%gmChap13.tex

\graphicspath{{./figures/chapter13/}}

\chapter[Localizability and the Maximum Principle]{Localizable Structuring Elements and the Local Maximum Principle}

Given a set of structuring elements $\mathcal B$, the scaled operators $IS_h$ defined by
\[
IS_hu(\fxx)=\inf_{B\in \mathcal B}\sup_{\fxy\in\fxx+hB} u(\fxy)
\]
are immediately translation invariant and contrast invariant. Furthermore, if the elements of $\mathcal B$ are uniformly bounded, then the operators satisfy an important local property that we have not yet emphasized: If two functions $u$ and $v$ are such that $u(\fxy)\leq v(\fxy)$ for all $\fxy$ in some disk $D(\fxx,r)$, then for sufficiently small $h$, $hB\in D(0,r)$, and $IS_hu(\fxx)\leq IS_hv(\fxx)$. This is a special case of the \textit{local maximum principle,} which for bounded structuring elements goes almost un-noticed. It might seem at first glance that we would not have a local maximum principle if the structuring elements were not bounded. It turns out, however, that for a large class of unbounded structuring elements, the operators $IS_h$ behave as if they were local operators---in the sense that they satisfy a local maximum principle. For example, if the operators $IS_h$ are affine invariant, then the affine-invariant structuring elements $\mathcal B$ cannot be bounded. Indeed, affine invariance allows an element $B\in\mathcal B$ to be stretched arbitrarily far in any direction: The matrix 
$A=
\Big(\begin{smallmatrix}
\varepsilon & 0\\
0 & 1/\varepsilon
\end{smallmatrix}
\Big)$,
where $\varepsilon$ is small, followed by a rotation, does the job. Nevertheless, there are affine-invariant operators that satisfy a local maximum principle. In particular, we will show that this is the case for $\mathcal B={\mathcal B}_{\rm aff}$. In general, the application of operators that satisfy this property involves an error term: 
\[
IS_hu(\fxx)\leq IS_hv(\fxx)+o(h^\beta).
\]

We are going to define a property of structuring elements $\mathcal B$ called \textit{localizability}, and even though $\mathcal B$ may contain arbitrarily large elements, or even unbounded elements, if it is localizable, then the inf-sup operators defined by $\mathcal B$ will satisfy a local maximum principle. The importance of the local maximum principle for our program will become clear in the chapter on viscosity solutions. Since our focus is on affine-invariant operators, we will apply the concept of localizability only to families of affine-invariant structuring elements, but the reader should keep in mind that concept is applicable to other situations. For example, the structuring elements associated with a median filter defined by a $k$ that does not have compact support, are unbounded, but they may be localizable. A case in point is the Gaussian: It does not have compact support, but it can be shown that the structuring elements are localizable. 

\section{Localizable sets of structuring elements}

Recall that the Euclidean distance $d$ between a point $\fxx$ and a set $Y$ is defined by 
\[
d(\fxx,Y)=\inf_{\fxy\in Y}|\fxx-\fxy|.
\]
It will be convenient to use the following notation: $D(0,\rho)$ will denote the open disk (or ball) $\{\fxx\mid |\fxx|<\rho\}$, and $D_a$ will denote the dilation operator defined by 
\[
D_a(X)=\{\fxx\mid d(\fxx,X)<a\}.
\]
In the notation of Chapter 8, this means that $D_a=\mathcal D_a$, where the structuring element for $\mathcal D_a$ is $D(0,1)$. Note that if $X$ is open and connected, then $D_a(X)$ is open and connected. We will write $\partial X$ to denote the boundary of $X$. These definitions and notation are used to define the concept of a set of structuring elements being localizable.

For convenience, we introduce two set operators:
\begin{enumerate}[$\bullet$]\setlength\itemsep{2pt}
\item $\mathcal C_{0}[X]=\text{ the connected component of $X$ that contains the origin.}$
\item $\mathcal C_{0}^\partial[X]=\text{ the connected component of $X$ that contains the origin in its boundary.}$
\end{enumerate}
The set $X$ will always be open and either contain the origin or contain the origin in its boundary. Note that these operators commute with scaling: For example, $\mathcal C_{0}[hX]=h\mathcal C_{0}[X]$. Note also that these operators are monotone.

\begin{definition}Let $\alpha>0$ be a positive constant. Assume that $\mathcal B$ is a set of structuring elements whose members are open and contain the origin. $\mathcal B$ is said to be $\alpha$-localizable if there are two constants $c>0$ and $R>0$, where $c$ and $R$ depend on $\mathcal B$, such that for every $\rho>R$ the following conditions holds: For each $B\in \mathcal B$, there is a $B'\in\mathcal B$ such that
\begin{enumerate}[$(i)$]\setlength\itemsep{2pt}
\item $B'\subset D(0,\rho);$
\item $B'\subset D_{c/\rho^\alpha}(\mathcal C_{0}[B_\rho])$, where $B_\rho=B\cap D(0,\rho)$.
\end{enumerate}
The constant $\alpha$ is called the exponent of localizability. 
\end{definition}

We wish to emphasize that our definition of localizable includes the assumption that the elements of $\mathcal B$ are open and contain the origin. Note also that a localizable set $\mathcal B$ contains bounded members. Also, it may happen  that $\mathcal C_0[D_{c/\rho^\alpha}(\mathcal C_0[B_\rho])\cap D(0,\rho)]$ is itself a member of $\mathcal B$ for all $B\in\mathcal B$; indeed, the proof of Proposition 13.4 shows that this is the case if $\mathcal B=\mathcal B_{\rm aff}$. The next result shows how the concept of localizability scales.

\begin{proposition}Let $h$, $0<h\leq1$, be a scaling factor and assume the notation of  Definition 13.1. A set of structuring elements $\mathcal B$ is  $\alpha$-localizable if and only if there are constants $c>0$ and $R>0$ such that for all $r>0$ and all $h<r/R$ the following conditions holds: For each $B\in h\mathcal B$, there is a $B'\in h\mathcal B$ such that
\begin{enumerate}[$(i')$]\setlength\itemsep{2pt}
\item $B'\subset D(0,r);$
\item $B'\subset D_{ch^{\alpha+1}/r^\alpha}(\mathcal C_{0}[B_r])$, where $B_r=B\cap D(0,r)$.
\end{enumerate}
\end{proposition}

\begin{proof}Assume the conditions of the proposition. Then $r/R>1$ for all $r>R$. Since $h$ can be any number in the range $0<h<r/R$, the conditions of the proposition are true for $h=1$. By letting $\rho =r$, we have the conditions of Definition 13.1. 

To prove that the conditions of Definition 13.1 imply the conditions of the proposition, let $\rho=r/h$. Then the statement ``all $\rho> R$'' is equivalent to the statement ``all $r>0$ and all $h<r/R$.'' Next, we need to see how  $D_{c/\rho^\alpha}(X)$ scales:   
\begin{align*}
hD_{c/\rho^{\alpha}}(X)&=\{h\fxx\mid d(\fxx,X)< c/\rho^{\alpha}\}\\
&=\{h\fxx\mid d(h\fxx,hX)< ch/\rho^{\alpha}\}\\
&=\{\fxy\mid d(\fxy,hX)< ch/\rho^{\alpha}\}\\
&=D_{ch/\rho^{\alpha}}(hX).
\end{align*}
If $X=\mathcal C_0[B_\rho]$, then  $hD_{c/\rho^{\alpha}}(\mathcal C_0[B_\rho])=D_{ch^{\alpha+1}/r^{\alpha}}(\mathcal C_0[hB_\rho])$, where  $hB_\rho=B_r$. In other words, $B'\in D_{c/\rho^\alpha}(\mathcal C_{0}[B_\rho])$ implies that $hB'\in D_{ch^{\alpha+1}/r^{\alpha}}(\mathcal C_0[B_r])$,
which shows that $(ii)$ implies $(ii')$.
\end{proof}

We will use Definition 13.1 and its scaled version, Proposition 13.1, to prove two results: The first is that if $\mathcal B$ is localizable, then the $IS_h$ satisfy the local maximum principle; the second is that ${\mathcal B}_{\rm aff}$ is 1-localizable.  


\section{The local maximum principle}

While the notion of $\alpha$-localizability has an important role in mathematical morphology, we are concerned in this book only with the 1-localizabable families of structuring elements. Thus, from this point, we assume that $\alpha=1$ and leave the general cases as exercises.

\begin{lemma}[local maximum principle]Let $\mathcal B$ be a $1$-localizable set of structuring elements with the associated constants $c>0$ and $R>0$. Assume that the functions $u$ and $v$ satisfy a Lipschitz condition on a disk $D(\fxx,r)$ with Lipschitz constant $L$. If $u(\fxy)\leq v(\fxy)$ for $\fxy\in D(\fxx, r)$ and if $h<r/R$, then
\[
IS_hu(\fxx)\leq IS_hv(\fxx)+Lc \frac{h^2}{r}\quad\text{and}\quad SI_hu(\fxx)\leq SI_hv(\fxx)+Lc \frac{h^2}{r}.
\]

\end{lemma}
\begin{proof}For notational convenience, we take $\fxx=0$. Then 
\begin{equation}\label{eq145}
IS_hv(0)=\inf_{B\in h\mathcal B}\sup_{\fxy\in B}v(\fxy)\geq\inf_{B\in h\mathcal B}\sup_{\fxy\in B\cap D(0,r)}v(\fxy)\geq\inf_{B\in h\mathcal B}\sup_{\fxy\in B\cap D(0,r)}u(\fxy).
\end{equation}
By Proposition 13.2, for all $B\in h\mathcal B$, there exists $B'\in h\mathcal B$ such that $B'\subset D(0,r)$ and $B'\subset D_{ch^2/r}(\mathcal C_0[B\cap D(0,r)])$. Thus, since $u$ is Lipschitz with constant $L$ and since for every point $b'\in B'$ there is a point $b\in B\cap D(0,r)$ such that $|b'-b|<ch^2/r$, we have
\begin{equation}\label{eq131bis}
\sup_{\fxy\in B\cap D(0,r)}u(\fxy)\geq \sup_{\fxy\in B'}u(\fxy)-Lc \frac{h^2}{r}.
\end{equation}
These last two inequalities imply that 
\[
IS_hv(0)+Lc \frac{h^2}{r}\geq\inf_{B\in h\mathcal B}\sup_{\fxy\in B'}u(\fxy),
\]
where $B'$ is any set such that $B'\in h\mathcal B$, $B'\subset D(0,r)$, and $B'\subset D_{ch^2/r}(\mathcal C_0[B\cap D(0,r)])$. If we denote the family of all such sets $B'$ associated with $B$ by $\mathcal B'=\mathcal B'(B)$, then a more precise statement is that
\[
IS_hv(0)+Lc \frac{h^2}{r}\geq\inf_{B\in h\mathcal B}\Big(\sup_{B'\in\mathcal B'}\sup_{\fxy\in B'}u(\fxy)\Big).
\]
Let  $\inf_{B\in h\mathcal B}\Big(\sup_{B'\in\mathcal B'}\sup_{\fxy\in B'}u(\fxy)\Big)=\lambda$. Then we claim that $\inf_{B\in h\mathcal B}\sup_{\fxy\in B}u(\fxy)\leq\lambda$. To see this, let $\varepsilon>0$ be arbitrary. By the definition of $\lambda$, there is some set $B\in h\mathcal B$ such that $\sup_{\fxy\in B}u(\fxy)\leq\lambda+\varepsilon$. Thus, $\inf_{B\in h\mathcal B}\sup_{\fxy\in B}u(\fxy)\leq\lambda+\varepsilon$, which, since $\varepsilon$ is arbitrary,  implies that $\inf_{B\in h\mathcal B}\sup_{\fxy\in B}u(\fxy)\leq\lambda$. This yields the result:
\[
IS_hu(\fxx)=\inf_{B\in h\mathcal B}\sup_{\fxy\in\fxx+B}u(\fxy)\leq IS_hv(\fxx)+Lc \frac{h^2}{r}. 
\]
The result for $SI_h$ follows from the relation $-IS_h(-u)=SI_h(u)$. 
\end{proof}

Note that the proof for $IS_h$ does not use the fact that $v$ is locally Lipschitz. In fact, the proof works for any $v$. The problem with this is that we would not have the result for $SI_h$ if we did not assume that $v$ is locally Lipschitz. Also, taking $\fxx=0$ in the proof is indeed only a notational convenience; knowing that the assumptions hold at $\fxx$ does not imply they hold elsewhere.  Thus, as stated, the result of Lemma 13.3 is strictly local. 

Lemma 13.3 and the next lemma provide the links between the localizability of structuring elements and the local properties of the associated operators. As such, they lie near the heart of our program. Their use is the key to  demonstrating the asymptotic behavior of inf-sup operators defined by 1-localizable families of affine-invariant structuring elements. The local maximum principle is also used in the proof of the Barles--Souganidis theorem, Proposition 15.13, which is essential for relating the inf-sup operators to their associated PDEs via viscosity solutions of the PDEs. 

\begin{exercise}Prove the general form of Lemma 13.3: Replace the hypothesis that $\mathcal B$ is $1$-localizable with the hypothesis that it is $\alpha$-localizable and conclude that 
\[
IS_hu(\fxx)\leq IS_hv(\fxx)+Lc \frac{h^{\alpha+1}}{r^\alpha}.\; \finex
\]
\end{exercise}

Lemma 13.3 compares the action of $IS_h$ on two functions $u$ and $v$. In the next lemma, we consider an operator $IS_h^r$ that approximates $IS_h$ and examine its action on a single function $u$. The approximate operator $IS_h^r$ is defined by truncating the structuring elements $\mathcal B$: We replace the family $\mathcal B$ with the family $\mathcal B_r=\{B_r\mid B_r=B\cap D(0,r), B\in\mathcal B\}$. Thus,
\[
IS_h^ru(\fxx)=\inf_{B_r\in h\mathcal B_r}\sup_{\fxy\in\fxx+B_r}u(\fxy)=\inf_{B\in h\mathcal B}\sup_{\fxy\in (\fxx+ B)\cap D(\fxx,r)}u(\fxy).
\]
The local properties of $IS_h^r$ have been imposed by definition. Later, when we apply this result, we will take $r=h^{1/2}$, so the error term will be $Lch^{3/2}$.

\begin{lemma}[localization lemma]Let $\mathcal B$ be a $1$-localizable set of structuring elements with constants $c>0$ and $R>0$ and assume that $h<r/R$.
If $u$ satisfies a Lipschitz condition in $D(\fxx,r)$ with constant $L$, then
\begin{enumerate}[$(i)$]\setlength\itemsep{2pt}
\item $IS_h^ru(\fxx)\leq IS_hu(\fxx)\leq IS_h^ru(\fxx)+Lch^2/r;$
\item $|IS_h^ru(\fxx)-IS_hu(\fxx)|\leq Lch^2/r;$
\item $|SI_h^ru(\fxx)-SI_hu(\fxx)|\leq Lch^2/r$.
\item $|SI^r_hIS^r_hu(\fxx)-SI_hIS_hu(\fxx)|\leq 2Lch^2/r$, if $u$ is $L$-Lipschitz on $\RR^N$
\end{enumerate}
\end{lemma}
\begin{proof}By taking $u=v$ in inequality \eqref{eq145}, we see that $IS_hu(\fxx)\geq IS_h^ru(\fxx)$. This half of $(i)$ does not depend on $u$ being Lipschitz on $D(\fxx,r)$, but the other half of $(i)$ does depend on $u$ being Lipschitz on $D(\fxx,r)$. To prove the other half of $(i)$, we are going to follow the proof of Lemma 13.3, including the notational convenience that $\fxx=0$. In particular, we use the 1-localizability of $\mathcal B$ to establish the inequality
\[
\sup_{\fxy\in B\cap D(0,r)}u(\fxy)\geq\sup_{\fxy\in B'}u(\fxy)+Lc\frac{h^2}{r},
\]
 which is \eqref{eq131bis}. The remainder of the proof shows that
\[
IS_h^ru(\fxx)+Lc\frac{h^2}{r}\geq IS_hu(\fxx),
\]
and this proves the other half of $(i)$.

Inequality $(ii)$ is just a restatement of $(i)$. Inequality $(iii)$ is deduced from  $(ii)$ by using the relation $IS_h(-u)=-SI_hu$. To prove $(iv)$, first recall from Lemma 6.5 that if $u$ is Lipschitz with constant $L$, then $IS_hu$ and $IS_h^ru$ are Lipschitz with constants no greater than $L$. By $(ii)$ and $(iii)$ we have 
\begin{align}
IS_h^ru(\fxx)&\leq IS_hu(\fxx)\leq IS_h^ru(\fxx)+Lch^2/r;\label{eq133}\\
SI_h^ru(\fxx)&\leq SI_hu(\fxx)\leq SI_h^ru(\fxx)+Lch^2/r.\label{eq134}
\end{align}
Replacing $u$ with $IS_hu$ in \eqref{eq134} and applying $SI_h^r$ to \eqref{eq133}
shows that  
\[
SI_h^rIS_h^ru(\fxx)\leq SI_h IS_hu(\fxx)\leq SI_h^rIS_h^ru(\fxx) +2Lch^2/r,
\]
which proves $(iv)$.
\end{proof}

The statements and proofs of Lemmas 13.3 and 13.4 are strictly local. There are, however, immediate global generalizations, and since these more general results are important for later applications, we give them a precise statement for future reference.

\begin{lemma}Let $K$ be an arbitrary set and assume that the function $u$ and $v$ in Lemmas 13.3, 13.4$(i)$, 13.4$(ii)$, and 13.4$(iii)$ are $L$-Lipschitz in $D(\fxx,r)$ for every $\fxx\in K$. Then the results of these lemmas are true uniformly for $\fxx\in K$.
\end{lemma} 

These uniform results need no special proofs. One merely rereads the proofs of the local lemmas and notes that, if the same hypotheses hold at each point $\fxx\in K$, then the results are true for each $\fxx$ with exactly the same error term. When we apply Lemma 13.5, $K$ will be compact, but clearly this is not a necessary condition for the lemma.

The next result is a direct consequence of Lemma 13.3. It allows us to fix an optimal relation between the localization scale $r$ and the operator scale $h$. Again, it is a local result that can easily be made uniform.

\begin{lemma}Let $\mathcal B$ be a $1$-localizable set of structuring elements with constants $c>0$ and $R>0$. Let $u$ and $v$ be two continuous functions that satisfy Lipschitz conditions with the same constant $L$ on a disk $D(0,r)$. If  
\[
|u(\fxx)-v(\fxx)|\leq C|\fxx|^3
\]
for $\fxx\in D(0,r)$, and if $h\leq r^2$ and $h<1/R^2$, then
\[
|IS_hu(0)-IS_hv(0)|\leq(C+Lc)h^{3/2}.
\]
\end{lemma}
\begin{proof}The relation $v(\fxx)-Cr^3\leq u(\fxx)\leq v(\fxx)+Cr^3$ is true for all $\fxx\in D(0,r)$, so we can apply Lemma 13.3 and conclude that 
\[
IS_hv(0)-Cr^3-Lc\frac{h^2}{r}\leq IS_hu(0)\leq IS_hv(0)+Cr^3+Lc\frac{h^2}{r}
\]
for $h<r/R$. This argument is also true for $0< s\leq r$, if we have $h<s/R$. So, in particular, if we take $s=h^{1/2}\leq r$ and $h<s/R$, that is, $h<1/R^2$, we have
\[
IS_hv(0)-Ch^{3/2}-Lch^{3/2}\leq IS_hu(0)\leq IS_hv(0)+Ch^{3/2}+Lch^{3/2},
\]
which proves the result. 
\end{proof}

Here, we have taken the point of view that $r$ is given, and we ask that $h=r^2$. In other situation, we may take the opposite view and ask that $r$ be determined by $h$. This is the case, for example, if we are able to choose the size of $r$ for the localized operator $IS_h^r$.

The main application of Lemma 13.4 is to reduce the asymptotic analysis of the operator $IS_h$ as $h\to0$ to the case where it is applied to quadratic polynomials. (We have seen in Chapters 10 and 11 how this kind of analysis works in the case of structuring elements that are bounded and isotropic.) 


\section[${\mathcal B_{\rm aff}}$ is 1-localizable]{$\bm{\mathcal B_{\rm aff}}$ is 1-localizable}

We are going to prove that $\mathcal B_{\rm aff}$ is 1-localizable, but to make things as transparent as possible, we first do some geometry. Thus, consider Figure \ref{dilateline}.

We are interested in the area of $D_{c/\rho}([0,z])\cap D(0,\rho)\cap \Delta^c$. This is the area of the figure $ABCD$. In what follows, $c>0$ is a constant. For the figure to make sense, we must assume that $\rho^2\geq c$. We want a lower bound for the area of the set $ABCD$ that will hold for any $\beta\in[0,\pi/2]$. (In the limit case $\beta=\pi/2$, and $z\in\Delta$.) We also wish to compare this area with that of another set, and the way we do this limits us to using the set $A0zD$. (The reason for this will become clear.) Denote the area of $A0zD$ by $\mathcal A=\mathcal A(\alpha,\beta)$.  Thus, assuming $\rho^2\geq c$, we have
\[
\mathcal A(\alpha,\beta)=\frac{c}{2}\Big(\beta\sin\alpha+\frac{\alpha}{\sin\alpha}+\cos\alpha\Big).
\]
This is written in terms of $\alpha$ and $\beta$ because it is easy to see what happens as $\alpha$ ranges from zero to $\pi/2$. This is equivalent to $\rho$ going from $+\infty$ to $\sqrt{c}$, which is just the range of interest. The smallest value of the function 
\[
f(\alpha,\beta)=\beta\sin\alpha+\frac{\alpha}{\sin\alpha}+\cos\alpha
\]
for $0\leq\alpha\leq\pi/2$, $0\leq\beta\leq\pi/2$ occurs at $\alpha=\pi/2$,  $\beta=0$, and $\mathcal A(\pi/2,0)=(\pi/4)c$ for these values. 

To avoid repeating it, we assume that the set $D(0,\rho)\cap\Delta^c$  always denotes the same open half-disk, and all of the sets we consider are understood to lie on the same side of $\Delta$ as $D(0,\rho)\cap\Delta^c$.

Parameterize the segment $[0,z]$ so the points are represented by $tz$, $t\in[0,1]$. Let $L(t)$ denote the line orthogonal to $[0,z]$ at $tz$. Let $y_t$ be any point on $L(t)$ such that $y_t\in D(0,\rho)\cap\Delta^c$, and consider the set $D(y_t,c/\rho)\cap D(0,\rho)\cap\Delta^c$. Then the open segment $L(t)\cap D(y_t,c/\rho)\cap D(0,\rho)\cap\Delta^c$ is always at least as long as the segment on the same line defined by $D(tz,c/\rho)\cap L(t)\cap (A0zD)$. Note that this is true for all $t\in[0,1]$, even for those $t$ close to one. 

\begin{figure}[!t]
\setkeys{Gin}{width=7cm}
\centerline{
\includegraphics{fig13.0b.1.eps}
}
\caption[Dilation of $[0,z)$]{Dilation of $[0,z]$.}
\label{dilateline}
\end{figure}

\begin{lemma}Assume the geometry and notation of Figure \ref{dilateline}. Let $\Gamma$ be any Jordan arc connecting the origin and $z$ and lying completely in the set $D(0,\rho)\cap\Delta^c$, except for the end points. Then
\[
{\rm area}\big(D_{c/\rho}(\Gamma)\cap D(0,\rho)\cap\Delta^c\big)\geq{\rm area}(A0zD)\geq\frac{\pi}{4}c
\]
whenever $\rho^2\geq c$.
\end{lemma}
\begin{proof}Let $f$ denote the characteristic function of $D_{c/\rho}(\Gamma)\cap D(0,\rho)\cap\Delta^c=G$. Then
\[
{\rm area}(G)=\int_{\RR^2}f(x,y)\,\rmd x\, \rmd y.
\]
By Fubini's theorem, 
\[
{\rm area}(G)=\int_{\RR}\Bigg(\int_{\RR}f(x,y)\,\rmd y\Bigg)\, \rmd x,
\]
and
\[
{\rm area}(G)\geq \frac{c}{2}\beta\sin\alpha+\int_{0}^1\Bigg(\int_{\RR}f(tz,y)\,\rmd y\Bigg)\, \rmd t.
\]
For fixed $t$, 
\[
\int_{\RR}f(tz,y)\,\rmd y= \int_{l(t)}f(tz,y)\,\rmd y,
\]
where $l(t)=G\cap L(t)$. We know that $l(t)$ contains a line segment at least as long as $(A0zD)\cap L(t)$, so 
\[
{\rm area}(G)\geq \frac{c}{2}\beta\sin\alpha+ {\rm area}(0zDE)={\rm area}(A0zD).
\]
Note that in adding the term $(c/2)\beta\sin\alpha$ to the area of $G$, we use the fact that the origin is a point of $\Gamma$.
\end{proof}

Ostensibly, this lemma has little to do with $\mathcal B_{\rm aff}$. 
The lemma only compares the area of $D_{c/\rho}(\Gamma)\cap D(0,\rho)\cap\Delta^c$, where $\Gamma$ is a Jordan arc that connects the origin to $z\in\partial D(0,\rho)$, with the area of $D_{c/\rho}([0,z])\cap D(0,\rho)\cap\Delta^c$. This is a purely geometric result, however, the application to ${\mathcal B}_{\rm aff}$ is direct.

Before stating and proving the theorem, we note that all of the connected sets involved in the proof are open and thus arcwise connected. As usual, $\Delta$  denotes a straight line through the origin, and $\Delta^c$ always denotes the same open half-plane. If $A$ is an open set that contains the origin, then the set $A\cap\Delta^c$ will contain a half-neighborhood $D(0,\varepsilon)\cap\Delta^c$ for some $\varepsilon>0$. 

\begin{proposition}${\mathcal B}_{\rm aff}$ is $1$-localizable.
\end{proposition}

\begin{proof}We must exhibit a $c>$ and an $R>0$ such that the conditions of Definition 13.1 hold. Taking a clue from Lemma 13.6, we wish to have $c>4/\pi$, so we take $c=2$ and $R=\sqrt{2}$. These are not the ``best'' constants; we only claim that they work.

Let $B$ be any element of $\mathcal B_{\rm aff}$. Then $B$ is open and connected, $B$ contains the origin, and $\delta(0,B^c)>1$, or equivalently, given any $\Delta$ through the origin, the two connected components of $B\cap\Delta^c$ that contain the origin in their boundaries always have areas greater than or equal to $b=\delta(0,B^c)$. 

\pagebreak[2]

\noindent Three open connected sets enter the proof: 
\begin{enumerate}[\indent]\setlength\itemsep{2pt}
\item $B'=\mathcal C_0[D_{c/\rho}(\mathcal C_0[B\cap D(0,\rho)])\cap D(0,\rho)]$.
\item $C'=\mathcal C_0^\partial[B'\cap\Delta^c]$. 
\item $C=\mathcal C_0^\partial[B\cap\Delta^c]$. 
\end{enumerate}
By our convention regarding the use of $\Delta^c$, the sets $C$ and $C'$ lie on the same side of $\Delta$. 

The plan is to show that $B'$ is in $\mathcal B_{\rm aff}$. Since $B'$ is open and connected, it remains to exhibit a $c'>1$ that does not depend on $\Delta$ such that ${\rm area}(C')\geq c'>1$.

There are two cases: $C\subset D(0,\rho)$ and $C\not\subset D(0,\rho)$. If $C\subset D(0,\rho)$, then, by the definition of $\mathcal B_{\rm aff}$, ${\rm area}(C)\geq b$. (This is the only place where the definition of $\mathcal B_{\rm aff}$ is used.) In this case, $C\subset C'$, and ${\rm area}(C')\geq b>1$. Thus, $c'=b$ works for this case, and we are done. (The proof that $C\subset C'$ is left as an exercise.)

If $C\not\subset D(0,\rho)$, then there is a point $\fxa\in C$ such that $|\fxa|>\rho$. There is a Jordan arc $\gamma\in C$ that connects the origin to $\fxa$. (In fact, we may assume that this Jordan arc is piecewise linear.) Let $t:[0,1]\to\gamma$ be a parameterization such that $\gamma(0)=0$ and $\gamma(1)=\fxa$. Then there is a smallest $t=t_0$ such that $\gamma(t_0)\in\partial D(0,\rho)$. Call this point $z$ and let $\Gamma$ denote the part of $\gamma$ defined by $0\leq t\leq t_0$. The arc $\Gamma$ lies in $C\cap D(0,\rho)\cap\Delta^c$, and in particular, $\Gamma\subset B\cap D(0,\rho)\cap\Delta^c$. It follows that $D_{c/\rho}(\Gamma)\cap D(0,\rho)\cap\Delta^c\subset C'$. 

By Lemma 13.6, ${\rm area}(D_{c/\rho}(\Gamma)\cap D(0,\rho)\cap\Delta^c)\geq(\pi/4)c$, so ${\rm area}(C')>\pi/2$, by the definition of $c$. Thus, by taking $c'=\min\{\pi/2,b\}$, it is always true that ${\rm area}(C')\geq c'>1$, and $c'$ does not depend on $\Delta$. It follows from Definition 13.1 that ${\mathcal B}_{\rm aff}$ is $1$-localizable.
\end{proof}


\begin{exercise}We skipped over two points in the proof that the reader should check. The first was when we stated that $C\subset C'$ (the case $C\subset D(0,\rho)$), and the second was when we claimed that $D_{c/\rho}(\Gamma)\cap D(0,\rho)\cap\Delta^c\subset C'$ (the case $C\not\subset D(0,\rho)$). (Hint: All of the sets in sight are open and connected, so they are arcwise connected.) \finex
\end{exercise}


The next two exercises show that there are other affine-invariant families of structuring elements that are 1-localizable.

\begin{exercise}Let $\mathcal B$ be an affine-invariant family of open convex sets, each of which contains the origin and has area less than one. The goal is to show that $\mathcal B$ is 1-localizable. Here is one way to do this. Suppose $\rho>R$, where $R>0$ is a constant to be determined, and that $B$ is an element of $\mathcal B$. If $B\subset D(0,\rho)$, take $B'=B$, and we are done. If not, let $\fxx$ be a vector such that $|\fxx|=\xi=\sup_{\fxy\in\overline{B}}|\fxy|$. Establish the coordinate system based on ${\bm i}=\fxx/|\fxx|$ and ${\bm j}= {\bm i}^\perp$. Consider the affine transformation defined by
$A=
\Big(\begin{smallmatrix}
\rho/\xi & 0\\
0 & \xi/\rho
\end{smallmatrix}
\Big)$, and show that if $\rho$ is greater than some constant, then $AB\subset D(\rho,0)$. Let $B'=AB$, which belongs to $\mathcal B$ by assumption. (Hint: Let $\eta$ be the longest perpendicular distance from the $x$-axis to $B^c$. Relate the product $\eta\xi$ to the area of $B$ and determine a value for $R$ that works.) Having found an $R$ such that $B'\subset D(0,\rho)$ for $\rho>R$, look for a $c>0$ such that $B'\subset D_{c/\rho}(B)$. \finex 
\end{exercise}

\begin{exercise}Let $B$ be a bounded, open, and connected set that contains the origin. Define the affine-invariant family $\mathcal B$ by $\mathcal B=\{AB\mid A\in SL(\RR^2)\}$. Use the methods of Exercise 3.2 to show that $\mathcal B$ is 1-localizable. \finex
\end{exercise}

\section{Comments and references}We were vague about the domains of the various functions that appear in section 13.2. In fact, the results are true in $\RR^N$ even though we spoke of ``disks'' rather than ``balls.'' The results in section 13.3 are, however, strictly limited to $\RR^2$.

The mathematical techniques for localizing a set of structuring elements developed in this chapter were first explained in \cite{Guichard:1996:PDE}, \cite{Guichard:1995:PDE}, and \cite{Guichard:1994:AAM}. The version presented here is much simpler. The doctoral dissertations by Fr\'ed\'eric Cao and Denis Pasquignon contain related techniques \cite{Cao:1998:AML,Cao:1998:PDE,Pasq1999}.
