\graphicspath{{./figures/}}
 \chapter{Iterated Linear Filters and the Heat Equation\label{ChapCOIFTTHE}}


The title of this chapter is self-explanatory.
 The next section
fixes fairly general conditions so that the difference of a
smoothed image and the original be proportional to the Laplacian.
The second section proves the main result, namely the convergence
of iterated linear filters to the heat equation. So the choice of
a smoothing convolution kernel is somewhat forced : Iterating the
convolution with a smoothing kernel is asymptotically equivalent
to the convolution with a Gauss function. This result is known in
Probability as the central limit theorem, where it has a quite
different interpretation.  In image processing, it justifies the
prominent role of Gaussian filtering.
 A last section is
devoted to linear directional filters and their associated
differential operators.

\section{Smoothing and the Laplacian}
 There  are minimal requirements
on the smoothing kernels $g$ which we state in the next
definition.
\begin{definition}\label{deflaplacianconsistency}
We say that a real-valued kernel $g\in L^1(\RR^N)$ is Laplacian
consistent if it satisfies the following moment conditions:
\begin{description}%[$(i)$]\setlength\itemsep{2pt}
\item [(\emph{i})] $\int_{\RR^N} g(\fxx)\,\rmd\fxx=1.$ \item [\emph(ii)] For
$i=1,2,\ldots,N$, $\int_{\RR^N} x_ig(\fxx)\,\rmd\fxx=0.$ \item [\emph(iii)] For each pair $i,j=1,2,\ldots,N$, $i\neq
j$, $\int_{\RR^N}x_i x_jg(\fxx)\,\rmd\fxx=0.$ \item [\emph(iv)] For $i=1,2,\ldots,N$, $\int_{\RR^N}
x_i^2g(\fxx)\,\rmd\fxx=\sigma,$ where $\sigma>0.$ \item [\emph(v)] $\int_{\RR^N}|\fxx|^3|g(\fxx)|\,\rmd\fxx<+\infty.$
\end{description}

\end{definition}
Note that we do not assume that $g\geq0$; in fact, many important
filters used in signal and image processing are not positive.
However, condition $(i)$ implies that $g$ is ``on average''
positive. A discussion of the necessity of the requirements
$(i)-(v)$  is performed in Exercise \ref{minimalrequirements}.


We say that a function $g$ is \textit{radial} if
$g(\fxx)=g(|\fxx|)$, $\fxx\in\RR^N$. This is equivalent to saying
that $g$ is invariant under all rotations around the origin in
$\RR^N$. As pointed out in Exercise \ref{exercise2.1}, any radial
function $g\in L^1(\RR^N)$ can be rescaled to be Laplacian
consistent if it decays fast enough at infinity and if
$\int_{\RR^N}\fxx_i^2g(\fxx)\,\rmd\fxx$ and
$\int_{\RR^N}g(\fxx)\,\rmd\fxx$ have the same sign.

We consider rescalings of a kernel $g$ defined by
\begin{equation}\label{gh}
g_h(\fxx)=\frac{1}{h^{{N}/{2}}}\,g\Big(\frac{\fxx}{h^{{1}/{2}}}\Big)
\end{equation}
for $h>0$ (see Figure \ref{fig21}). Notice that this rescaling
differs slightly from the one used in Section 1.2. We have used
the factor $h^{1/2}$ here because it agrees with the factor
$t^{1/2}$ in the Gaussian. We denote the convolution of $g$ with
itself $n$ times by $g^{n*}$. The main result of this section
concerns the behavior of $g_h^{n*}$ as $n\to+\infty$ and $h\to0$.

\begin{exercise}\label{exercise2.2}
Prove the following two statements:
\begin{enumerate}[(i)]\setlength\itemsep{2pt}
\item $g_h$ is Laplacian consistent if and only if $g$ is
Laplacian consistent. \item If $g\in L^1(\RR^N)$, then
$(g_h)^{n*}=(g^{n*})_h$. \finex
\end{enumerate}
\end{exercise}

\begin{figure}[!t]
\setkeys{Gin}{width=10cm} \centerline{
\includegraphics{fig02.01.1.eps}
} \caption[Kernel rescaling]{\label{fig21}The rescalings
$g_t(\fxx)= ({1}/{t^2}) g({\fxx}/{t})$ of a kernel for t=4, 3, and
2.}
\end{figure}

\begin{theorem}\label{theorem2.2} If $g$ is Laplacian consistent, then for every $u\in {\cal F}\cap C^3(\real^N)$,
\begin{equation}\label{LC}
g_h*u(\fxx)-u(\fxx)=h\frac{\sigma}{2}\Delta
u(\fxx)+\varepsilon(h,\fxx)
\end{equation}
where $|\varepsilon(h,\fxx)|\leq Ch^{3/2}$.
\end{theorem}
\begin{proof}We use condition $(i)$, the definition of $g_h$, and rescaling $\fxy=h^{1/{2}}\fxz$ inside the integral to see that
\begin{align*}
g_h*u(\fxx)-u(\fxx)=&\int_{\RR^N}\frac{1}{h^{{N}/{2}}}\,g\Big(\frac{\fxy}{h^{{1}/{2}}}\Big)\big(u(\fxx-\fxy)-u(\fxx)\big)\,\rmd\fxy\\
=&\int_{\RR^N}g(\fxz)\big(u(\fxx-h^{1/{2}}\fxz)-u(\fxx)\big)\,\rmd\fxz.
\end{align*}
Using Taylor's formula with the Lagrange remainder \eqref{TaylorLagrange}, we have


\begin{eqnarray*}
% \nonumber to remove numbering (before each equation)
  u(\fxx-h^{1/{2}}\fxz)-u(\fxx) &=& -h^{1/{2}}Du(\fxx)\cdot\fxz+\frac{h}{2}D^2u(\fxx)(\fxz,\fxz)\\
   & & -\frac{1}{6}h^{3/{2}}D^3u(\fxx-h^{1/{2}}\theta\fxz)(\fxz,\fxz,\fxz),
\end{eqnarray*}
where $\theta=\theta(\fxx, \fxz,h)\in [0,1]$. By condition $(ii)$, $\int_{\RR^N}g(\fxz)Du(\fxx)\cdot\fxz\,\rmd\fxz=0$;
by conditions $(iii)$ and $(iv)$, $\int_{\RR^N}g(\fxz)D^2u(\fxx)(\fxz,\fxz)\,\rmd\fxz=\sigma\Delta u(\fxx)$. Thus,
\[
g_h*u(\fxx)-u(\fxx)=h\frac{\sigma}{2}\Delta
u(\fxx)-\frac{1}{6}h^{3/{2}}\int_{\RR^N}g(\fxz)D^3u(\fxx-h^{1/{2}}\theta\fxz)(\fxz,\fxz,\fxz)\,\rmd\fxz.
\]
We denote the error term by $\varepsilon(h,\fxx)$. Then we have
the following estimate:
\begin{align*}
|\varepsilon(h,\fxx)|\leq&\frac{1}{6}h^{{3}/{2}}\int_{\RR^N}|g(\fxz)D^3u(\fxx-h^{1/{2}}\theta\fxz)(\fxz,\fxz,\fxz)|\,\rmd\fxz\\
\leq&\frac{1}{6}h^{3/{2}}N^{3}\sup_{\alpha,\fxx}|\partial^{\alpha}u(\fxx)|\int_{\RR^N}|\fxz|^3|g(\fxz)|\,\rmd\fxz,
\end{align*}
where the supremum is taken over all vectors $\alpha=(\alpha_1,\alpha_2,\cdots,\alpha_N)$, $\alpha_j\in\{1,2,3\}$, such
that $|\alpha|=3$ and over all $\fxx\in \RR^N$.
\end{proof}

The preceding theorem shows a direct relation between smoothing
with a Laplacian-consistent kernel and the heat equation. It also
shows why we require $\sigma$ to be positive: If it is not
positive, the kernel is associated with the inverse heat equation
(see Exercise \ref{minimalrequirements}.)

\section{The convergence theorem}

The result of the next theorem is illustrated in Figure \ref{22}.

\begin{theorem}\label{theorem2.3} Let $g$ be a nonnegative Laplacian-consistent kernel
with $\sigma=2$ and define $g_h$ by \eqref{gh}. Write
$T_hu_0=g_h*u_0$ for $u_0\in \mathcal F$, and let
$u(t,\cdot)=G_t*u_0$ be the solution of the
 heat equation \eqref{heatsol}. Then, for each $t>0$,
\begin{equation}\label{37}\addtolength\belowdisplayskip{-12pt}
(T_h^nu_0)(\fxx)\to u(t,\fxx)\,\text{ uniformly in }\fxx \text{ as
}\, n\to+\infty\,\text{ and }\, nh\to t.
\end{equation}
\end{theorem}

\begin{proof}
Let us start with some preliminaries. We have $(g_h\ast u_0)(\infty)=u_0(\infty)$ and therefore
$T^n_hu_0(\infty)=u_0(\infty)$. The norm in $\mathcal F$ is $||u||_{\mathcal F}=\sup_{\fxx\in
\SN}|u(\fxx)|=\sup_{\fxx\in\RR^N}|u(\fxx)|$. The first order of business is to say precisely what is meant by the
asymptotic limit \eqref{37}: Given $t>0$ and given $\varepsilon>0$, there exists an $n_0=n_0(t,\varepsilon)$ and a
$\delta=\delta(t,\varepsilon)$ such that $\|T_h^nu_0-u(t,\cdot)\|_{\mathcal F}\leq\varepsilon$ if $n>n_0$ and
$|nh-t|\leq\delta$. This is what we must prove.  We will first prove the result when $h=t/n$. We will then show that
the result is true when $h$ is suitably close to $t/n$.

\begin{figure}[!t]
\setkeys{Gin}{width=12cm} \centerline{\includegraphics{fig02.02.1.eps}} \caption[Iterated linear smoothing converges to
the heat equation]{\label{22}Iterated linear smoothing converges to the heat equation. In this experiment with
one-dimensional functions, it can be appreciated how fast an iterated convolution of a positive kernel converges to a
Gaussian. On the left are displayed nine iterations of the convolution of the characteristic function of an interval
with itself, with appropriate rescalings. On the right, the same experiment is repeated with a much more irregular
kernel. The convergence is almost as fast as the first case.}
\end{figure}


We begin with comments about the notation. By Exercise \ref{exercise2.2}, $(T_h)^n=(T^n)_h$, so there is no ambiguity
in writing $T^n_h$. We will be applying $T^n_h$ to the solution $u$ of the heat equation, which is $C^{\infty}$ on
$(0,+\infty)\times\RR^N.$ In this situation, $t$ is considered to be a parameter, and we write $T^n_hu(t,\fxx)$ as
shorthand for $T^n_hu(t,\cdot)(\fxx).$ Throughout the proof, we will be dealing with error terms that we write as
$O(h^r)$. These terms invariably depend on $h$, $t$, and $\fxx$. However, in all cases, given a closed interval
$[t_1,t_2]\subset(0,+\infty)$, there will be a constant $C$ such that $|O(h^r)|\leq Ch^r$ uniformly for $t\in[t_1,t_2]$
and $\fxx\in \RR^N$. Finally, keep in mind that all functions of $\fxx$ tend to $u_0(\infty)$ as $\fxx\to \infty$.

We wish to fix an interval $[t_1,t_2]$, but since this depends on the point $t$ in \eqref{37} and on $\varepsilon$, we
must first choose these numbers. Thus, choose $\tau>0$ and keep it fixed. This will be the ``$t$'' in \eqref{37}. Next,
choose $\varepsilon>0$. Here are the conditions we wish $t_1$ and $t_2$ to satisfy:
\begin{enumerate}[(1)]\setlength\itemsep{2pt}

\item $t_1$ is small enough so $\|u(t_1,\cdot)-u_0\|_{\mathcal
F}<\varepsilon$. (This is possible by Theorem
\ref{existenceuniquenessheat}.)

\item $t_1$ is small enough so
$\|u(t_1+\tau,\cdot)-u(\tau,\cdot)\|_{\mathcal F}<\varepsilon$.
(Again, by Theorem \ref{existenceuniquenessheat}.)

\item $t_2$ is large enough so $t_1+\tau<t_2$.
\end{enumerate}

There is no problem meeting these conditions, so we fix the
interval $[t_1,t_2]\subset(0,+\infty)$.


\medskip  \noindent {\bf Step 1, main argument :  proof that}
\begin{equation}\label{premiereetape}
\lim_{n\to+\infty\atop nh=\tau}T^n_hu(t_1,\fxx)=u(t_1+\tau,\fxx),
\end{equation}
where the convergence is uniform for  $\fxx\in \RR^N$.
\newline We can use
Theorem \ref{theorem2.2} to write
\begin{equation}\label{38}
T_hu(t,\fxx)-u(t,\fxx)=h\Delta u(t,\fxx)+O(h^{{3}/{2}}),
\end{equation}
where $t\in[t_1,t_2]$. That the error function is bounded uniformly  by $Ch^{3/2}$ on  $[t_1,t_2]\times \RR^N$ follows
from the fact that $\sup_{\alpha,t,\fxx}|\partial^{\alpha}u(t,\fxx)|$ is finite for $(t,\fxx)\in[t_1,t_2]\times \RR^N$
(see the proof of Theorem \ref{theorem2.2}). Since $u$ is a solution of the heat equation, we also have by
Taylor-Lagrange formula \eqref{TaylorLagrange},
\begin{equation}\label{39}
u(t+h,\fxx)-u(t,\fxx)=h\frac{\partial u}{\partial t}(t,\fxx) + \frac{h^2}2\frac{\partial^2}{\partial t^2}u(t+\theta
h,\fxx)=h\Delta u(t,\fxx) + O(h^2).
\end{equation}
This time the error term is bounded uniformly by $Ch^2$ on $[t_1,t_2]\times \RR^N$ because $\frac{\partial^2
u}{\partial t^2}(t,\fxx)$ is bounded on   $[t_1,t_1]\times\RR^N.$ By subtracting \eqref{39} from \eqref{38} we see that
\begin{equation}\label{310}
T_hu(t,\fxx)=u(t+h,\fxx)+O(h^{{3}/{2}}).
\end{equation}
This shows that applying $T_h$ to a solution of the heat equation
at time $t$ advances the solution to time $t+h$, plus an error
term.

So far we have not used the assumption that $g$ is nonnegative.
Thus, \eqref{310} is true for any Laplacian-consistent kernel $g$
with $\sigma=2$. However, we now wish to apply the linear operator
$T_h$ to both sides of equation \eqref{310}, and in doing so we do
not want the error term to increase. Since $g\geq0$, this is not a
problem:
\[
|T_hO(h^{{3}/{2}})|\leq
\int_{\RR^N}|O(h^{{3}/{2}})|g_h(\fxx-\fxy)\,\rmd\fxy\leq
\int_{\RR^N}Ch^{{3}/{2}}g_h(\fxx-\fxy)\,\rmd\fxy=Ch^{{3}/{2}}.
\]
With this in hand, we can apply $T_h$ to both sides of \eqref{310}
and obtain
\begin{equation}\label{311}
T^2_hu(t,\fxx)=T_hu(t+h,\fxx)+O(h^{{3}/{2}}).
\end{equation}
If we write equation \eqref{310} with $t+h$ in place of $t$ and
substitute the expression for $T_hu(t+h,\fxx)$ in equation
\eqref{311}, we have
\begin{equation}
T^2_hu(t,\fxx)=u(t+2h,\fxx)+2O(h^{{3}/{2}}).
\end{equation}
We can iterate this process and get
\begin{equation}
T^n_hu(t,\fxx)=u(t+nh,\fxx)+nO(h^{{3}/{2}})
\end{equation}
with the same constant $C$ in the estimate $|O(h^{{3}/{2}})|\leq
Ch^{{3}/{2}}$ as long as $t+nh\in[t_1,t_2]$. To ensure that this
happens, we take $t=t_1$ and $h=\tau/n$. Then
\begin{equation}\label{315}
T^n_hu(t_1,\fxx)=u(t_1+\tau,\fxx)+O\Big(\Big(\frac{\tau}{n}\Big)^{{1}/{2}}\Big)
\end{equation}
and we obtain \eqref{premiereetape}. If we could take $t_1=0$,
this would end the proof. This is not possible because all of the
$O$ terms were based on a fixed interval $[t_1,t_2]$. However, we
have taken $t_1$  small enough to finish the proof .


\medskip \noindent {\bf Step 2 : getting rid of $t_1$.}\newline
Since $\int_{\RR^N}g(\fxx)\,\rmd\fxx=1$, $\|g_h\|_{L^1(\RR^N)}=1$,
and thus
\[
\|g_h^{n*}*v\|_{\mathcal F}\leq\|v\|_{\mathcal F}.
\]
If we take $v=u(t_1,\cdot)-u_0$, then this inequality and
condition (1) imply that
\begin{equation}\label{317}
\|T_h^{n}u(t_1,\cdot)-T_h^{n}u_0\|_{\mathcal F}<\varepsilon.
\end{equation}
  Relations
\eqref{317} and \eqref{315} imply that
\begin{equation}\label{319}
\|T_h^{n}u_0-u(t_1+\tau,\cdot)\|_{\mathcal F}<2\varepsilon.
\end{equation}
This inequality and condition (2) show that
\begin{equation}\label{215}
\|T_h^{n}u_0-u(\tau,\cdot)\|_{\mathcal F}<3\varepsilon
\end{equation}
for $n>n_0$ and $h=\tau/n$. This proves the theorem in the case
$h=\tau/n$.

\medskip\noindent{\bf Conclusion.} It is a simple matter to obtain the more
 general result.
Again, by Theorem 1.10, there is a
$\delta=\delta(\tau,\varepsilon)$ such that $|nh-\tau|<\delta$
implies that $\|u(nh,\cdot)-u(\tau,\cdot)\|_{\mathcal
F}<\varepsilon$ and that $nh\in[t_1,t_2]$ (by condition (3)).
Combining this with \eqref{215} shows that
\[
\|T_h^{n}u_0-u(nh,\cdot)\|_{\mathcal F}<4\varepsilon
\]
if $n>n_0$ and $|nh-\tau|<\delta$, and this completes the
proof.\hfill
\end{proof}


\section{Directional averages and directional heat equations}
In this section, we list easy  extensions of Theorem \ref{theorem2.2}. They analyze local averaging processes which
take averages at each point in a singular neighborhood made of a segment.  In that way, we will make appear several
nonlinear generalizations of the Laplacian which will accompany  us throughout the book.
 Consider a
$C^2$ function from $\RR^N$ into $\RR$ and a vector $\fxz\in\RR^N$
with $|\fxz|=1$. We wish to compute the mean value of $u$ along a
segment of the line through $\fxx$ parallel to the vector $\fxz$.
To do this, we define the operator $T^{\fxz}_h$, $h\in[-1,1]$, by
\[
T^{\fxz}_hu(\fxx)=\frac{1}{2h}\int_{-h}^h u(\fxx+s\fxz)\,\rmd s.
\]
This operator is the directional counterpart of the isotropic operator $M_h$ defined by equation (1.1).
\begin{proposition}\label{proposition2.4}
\[
T^{\fxz}_hu(\fxx)=u(\fxx)+\frac{h^2}{6}D^2u(\fxx)(\fxz,\fxz)+o(h^2).
\]
\end{proposition}
Proposition \ref{proposition2.4} is deduced from Theorem \ref{theorem2.2}, and it suggests that iterations of the
operator $T^{\fxz}_h$ are associated with the directional heat equation
\begin{equation}\label{dirheateqn1}
\frac{\partial u}{\partial
t}(t,\fxx)=\frac{1}{6}D^2u(t,\fxx)(\fxz,\fxz)
\end{equation}
in the same way that the iterations  of the operator $T_h$ in Theorem \ref{theorem2.3} are associated with the ordinary
heat equation. If $\fxz$ is fixed, then the operator $T^{\fxz}_h$ and equation \eqref{dirheateqn1} act on $u$ along
each line in $\RR^N$ parallel to $\fxz$ separately; there is no ``cross talk'' between lines. Exercise
\ref{exercise2.4} formalizes and clarifies these comments when $\fxz$ is fixed. However, Proposition
\ref{proposition2.4}  remains true when $\fxz$ is a function of $\fxx$. This means that we are able to approximate the
directional second derivative by taking directional averages where $\fxz$ varies from point to point. The main choices
considered in the book are $\fxz={Du}/{|Du|}$ and $\fxz={Du^{\perp}}/{|Du|}$, where $Du=(u_x,u_y)$ and
$Du^{\perp}=(-u_y,u_x)$. Then by Proposition \ref{proposition2.4} we have the following limiting relations:

\begin{enumerate}[$\bullet$]\setlength\itemsep{2pt}
\item Average in the direction of the gradient. By choosing
$z={Du}/{|Du|}$,
\[
\frac{1}{|Du|^2}D^2u(Du,Du)=6\lim_{h\to0}\frac{T^{Du/|Du|}_hu-u}{h^2}.
\]
We will interpret this differential operator as Haralick's edge
detector in section \ref{sectionTEDD}.

\item Average in the direction orthogonal to the gradient. By
choosing $z={Du}/{|Du^\perp|}$,
\[
\frac{1}{|Du|^2}D^2u(Du^\perp,Du^\perp)=6\lim_{h\to0}\frac{T^{Du^\perp/|Du|}_hu-u}{h^2}.
\]
This differential operator appears as the second term of the curvature equation. (See Chapter \ref{Chap5}.)
\end{enumerate}
Although we have not written them as such, the limits are
pointwise in both cases.

\section{Exercises}

\begin{exercise}
We will denote the characteristic function of a set
$A\subset\RR^N$ by $\bmone_A$. Thus, $\bmone_A(\fxx)=1$ if
$\fxx\in A$ and $\bmone_A(\fxx)=0$ otherwise. Consider the kernel
 $g=(1/\pi)\bmone_{D(0,1)}$, where $D(0,1)$ is the disk of
radius one centered at zero. In this case, $g$ is a radial
function
 and it is clearly Laplacian consistent. For $N=2$, let
  $A=[-1/2,1/2]\times[-1/2,1/2]$. Then $g=\bmone_A$ is not radial.
  Show that it is, however, Laplacian consistent. If we take
  $B=[-1,1]\times[-1/2,1/2]$, then $g=(1/2)\bmone_B$ is no longer
   Laplacian consistent because it does not satisfy condition $(iv)$.
   Show that this kernel does, however, satisfy a relation similar to
   \eqref{LC}. \finex
\end{exercise}

\begin{exercise} \label{exercise2.1}  The aim of the exercise
 is to prove roughly that radial
functions with fast decay are Laplacian consistent. Assume $g\in L^1(\RR^N)$ is radial with finite first second
moments, $\int_{\RR^N}|\fxx|^k|g(\fxx)|\,\rmd\fxx<+\infty,$ $k=0, 1, 2, 3$ and such that
$\int_{\RR^N}\fxx_i^2g(\fxx)\,\rmd\fxx>0.$ Show that $g$ satisfies conditions $(ii)$ and $(iii)$ of Definition
\ref{deflaplacianconsistency} and that, for suitably chosen $a,b\in\RR$, the rescaled function $\fxx\mapsto ag(\fxx/b)$
satisfies conditions $(i)$ and $(iv)$, where $\sigma$ can be taken to be an arbitrary positive number. \finex
\end{exercise}

\begin{exercise}\label{minimalrequirements} The aim of the
exercise is to illustrate by simple examples what happens to the
iterated filter $g^{n\ast}$, $n\in \NN$ when $g$ does not satisfy
some of the requirements of the Laplacian consistency (Definition
\ref{deflaplacianconsistency}). We recall the notation \eqref{gh},
$g_h(\fxx)=\frac{1}{h^{{N}/{2}}}\,g\Big(\frac{\fxx}{h^{{1}/{2}}}\Big).$

\medskip \noindent 1)  Take on $\RR$, $g(x)=1$ on $[-1,1]$,
$g(x)=0$ otherwise. Which one of the assumptions $(i)-(v)$ is not
satisfied in Definition \ref{deflaplacianconsistency} ?  Compute
$g_\frac 1n^{n\ast}\ast u$, where  $u=1$ on $\RR$. Conclude : the
iterated filter blows up.

\medskip \noindent 2) Take on $\RR$, $g(x)=1$ on $[0,1]$,
$g(x)=0$ otherwise. Which one of the assumptions $(i)-(v)$ is not satisfied in Definition \ref{deflaplacianconsistency}
  Compute $g^{n\ast}\ast u$, where  $u(x)=x$ on $\RR$. Conclude : the iterated filter ``drifts".

\medskip \noindent 3) Assume that the  assumptions
$(i)-(v)$ hold, except $(iii)$. By a simple adaptation of its
proof, draw a more general form of Theorem \ref{theorem2.2}.

\medskip  \noindent 4) Perform the same analysis as in 3) when all
assumptions hold but $(iv)$.

\medskip \noindent 5) Take the case of dimension $N=1$ and assume
 that $(i)$  hold but $(ii)$ does not  hold. Set
 $g_h(\fxx)=\frac{1}{h}\,g\Big(\frac{\fxx}{h}\Big)$ and give a
 version of Theorem \ref{theorem2.2} in that case (make an order 1 Taylor
 expansion of $u$).
\finex
\end{exercise}

\begin{exercise}\label{exercise2.4}
  Let $\fxz$ be a
fixed vector in $\RR^N$ with $|\fxz|=1$ and let $u_0$ be in
$\mathcal  F$. Define  a one-dimensional kernel $g$ by
$g(s)=\frac{1}{2}{\bmone}_{[-1,1]}(s)$.
\begin{enumerate}[(i)]\setlength\itemsep{2pt}
\item Show that $g$ is Laplacian consistent. Compute the variance
$\sigma$ of $g$. \item Show that
\[
u(t,\fxx)=\int_\RR u_0(\fxx+s\fxz)G_t(s)\,\rmd s
\]
is a solution of the directional heat equation
\begin{equation}\label{dirheateqn2}
\frac{\partial u}{\partial t}(t,\fxx)=D^2u(t,\fxx)(\fxz,\fxz),\quad u(0, \fxx)=u_0(\fxx).
\end{equation}

Give an example to show that $u(t,\cdot)$ is not necessarily
$C^2$. This being the case, how does one interpret the right-hand
side of \eqref{dirheateqn2}? \item Let
$g_h(s)=(6h)^{-1/2}g(s/(6h)^{-1/2})$ and
$T_hu(\fxx)=\int_{\RR}u(\fxx+s\fxz)g_h(s)\,\rmd s$. By applying
Theorem \ref{theorem2.3} for $N=1$, show that, for each $t>0$,
\begin{equation}
T_h^nu_0\to u(t,\cdot)\,\text{ in }\, \mathcal F\,\text{ as }\,
n\to+\infty\,\text{ and }\, nh\to t.\, \finex
\end{equation}
\end{enumerate}
\end{exercise}


\begin{exercise}
The Weickert equation can be viewed  as a variant of the curvature equation \cite{Weickert:1998:ADI}. It uses a
nonlocal estimate of the direction orthogonal to the gradient for the diffusion direction. This direction is computed
as the direction $v$ of the eigenvector corresponding to the smallest eigenvalue of $k\ast(Du\otimes Du)$, where
$(\bm{y}\otimes\bm{y})(\bm{x})=(\bm{x}\cdot\bm{y})\bm{y}$ and $k$ is a smooth kernel, typically a gaussian. Prove that
if the convolution kernel is removed, then this eigenvector is simply $Du^\perp$.  So the equation writes
\begin{equation}\label{diffusiondansunedirection}
\frac {\partial u}{\partial t}  = u_{\eta\eta},
\end{equation}
where $\eta$ denotes the coordinate in the direction $v$. \finex
\end{exercise}

\begin{exercise} Suppose that $u\in C^2(\RR)$. Assuming that $u'(x)\not=0$, show that
\begin{equation}\label{ex251}
u''(x)=\lim_{h\to0}\frac{1}{h^2}\Big(\max_{s\in[-h,h]}u(x+s)+\min_{s\in[-h,h]}u(x+s)-2u(x)\Big).
\end{equation}
What is the value of the right-hand side of \eqref{ex251} if
$u'(x)=0$?

Now consider $u\in C^2(\RR^2)$. We wish to establish an algorithm
similar to \eqref{ex251} to compute the second derivative of $u$
in the direction of the gradient $Du=(u_x,u_y)$. For this to make
sense, we must assume that $Du(\fxx)\not=0$. With these
assumptions, we know from \eqref{ex251} that
\begin{equation}\label{ex252}
u_{\xi\xi}(\fxx)=\frac{\partial^2 v}{\partial
\xi^2}(\fxx,0)=\lim_{h\to0}\frac{1}{h^2}\Big(\max_{s\in[-h,h]}u(\fxx+s\fxz)+\min_{s\in[-h,h]}u(\fxx+s\fxz)-2u(\fxx)\Big),
\end{equation}
where $v(\fxx,\xi)=u(\fxx+\xi\fxz)$ and $\fxz=Du/|Du|$. The second
part of the exercise is to prove that, in fact,
\begin{equation}\label{ex253}
u_{\xi\xi}(\fxx)=\lim_{h\to0}\frac{1}{h^2}\Big(\max_{\fxy\in
D(0,h)}u(\fxx+\fxy)+\min_{\fxy\in
D(0,h)}u(\fxx+\fxy)-2u(\fxx)\Big),
\end{equation}
where $D(0,h)$ is the disk of radius $h$ centered at the origin.
Intuitively, \eqref{ex253} follows from \eqref{ex252} because the
gradient indicates the direction of maximal change in $u(\fxx)$,
so in the limit as $h\to0$, taking max and min in the direction of
the gradient is equivalent to taking max and min in the disk. The
point of the exercise is to formalize this. \finex
\end{exercise}


\section{Comments and references}
\paragraph*{Asymptotics.}
Our proof that iterated and rescaled convolutions of a Laplacian-consistent kernel tend asymptotically to the Gaussian
is a version of the De Moivre--Laplace formula, or the central limit theorem, adapted to image processing
\cite{Breiman:1968:P}. This result is particularly relevant to image analysis, since it implies that iterated linear
smoothing leads inevitably to convolution with the Gaussian, or equivalently, to the application of the heat equation.
We do not wish to imply, however, that the Gaussian is the only important kernel for image processing. The Gaussian
plays a significant role in our form of image analysis, but there are other kernels that, because of their spectral and
algebraic properties, have equally important roles in other aspects of signal and image processing. This is
particularly true for wavelet theory which combines recursive filtering and sub-sampling.


\paragraph*{Directional diffusion.}
Directional diffusion has a long history that began when Hubel and
Wiesel showed the existence of direction-sensitive cells in the
visual areas of the neocortex \cite{Hubel}. There has been an
explosion of publication on directional linear filters, beginning,
for example, with influential papers such as that by Daugman
\cite{Daugman:1985:URR}. We note again that Gabor's contribution
to directional filtering is described in
\cite{Lindenbaum:1994:GCI}.


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