

%
%  Ce fichier contient les macros utilisees dans le texte.
%
%
%


\hyphenation{two brac-kets}
\def\displayskip{\vskip6pt\relax}
\def\enddisplayskip{\vskip7pt\relax}

\newcommand{\gfigbox}[2]{\epsfxsize=#1
  \epsfbox{#2.ps}}
%\newcommand \D[1] {\left[ #1 \right]}
%\newcommand \deriva[2]{\frac{\partial #1}{\partial #2}}
%\newcommand \bigh{\frac{}{}\!\!}
\newcommand \Binom[2]{\binom{#1}{#2}}


\def\fInt{{\rm Int}}
\def\interior{{\rm interior}}
\def\AMSS{AMSS}
\def\conv{\ast}
\def\setT{{\tt T}}
\def\setF{{{I\!\!F}}}
%%\def\threshold{{\tt X}}
\def\sgn{sgn}
\def\trace{\hbox{Tr}}
\newtheorem{th1}{Theorem}[chapter]
\newtheorem{lem}[th1]{Lemma}
\newtheorem{cor}[th1]{Corollary}
\newtheorem{pro}[th1]{Proposition}
\newtheorem{fredDef}[th1]{Definition}
\newtheorem{Exa}[th1]{Example}
%\newtheorem{Remark}[th1]{Remark}
\newtheorem{thdef}[th1]{Theorem and Definition}
%\newtheorem{remark}[th1]{Remark}
\newtheorem{Defi}[th1]{Definition}
\newtheorem{rem}[th1]{Remark}%[chapter]%  remarque
\newtheorem{Exercice}{Exercise}[chapter]
\def\drem{\begin{rem}\rm}%  sans
\def\frem{\end{rem}}%           italique !

\def\gmoins{g^{(-1)}}
\def\notation{ }

\def\supp{\hbox{supp}}
\def\fb& {\begin{equation}}       % debut d'une equation
\def\fe& {\end{equation}}         % fin d'une equation
\def\endproof{\hfill $\Box$\newline}     % fin de preuve
\def\enddemo{\hfill $\Box$\newline}

\def\union{\cup}                  % l'union
\def\inter{\cap}                  % l'intersection
\def\nin{\not\in}                 % la negation de \in
\def\indic{1\!\!1}                % fonction indicatrice
\def\levelset{{\cal X}}           % la surface de niveau


\def\curv{\hbox{{\rm curv}}}
\def\div{\hbox{{\rm div}}}
\def\med{\hbox{{\rm med}}}
\def\fgra{|\nabla u|}
\def\G{G}
%\def\med{\mathop{med}}               % le median
\def\meas{\mathop{meas}}      % mesure de
\def\SetB{{I\!\!B}}               % l'ensemble grand B
\def\setB{\SetB}
\def\setbs{{I\!\!B_s}}
\def\plusoumoins{\pm}
\def\real{I\!\!R}
\def\entier{I\!\!N}
\def\natural{I\!\!N}
\def\relative{I\!\!\!Z}
\def\ut& {\frac{\partial u}{\partial t}}
\def\entier{I\!\!N}
\def\rational{Z\!\!Q}

\def\equi&{\Leftrightarrow}

\def\tens{\otimes}
\def\AA{{\hbox{A}}}
\def\fxx{{\hbox{\bf x}}}          % point x dans R2
\def\fyx{\hbox{\bf y}}          % point y dans R2
\def\fxy{\hbox{\bf y}}
\def\fxc{\hbox{\bf c}}
\def\fxz{\hbox{\bf z}}
\def\fxz{\hbox{\bf z}}
\def\fxq{\hbox{\bf q}}
\def\fxp{\hbox{\bf p}}
\def\fxn{\hbox{\bf n}}
\def\fxm{\hbox{\bf m}}
\def\fxv{\hbox{\bf v}}
\def\fxi{\hbox{\bf i}}
\def\fxj{\hbox{\bf j}}
\def\fxk{\hbox{\bf k}}
\def\fxw{\hbox{\bf w}}
\def\fxV{\hbox{\bf V}}
\def\fxb{\hbox{\bf b}}
\def\ffcap{\subset}

% des infsup habituels

\def\supinf&{\sup_{B \in \SetB} \inf_{\fxy\in \fxx + B}}
\def\infsup&{\inf_{B \in \SetB} \sup_{\fxy\in \fxx + B}}
\def\supinfh&{\sup_{B \in h\SetB} \inf_{\fxy\in \fxx + B}}
\def\infsuph&{\inf_{B \in h\SetB} \sup_{\fxy\in \fxx + B}}
\def\supif&{\sup_{B \in \SetB_s} \inf_{\fxy\in B}}
\def\supig&{\sup_{B \in \SetB_s +\fxx_0} \inf_{\fxy\in B}}
\def\infsg&{\inf_{B \in \SetB_s +\fxx_0} \sup_{\fxy\in B}}


% Autres definitions

\def\BB{{\cal B}}
\def\su&{u_+}
\def\sv&{v_+}
\def\iu&{u_-}
\def\iv&{v_-}
\def\fequi&{\sim}              % equivalent �
\def\gequi&{\fequi&}               % idem
\def\skipaline {\vspace{4 mm}}
\def\zprec{\prec\!\!}
\def\zsucc{\!\!\succ}
\def\vfp{\vspace{5mm}}
\def\vff{\vspace{3mm}}
\def\vfd{\vspace{8mm}}


\hyphenation{e-qua-tion dif-fer-en-tial in-tel-li-gen-ce l-li-gen-ce
mul-ti e-chel-le quen-ce con-se-quen-ce i-ma-ge in-va-rian-ce
In-va-rian-ce pre-ser-ving re-gu-la-ri-ty for-ma-li-sa-tion mil-li-se-con-de}

\def\skipaline {\vspace{4 mm}}
\def\Xbol{\hbox{\bf X}}
\def\Ybol{\hbox{\bf Y}}
\def\Tbol{\hbox{\bf T}}
\def\Gbol{\hbox{\bf G}}
\def\ybol{\hbox{\bf y}}
\def\zbol{\hbox{\bf z}}
\def\xbol{\hbox{\bf x}}
\def\nbol{\hbox{\bf n}}
\def\Nbol{\hbox{\bf N}}
\def\Cbol{\hbox{\bf C}}
\def\Abol{\hbox{\bf A}}
\def\Abil{\hbox{ A}}
\def\Gaussiancurv{\hbox{G}}
\def\tbol{\hbox{\bf t}}

%LINFINI STABILITE
%\begin{lem}\label{bound} %[L^\infty stanility]
%If $u_0$ is bounded on $\real^2$ then for all $h$ and all $n$ the
%functions $u_h(.,nh)$ are bounded and $|| u_h(.,nh) ||_\infty \leq ||
%u_0 ||_\infty$.
%\end{lem}
%Evident, by the comparaison principle, and by the fact that constant
%functions are invariant by $T_h$.


%FICHIER MACRO.TEX:
%�������������������������������������������������


\def\comp{\circ}

%
%  Ce fichier contient les macros utilisees dans le texte.
%
%
%

\def\R{I\!\!R}
\def\B{I\!\!B}

\def\threshold{\gamma}

%\sim
%\chi
%\newenvironement{nom}[#arg]{defbegin}{defend}
%\newcounter{Hyp}  %[autrecompteur]
%\setcounter{Hyp}{0}
%\addtocounter{nom}{valeur}


\def\fredstyleaxiom{\bf}
\def\CAUS{{\fredstyleaxiom Pyramidal architecture}}
\def\RECU{{\fredstyleaxiom Recursive}}
\def\REG2{{\fredstyleaxiom Regular}}
\def\MAX{{\fredstyleaxiom Maximum principle}}
\def\MAXP{{\fredstyleaxiom local maximum principle}}
\def\GLSI{{\fredstyleaxiom Grey level shift inv.}}
\def\MORP{{\fredstyleaxiom Morphological Inv.}}
\def\TRAN{{\fredstyleaxiom Transl. Inv.}}
\def\ISOM{{\fredstyleaxiom Euclidian inv.}}
\def\SCAL{{\fredstyleaxiom Scale Inv.}}
\def\AFFI{{\fredstyleaxiom Affine Inv.}}
\def\EUCL{{\fredstyleaxiom Euclidian Inv.}}
\def\ROTA{{\fredstyleaxiom Isom. invariance}}
\def\REGU{{\fredstyleaxiom Regular}}
\def\MORp{{\fredstyleaxiom Morph. Inv.2}}
\def\GALI{{\fredstyleaxiom Galilean Inv.}}
\def\TIME{{\fredstyleaxiom Time acceleration inv.]}}
\def\RSI{{\fredstyleaxiom Regular scale invariance]}}
\def\ffe&{{{\hbox{\bf e}}^\pm}}
\def\fff&{{{\hbox{\bf e}}^\perp}}
\def\deriv& {\frac{\hbox{ \rm D}v_1}{\hbox{ \rm D}\theta}}

\def\fW{{\cal {W}}}
\def\fV{{\cal {V}}}
\def\fde&{\Delta \theta}
\def\arginf{\hbox{\rm arginf}}
\def\zbibnote{\vspace{8mm} \noindent {\Large \bf References.} \par \vspace{3mm}}
