%% bare_conf.tex
%% V1.3
%% 2007/01/11
%% by Michael Shell
%% See:
%% http://www.michaelshell.org/
%% for current contact information.
%%
%% This is a skeleton file demonstrating the use of IEEEtran.cls
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%%*************************************************************************

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% *** not appear when using other class files.                            ***
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% Add the compsocconf option for Computer Society conferences.
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 \usepackage[latin1]{inputenc}
 \usepackage[cyr]{aeguill}
 \usepackage[francais]{babel}

% Some very useful LaTeX packages include:
% (uncomment the ones you want to load)


% *** MISC UTILITY PACKAGES ***
%
%\usepackage{ifpdf}
% Heiko Oberdiek's ifpdf.sty is very useful if you need conditional
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% usage:
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% *** CITATION PACKAGES ***
%
\usepackage{cite}
% cite.sty was written by Donald Arseneau
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% *** GRAPHICS RELATED PACKAGES ***
%
\usepackage{transparent}
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  % declare the path(s) where your graphic files are
  \graphicspath{{img/}}
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\else
  % or other class option (dvipsone, dvipdf, if not using dvips). graphicx
  % will default to the driver specified in the system graphics.cfg if no
  % driver is specified.
  % \usepackage[dvips]{graphicx}
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\fi
% graphicx was written by David Carlisle and Sebastian Rahtz. It is
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%
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%
% You can find documentation about the pdfTeX application at:
% http://www.tug.org/applications/pdftex





% *** MATH PACKAGES ***
%
\usepackage[cmex10]{amsmath}
% A popular package from the American Mathematical Society that provides
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%
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% *** SPECIALIZED LIST PACKAGES ***
%
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% Also of interest may be the (relatively newer and more customizable)
% algorithmicx.sty package by Szasz Janos:
% http://www.ctan.org/tex-archive/macros/latex/contrib/algorithmicx/




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%
\usepackage{array}
% Frank Mittelbach's and David Carlisle's array.sty patches and improves
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% generation code is lacking to the point of almost being broken with
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\usepackage{mdwmath}
\usepackage{mdwtab}
% Also highly recommended is Mark Wooding's extremely powerful MDW tools,
% especially mdwmath.sty and mdwtab.sty which are used to format equations
% and tables, respectively. The MDWtools set is already installed on most
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% IEEEtran contains the IEEEeqnarray family of commands that can be used to
% generate multiline equations as well as matrices, tables, etc., of high
% quality.


%\usepackage{eqparbox}
% Also of notable interest is Scott Pakin's eqparbox package for creating
% (automatically sized) equal width boxes - aka "natural width parboxes".
% Available at:
% http://www.ctan.org/tex-archive/macros/latex/contrib/eqparbox/





% *** SUBFIGURE PACKAGES ***
%\usepackage[tight,footnotesize]{subfigure}
% subfigure.sty was written by Steven Douglas Cochran. This package makes it
% easy to put subfigures in your figures. e.g., "Figure 1a and 1b". For IEEE
% work, it is a good idea to load it with the tight package option to reduce
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% subfigure.sty has been superceeded by subfig.sty.



%\usepackage[caption=false]{caption}
%\usepackage[font=footnotesize]{subfig}
% subfig.sty, also written by Steven Douglas Cochran, is the modern
% replacement for subfigure.sty. However, subfig.sty requires and
% automatically loads Axel Sommerfeldt's caption.sty which will override
% IEEEtran.cls handling of captions and this will result in nonIEEE style
% figure/table captions. To prevent this problem, be sure and preload
% caption.sty with its "caption=false" package option. This is will preserve
% IEEEtran.cls handing of captions. Version 1.3 (2005/06/28) and later 
% (recommended due to many improvements over 1.2) of subfig.sty supports
% the caption=false option directly:
\usepackage[caption=false,font=footnotesize]{subfig}
%
% The latest version and documentation can be obtained at:
% http://www.ctan.org/tex-archive/macros/latex/contrib/subfig/
% The latest version and documentation of caption.sty can be obtained at:
% http://www.ctan.org/tex-archive/macros/latex/contrib/caption/




% *** FLOAT PACKAGES ***
%
\usepackage{fixltx2e}
% fixltx2e, the successor to the earlier fix2col.sty, was written by
% Frank Mittelbach and David Carlisle. This package corrects a few problems
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%\usepackage{stfloats}
% stfloats.sty was written by Sigitas Tolusis. This package gives LaTeX2e
% the ability to do double column floats at the bottom of the page as well
% as the top. (e.g., "\begin{figure*}[!b]" is not normally possible in
% LaTeX2e). It also provides a command:
%\fnbelowfloat
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% not format its papers in such ways.



% correct bad hyphenation here
% \hyphenation{op-tical net-works semi-conduc-tor}


\begin{document}
%
% paper title
% can use linebreaks \\ within to get better formatting as desired
\title{Isolines}


% author names and affiliations
% use a multiple column layout for up to two different
% affiliations

\author{
}



% use for special paper notices
%\IEEEspecialpapernotice{(Invited Paper)}


% make the title area
\maketitle


\section{\label{isolines}Isolines} 
Soit $I$ l'image de reference non bruitee, $I'$ l'image bruitee par un bruit blanc gaussien additif de valeur moyenne nulle et d'ecart type $\sigma$. 

Soit $\widehat{I}$ l'image reconstruite. 
Le niveau de gris de chaque pixel de coordonnees $(i,j)$ est note $z(i,j)$.

Dans une fenetre rectangulaire $\omega$, soit $S^n$ la ligne de niveau partielle a laquelle appartient le pixel central de la fenetre.\\
$S^n$ est un ensemble de $n$ pixels dont les niveaux de gris possedent la meme valeur moyenne $\mu_{S^n}$ et dont la variance est $\sigma^2$.\\

Soit $\overline{S^n}$ definie par $\omega = S^n \cup \overline{S^n}$.\\
Les valeurs moyennes $\mu_{ij}$ des niveaux de gris des pixels de $\overline{S^n}$ sont inconnues et supposees differentes en tout point de $\overline{S^n$}. On note  $\left\{\mu_{ij}\right\}_{\overline{S^n}}$ l'ensemble de ces valeurs moyennes.

Soit $Z$ est l'ensemble des niveaux de gris des pixels de $\omega$.\\

La vraisemblance est donnee par :
\begin{eqnarray}
\displaystyle P \left[Z | S^n, \mu_{S^n}, \left\{\mu_{ij}\right\}_{\overline{S^n}}, \sigma \right]
\label{LL1}
\end{eqnarray}

\begin{eqnarray}
\displaystyle \prod_{(i,j)\in S^n}{P\left[z(i,j) | \mu_{S^n}, \sigma \right]} \times \displaystyle\prod_{(i,j)\in \overline{S^n}}{P\left[z(i,j) | \left\{\mu_{ij}\right\}_{\overline{S^n}}, \sigma \right]}
\label{LL2}
\end{eqnarray}


on recherche $\left\{\mu_{ij}\right\}_{\overline{S^n}}$ tel que 
\begin{eqnarray}
\displaystyle \widehat{\mu_{ij}} = z(i,j)
\end{eqnarray}
ET
\begin{eqnarray}
\displaystyle\prod_{(i,j)\in \overline{S^n}}{P\left[z(i,j) | \left\{\widehat{\mu_{ij}}\right\}_{\overline{S^n}}, \sigma \right]=1}
\end{eqnarray}

Il reste la vraisemblance generalisee 
\begin{eqnarray}
\displaystyle \prod_{(i,j)\in S^n}{P\left[z(i,j) | \mu_{S^n}, \sigma \right]}
\end{eqnarray}

% that's all folks
\end{document}