx_0)\in\RR\times\RR^2$,
  and assume that $Du(t_0,\fxx_0)\not=0$. Then there exists
  an open interval $J$ centered at $t_0$, an open disk $V$ centered at
   $\fxx_0$, and a unique $C^1$ function $\fxx:J\times V\to\RR^2$ that
   satisfy the following properties:
\begin{enumerate}[$(i)$]\setlength\itemsep{2pt}
\item $u(t,\fxx(t,\fxy))=u(t_0,\fxy)$ and $\fxx(t_0,\fxy)=\fxy$ for all $(t,\fxy)\in J\times V$.
\item The vectors $(\partial\fxx/\partial t)(t,\fxy)$ and $Du(t,\fxx(t,\fxy))$ are collinear.
\end{enumerate}
In addition, the function $\fxx$ satisfies the following
differential equation:
\begin{equation}\label{eq66x}
\frac{\partial\fxx}{\partial t}(t,\fxy)=-\Big(\frac{Du}{|Du|^2}\frac{\partial u}{\partial t}\Big)(t,\fxx(t,\fxy)).
\end{equation}
  The
trajectory $t\mapsto\fxx(t,\fxy)$ is called the {\rm normal flow
starting from $(t_0,\fxy)$.}
\end{lemma}

\begin{proof}
Differentiating the relation $u(t, \fxx(t))=0$ with respect to $t$ yields $\frac{\partial u}{\partial t} +
Du.\frac{\partial \fxx}{\partial t} = 0$. By multiplying this equation by the vector $Du$ we see that $\frac{\partial
\fxx}{\partial t}$ is collinear to $Du$ if and only if \eqref{eq66x} holds. Now,
 this relation defines $\fxx(t)$ as the solution of an
ordinary differential equation, with initial condition $ \fxx(t_0) = \fxy$.  Since $u$ is $C^2$,  the second member of
(\ref{eq66x}) appears to be a Lipschitz function of $(t,\fxx)$ provided $Du(t, \fxx)\neq 0$, which is ensured for $(t,
\fxx)$ close enough to $(t_0, \fxx_0)$. Thus, by Cauchy-Lipschitz Theorem, there exists an open interval $J$ such that
the O.D.E. (\ref{eq66x}) has a unique solution $\fxx(t, \fxy)$ for all $\fxy$ in a neighborhood of $\fxx_0$ and $t\in
J$.
\end{proof}


\begin{proposition}\label{corollary5.9}
Assume that the function $(t,\fxx)\mapsto u(t,\fxx)$ is $C^2$ in a
neighborhood of $(t_0,\fxx_0)$ and that $Du(t_0,\fxx_0)\not=0$.
Then $u$ satisfies the curvature motion equation
\begin{equation}\label{eq68x}
\frac{\partial u}{\partial t}(t,\fxx)= {\rm
curv}(u)(t,\fxx)|Du|(t,\fxx)
\end{equation}
in a neighborhood of $(t_0,\fxx_0)$ if and only if the normal flow $\fxx(t,\fxy)$  of $u$  in this neighborhood
satisfies the intrinsic heat equation
\begin{equation} \label{nonparametricintrinsic}
\frac{\partial\fxx}{\partial t}(t,\fxy)=\bm{\kappa}(\fxx(t,\fxy)),
\end{equation}
where $\bm{\kappa}(\fxx(t,\fxy))$ denotes the curvature vector of the level line of $u(t)$ passing by $\fxx(t,\fxy)$.
\end{proposition}

 \begin{proof}

Assume first that $\fxx(t,\fxy)$ satisfies \eqref{nonparametricintrinsic}. Applying \eqref{eq62x}
 for all $t$ in a neighborhood of
 $t_0$ to each image $u(t): \fxx\to u(t,\fxx)$
  yields
$$ \bm{\kappa}(\fxx(t,\fxy))=-{\rm curv}(u)\frac{Du}{|Du|}(t, \fxx(t,\fxy)).
$$
Substituting \eqref{nonparametricintrinsic} in this last relation we obtain

$$
\frac{\partial\fxx}{\partial t}(t,\fxy)=-{\rm curv}(u)\frac{Du}{|Du|}(t, \fxx(t,\fxy))
$$
and by the normal flow equation \eqref{eq66x},

$$(\frac{\partial u}{\partial t}\frac{Du}{|Du|^2})(t,\fxx(t,\fxy))=
{\rm curv}(u)\frac{Du}{|Du|}(t,\fxx(t,\fxy)).$$ Multiplying this equation by $Du(t, \fxx(t,\fxy))$ yields the curvature
motion equation (\ref{eq68x}).

 The converse statement follows exactly the same lines backwards.
\end{proof}

\begin{exercise} Write the proof of the converse statement of Proposition \ref{corollary5.9}.
\finex \end{exercise}

 The preceding proof is immediately adaptable to all curvature equations :

\begin{proposition} \label{proposition5.8} Assume that the function $(t,\fxx)\mapsto u(t,\fxx)$ is $C^2$ in a
neighborhood of $(t_0,\fxx_0)$ and that $Du(t_0,\fxx_0)\not=0$. Let $g:\RR\times\RR^+\to\RR$ be continuous and
nondecreasing with respect to $\kappa$ and such that $g(-\kappa,t)=-g(\kappa,t)$. Then $u$ satisfies the curvature
motion equation
\begin{equation}\label{P581}
\frac{\partial u}{\partial t}(t,\fxx)=g({\rm curv}(u)(t,\fxx),t)|Du|(t,\fxx)
\end{equation}
in a neighborhood of $(t_0,\fxx_0)$ if and only if the normal flow $t\mapsto\fxx(t,\cdot)$ satisfies the curvature equation
\begin{equation}\label{P582}
\frac{\partial\fxx}{\partial t}(t,\fxy)=
g(|\bm{\kappa}(\fxx(t,\fxy))|)\frac{\bm{\kappa}(\fxx(t,\fxy))}{|\bm{\kappa}(\fxx(t,\fxy))|}.
\end{equation}
\end{proposition}


\subsection{Introduction to the affine curve and function equations}

There are two curvature equations that are affine invariant and are therefore  particularly well suited for use in
shape recognition. In their definition, for $x\in\RR$, $x^{1/3}$ stands for ${\rm sign}(x)|x|^{1/3}$.

\begin{definition}The image evolution equation
\begin{equation}\label{eq69x}
\frac{\partial u}{\partial t}(t,\fxx)= ({\rm curv}(u)(t,\fxx))^{1/3}|Du(t,\fxx)|
\end{equation}
is called affine morphological scale space (AMSS). The curve evolution equation
\begin{equation}\label{eq610x}
\frac{\partial\fxx}{\partial
t}(t,s)=|\bm{\kappa}(\fxx(t,s))|^{1/3}\fxn(t,s)\quad
\Big(=\frac{\bm{\kappa}(\fxx(t,s))}{|\bm{\kappa}(\fxx(t,s))|^{2/3}}\Big)
\end{equation}
is called affine scale space (ASS).
\end{definition}

It is clear that AMSS and ASS are equivalent in the sense of
Proposition  \ref{proposition5.8}. As one would expect from the
names of these equations, they both have some sort of affine
invariance. This is the subject of the next definition, Exercises
\ref{exercise5.2} and \ref{exercise5.3} and the next section.

\begin{definition}We say that  a curvature equation $(E)$
(image evolution equation) is affine invariant, if for
 every linear map $A$ with positive determinant, there is a positive
  constant $c=c(A)$ such that $(t,\fxx)\mapsto u(t,\fxx)$ is a solution
   of $(E)$ if and only if $(ct, A\fxx)\mapsto u(ct, A\fxx)$ is a solution of $(E)$.
\end{definition}


\subsection{The affine scale space as an intrinsic heat equation}

Suppose that for each scale $t$, $\sigma\mapsto\fxx(t,\sigma)$ is
a Jordan arc (or curve) parameterized by $\sigma$, which is not in
general an arc length. As in Chapter \ref{Chapter4}, we will
denote the curvature of $\fxx$ by ${\bm \kappa}$. We wish to
demonstrate a formal equivalence between the affine scale space,
\begin{equation}\label{eq615x}
\frac{\partial\fxx}{\partial t}=|{\bm \kappa}|^{1/3}\fxn(\fxx),
\end{equation}
and an ``intrinsic heat equation''
\begin{equation}\label{eq616x}
\frac{\partial\fxx}{\partial t}=\frac{\partial^2\fxx}{\partial\sigma^2},
\end{equation}
where $\sigma$ is a special parameterization called \textit{affine
length}.  We define an \textit{affine length parameter} of a
Jordan curve (or arc) to be any parameterization
$\sigma\mapsto\fxx(\sigma)$ such that
\begin{equation}\label{eq617x}
[\fxx_\sigma,\fxx_{\sigma\sigma}]=1,
\end{equation}
where $[\fxx,\fxy]=\fxx^\perp\cdot\fxy$. If $s$ is an arc-length
parameterization, then we have (Definition \ref{definition4.2})
\begin{equation}\label{eq618x}
{\bm\tau}=\fxx_s\quad\fxn=|{\bm\kappa}|^{-1}\fxx_{ss}\quad\Big(=
\frac{\bm{\kappa}(\fxx)}{|\bm{\kappa}(\fxx)|}\Big).
\end{equation}
We also have
\begin{equation}\label{eq619x}
\fxx_\sigma=\fxx_s\frac{\partial s}{\partial\sigma}\quad\text{and}\quad\fxx_{\sigma\sigma}=\fxx_{ss}\Big(\frac{\partial s}{\partial\sigma}\Big)^2+\fxx_s\frac{\partial^2 s}{\partial\sigma^2}.
\end{equation}
Thus,
\[
[\fxx_\sigma,\fxx_{\sigma\sigma}]=[\fxx_s,\fxx_{ss}]\Big(\frac{\partial s}{\partial\sigma}\Big)^3,
\]
and if \eqref{eq617x} holds, then
\[
[\fxx_s,\fxx_{ss}]\Big(\frac{\partial s}{\partial\sigma}\Big)^3=1.
\]
Since by \eqref{eq618x} $[\fxx_s,\fxx_{ss}]={\rm sign}([\fxx_s,\fxx_{ss}])|{\bm\kappa}|$, we conclude that
\begin{equation}\label{eq621x}
\frac{\partial s}{\partial\sigma}=({\rm sign}([\fxx_s,\fxx_{ss}])|{\bm\kappa}|)^{-1/3}.
\end{equation}

Substituting this result in the expression for
$\fxx_{\sigma\sigma}$ shown in \eqref{eq619x} and writing
$\fxx_s={\bm\tau}$, we see that
\[
\fxx_{\sigma\sigma}=|{\bm\kappa}|^{1/3}\fxn+\Big(\frac{\partial^2 s}{\partial\sigma^2}\Big){\bm\tau}.
\]
This tells us that equation \eqref{eq616x} is equivalent to the
following equation:
\begin{equation}\label{eq622x}
\frac{\partial\fxx}{\partial t}=|{\bm\kappa}|^{1/3}\fxn+\Big(\frac{\partial^2 s}{\partial\sigma^2}\Big){\bm\tau}.
\end{equation}

Now it turns out that the graphs of the functions $\fxx$ that you
get from one time to another do not depend on the term involving
${\bm\tau}$; you could drop this term and get the same graphs.
More precisely, Epstein and Gage \cite{Epstein:1987:CSF} have
shown that the tangential component of an equation like
\eqref{eq622x} does not matter as far as the geometric evolution
of the curve is concerned. In fact, the tangential term just moves
points along the curve itself, and the total curve evolution is
determined by the normal term. As a consequence, equation
\eqref{eq615x} is equivalent to equation \eqref{eq616x} in any
neighborhood that avoids an inflection point, that is, in any
neighborhood where $\fxn(\fxx)\not=0$. At an inflection point,
${\bm\kappa}=0$, and the two equations give the same result.

\section{Curvature motion in N dimensions}

We consider an evolution  $(t,\fxx)\mapsto u(t,\fxx)$, where $\fxx\in\RR^N$ and $u(0,\cdot)=u_0$ is an initial
$N$--dimensional image. Let $\kappa_i(u)(t,\fxx)$, $i=1,\ldots,N-1$, denote the $i^{\rm th}$ principal curvature at the
point $(t,\fxx)$.   By definition \ref{definitionmeancurvatureoffunction} the mean curvature is  ${\rm
curv}(u)=\sum_{i=1}^{N-1}\kappa_i$. We will now define three curvature motion flow equations in $N$ dimensions.

\paragraph*{Mean curvature motion.} This equation is a direct translation
of equation \eqref{eq68x} in $N$  dimensions:
\[
\frac{\partial u}{\partial t}=|Du|{\rm curv}(u).
\]
This says that the motion of a level hypersurface of $u$ in the
normal direction is proportional to its mean curvature.

\paragraph*{Gaussian curvature motion for convex functions.} We say that a function
is convex if all of its principal curvatures have the same sign. An example of such
a function is the signed distance function to a regular convex shape. The equation
 is
\[
\frac{\partial u}{\partial t}=|Du|\prod_{i=1}^{N-1}\kappa_i.
\]
The motion of a level hypersurface is proportional to the product
of its principal curvatures, which is the Gaussian curvature. As
we will see in Chapter \ref{Chap18}, this must be modified before
it can be applied to a nonconvex function.

\paragraph*{Affine-invariant curvature motion.} The equation is
\[
\frac{\partial u}{\partial t}=|Du|\Big|\prod_{i=1}^{N-1}\kappa_i\Big|^{1/(N+1)}H\Big(\sum_{i=1}^{N-1}{\rm sign}(\kappa_i)\Big),
\]
where $H(N-1)=1$, $H(-N+1)=-1$, and $H(n)=0$ otherwise. The motion is similar to Gaussian curvature motion, but the
affine invariance requires that the Gaussian curvature be raised to the power $1/(N+1)$. There is no motion at a point
where the principal curvatures have mixed signs. This means that only concave or convex parts of level surfaces get
move by such an equation.

\section{Exercises}

\begin{exercise}Check that all of the curvature  equations \eqref{eq65x}
 are contrast invariant. That is, assuming that $h$ is a real-valued $C^2$ increasing function
 defined on $\RR$ and $u$ is  $C^2$, show that the function $v$
  defined by $v(t,\fxx)=h(u(t,\fxx))$
  satisfies one of these equations  if and only if $u$ satisfies the same equation.
  \finex
\end{exercise}


\begin{exercise}\label{exercise5.2}Assume that $(t,\fxx)\mapsto u(t,\fxx)$ is a $C^2$ function and that $A$ is a $2\times2$ matrix with positive determinant, which we denote by $|A|$. Define the function $v$ by $v(t,\fxx)=u(ct,A\fxx)$, where $c=|A|^{-2/3}$.
\begin{enumerate}[(i)]\setlength\itemsep{2pt}
\item Prove that for each point $\fxx$ such that
$Du(\fxx)\not={0}$ one has the relation
\[
{\rm curv}(v)(\fxx)|Dv(\fxx)|^3=|A|^2{\rm
curv}(u)(A\fxx)|Du(A\fxx)|^3.
\]
\item Use (i) to deduce that the AMSS equation \eqref{eq69x} is
affine invariant, that is, $(t,\fxx)\mapsto u(t,\fxx)$ is a
solution of AMSS if and only $(t,\fxx)\mapsto v(t,\fxx)$ does.
\finex
\end{enumerate}
\end{exercise}

\begin{exercise}\label{exercise5.3}
This exercise is to show that the affine scale space  (equation \eqref{eq610x}) is affine invariant. It relies directly
on results from Exercise \ref{calculcourburesparametrees}. Let $\sigma\mapsto \fxc(\sigma)$ be a $C^2$ curve, and
assume that $|\fxc'(\sigma)|>0$. Then we know from Exercise \ref{calculcourburesparametrees} that
\begin{equation}\label{eq611x}
\bm{\kappa}(\fxc)(\sigma)=\frac{1}{|\fxc'(\sigma)|^2}\left[\fxc''(\sigma)-\Bigg(\fxc''(\sigma)\cdot\frac{\fxc'(\sigma)}{|\fxc'(\sigma)|}\Bigg)\frac{\fxc'(\sigma)}{|\fxc'(\sigma)|}\right].
\end{equation}
Now assume that we have a  family of $C^2$ Jordan arcs
$(t,\sigma)\mapsto\fxc(t,\sigma)$. By projecting both sides of the
intrinsic heat equation onto the unit vector $\fxc'^\perp/|\fxc'|$
and by using \eqref{eq611x}, we have the following equation:
\begin{equation}\label{eq612x}
\frac{\partial\fxc}{\partial
t}\cdot\frac{\fxc'^\perp}{|\fxc'|}=\frac{\fxc''\cdot\fxc'^\perp}{|\fxc'|^3}
\end{equation}
We say that $\fxc$ satisfies  a parametric curvature equation if
it satisfies equation \eqref{eq612x}. In the same spirit, we say
that $\fxc$ satisfies a parametric affine equation if for some
constant $\gamma>0$
\begin{equation}\label{eq613x}
\frac{\partial\fxc}{\partial
t}\cdot\fxc'^\perp=\gamma(\fxc''\cdot\fxc'^\perp)^{1/3}.
\end{equation}
\begin{enumerate}[(i)]\setlength\itemsep{2pt}
\item Suppose that $\sigma=s$, an arc-length parameterization of
$\fxc$.  Show that equation \eqref{eq612x} can be written as
\[
\frac{\partial\fxc}{\partial t}=\bm{\kappa}(\fxc)+\lambda\tau,
\]
where $\lambda$ is a real-valued function and $\tau$ is  the unit
tangent vector $\partial\fxc/\partial s$. (See the remark
following equation \eqref{eq622x}.) \item Let $A$ be a $2\times2$
matrix with positive determinant, and define the curve $\fxy$ by
$\fxy(t,\sigma)=A\fxc(t,\sigma)$. We wish to show that if $\fxc$
satisfies a parametric affine motion, then so does $\fxy$. As a
first step, show that $A\fxx\cdot(A\fxy)^\perp=|A|\fxx\cdot\fxy$
and hence that $A(\fxx^\perp)\cdot(A\fxx)^\perp=|A||\fxx|^2$ for
any $\fxx,\fxy\in\RR^2$. \item Show that if $\fxc$ satisfies
equation \eqref{eq613x}, then $\fxy$ satisfies
\[
\frac{\partial\fxy}{\partial
t}\cdot\fxy'^\perp=\gamma|A|^{2/3}(\fxy''\cdot\fxy'^\perp)^{1/3}.
\finex
\]
\end{enumerate}
\end{exercise}

\section{Comments and references}

\paragraph*{Our definition of shape.}

The Italian mathematician Renato Caccioppoli proposed a theory of sets whose boundaries have finite length (finite
Hausdorff measure). From his theory, it can be deduced that the boundary of a Caccioppoli set is composed of a
countable number of Jordan curves, up to a set with zero length. This decomposition can even be made unambiguous. In
other words, the set of Jordan curves associated with a given Caccioppoli set is unique and gives enough information to
reconstruct the set \cite{ACMM2001}. This result justifies our focus on Jordan curves as the representatives of shapes.

\paragraph*{The role of curvature in shape analysis.} After Attneave's
founding paper \cite{attneave1954sia}, let us mention the thesis by G. J. Agin \cite{Agin:1972:RDC} as being one of the
first references dealing with the use of curvature for the representation and recognition of objects in computer
vision. The now-classic paper by Asada and Brady \cite{Asada:1986:CPS} entitled ``The curvature primal sketch''
introduced the notion of computing a ``multiscale curvature'' as a tool for object recognition. (The title is an
allusion to David Marr's famous ``raw primal sketch," which is a set of geometric primitives extracted from and
representing an image.) The Asada--Brady paper led to a long series of increasingly sophisticated attempts to represent
shape from curvature \cite{Dudek:1990:SRC,Dudek:1991:SRR} and to compute curvature correctly \cite{Mundy:1995:ORS}. The
shape recognition programme we sketched in the beginning of this chapter was anticipated in a visionary paper by
Attneave \cite{attneave1954sia} and has been very recently fully developed in the works of Jos\'{e} Luis Lisani, Pablo
Mus\'{e}, Fr\'{e}d\'{e}ric Sur, Yann Gousseau and Fr\'{e}d\'{e}ric  Cao \cite{muse2003uts}, \cite{muse2006cdm},
\cite{cao1746ufd}, \cite{cao:aef}.

\paragraph*{Curve shortening.} The mathematical study of the
intrinsic heat equation (or curvature motion in two dimensions)
was done is a series of brilliant papers in differential geometry
between 1983 and 1987. We repeat a few of the titles, which
indicate the progress: There was Gage \cite{Gage:1983:IIA} and
Gage \cite{Gage:1984:CSM}: ``Curve shortening makes convex curves
circular.'' Then there was Gage and Hamilton
\cite{Gage:1986:HESa}: ``The heat equation shrinking convex plane
curves.'' In this paper the authors showed that a plane convex
curve became asymptotically close to a shrinking circle. In 1987
there was the paper by Epstein and Gage \cite{Epstein:1987:CSF},
and, in the same year, Grayson removed the convexity condition and
finished the job \cite{Grayson:1987:HESb}: ``The heat equation
shrinks embedded plane curves to round points.'' As the reviewer,
U. Pinkall, wrote, ``This paper contains the final solution of the
long-standing \textit{curve-shortening problem} for plane
curves.''

The first papers that brought curve shortening (and some variations) to image analysis were by Kimia, Tannenbaum, and
Zucker \cite{Kimia:1992:ECF} and by Mackworth and Mokhtarian \cite{Mackworth:1992:TMC}. Curve shortening was introduced
as a way to do a multiscale analysis of curves, which were considered as shapes extracted from an image. In the latter
paper, curve shortening was proposed as an efficient numerical tool for multiscale shape analysis.


\paragraph*{Affine-invariant curve shortening.} Affine-invariant
geometry seems to have been founded by W. Blaschke. His
three-volume work ``Vorlesungen {\"u}ber Differentialgeometrie''
(1921--1929) contains definitions of affine length and affine
curvature. Curves with constant affine curvature are discussed in
\cite{Maeda:1941:CPS}. The term ``affine shortening'' and the
corresponding curve evolution equation were introduced by Sapiro
and Tannenbaum in \cite{Sapiro:1993:AIS}. Several mathematical
properties were developed by the same authors in
\cite{Sapiro:1993:ICE} and \cite{Sapiro:1994:APC}. Angenent,
Sapiro, and Tannenbaum gave the first existence and uniqueness
proof of affine shortening in \cite{Angenent:1998:AHF} and prove a
theorem comparable to Grayson's theorem : they prove that a shape
eventually becomes convex and thereafter evolves towards an
ellipse before collapsing.

\paragraph*{Mean curvature motion.} In his famous paper entitled
``Shapes of worn stones,'' Firey proposed a model for the natural
erosion of stones on a beach \cite{Firey:1974:SWS}. He suggested
that the rate of erosion of the surface of a stone was
proportional to the Gaussian curvature of the surface, so that
areas with high Gaussian curvature eroded faster than areas with
lower curvature, and he conjectured that the final shape was a
sphere. The first attempt at a mathematical definition of the mean
curvature motion is found in Brakke \cite{Brakke:1978:MSM}. Later
in the book, we will discuss the Sethian's clever numerical
implementation of the same equation \cite{Sethian:1985:CEF}.
Almgren, Taylor, Wang proposed a more general formulation of mean
curvature motion that is applicable to crystal growth and, in
general, to the evolution of anisotropic solids
\cite{Fred:1993:CDF}.


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