Dans ce chapitre nous pr\'esentons le logiciel de segmentation et nous discutons les segmentations d'images obtenues.

\section{Pr\'esentation du logiciel.}

On a impl\'ement\'e un algorithme de croissance de r\'egions pyramidal en langage C (2000 lignes de code informatique) en utilisant des primitives du logiciel {\em Megawave} (auteur J. Froment \cite{Fro}). On initialise l'algorithme avec une ``segmentation initiale'' qui n'est autre chose qu'une grille sur l'image et dont les mailles sont les r\'egions initiales (on prendra souvent 1 r\'egion = 1 pixel, ce que l'on appelle la segmentation triviale). Notons $D$ le c\^{o}t\'e d'une maille triviale. On fait ensuite la segmentation multi\'echelle c'est \`a dire qu'on d\'etermine des segmentations 2-normales pour diff\'erentes valeurs de $\lambda$ \`a partir de la segmentation donn\'ee (pour cel\`a on se donne une suite croissante en $\lambda$).

A partir d'un fichier image $g$ on g\'en\`ere une segmentation qui est fournie par le logiciel sous forme d'une image (``bitmap'') des bords $K$ et d'une image repr\'esentant l'estimation constante par morceaux $u$. La surface d'une r\'egion est le nombre de pixels y contenus et la longueur d'un bord est le nombre de pixels qui constituent ce bord.

La construction des noms des fichiers r\'esultats utilise les conventions suivantes: \verb-B- indique que l'on a dessin\'e les bords des r\'egions (noir ou pointill\'e), \verb-u- repr\'esente la reconstruction de l'image et l'image originale est indiqu\'e par \verb-o-.\\
Ainsi \verb-SegCt1_B_2000_image.img- est la repr\'esentation des bords de la segmentation 2-normale de l'image \verb-image.img- \`a l'\'echelle \verb-2000-.

%-----Exemple-----
\begin{figure}[htb]
  \insertps{figures/Exemple.PS}{120mm}{20mm}
  \caption{De gauche \`a droite: (en haut) original $g$, reconstruction $u$, \mbox{ \hspace{3cm}} (en bas) bords $K$, original avec bords. }
  \label{fig_Exemple}
\end{figure}
%------------------

Pour illustrer l'utilisation du programme donnons un exemple. On se propose de segmenter l'image ``classique'' du bureau de l'INRIA, stock\'e sous le format {\em Megawave} \verb-img-, le param\`etre d'\'echelle final est 2304 et la suite des $\lambda$ est : 2, 4, \ldots, 1024, 1280, \ldots, 2304 . Il faut rentrer la ligne de commande suivante
\begin{center}
\verb+SegCt1 -rg bureau.img 10 256 2400+
\end{center}
Apr\`es cette ligne de commande le programe va g\'en\'erer la sortie suivante (`` $\vdots$ '' indique que l'on saute des lignes de l'affichage complet).
\begin{verbatim}

      SEGMENTATION OF AN IMAGE USING THE ENERGY CRITERION OF
                 M U M F O R D  &  S H A H .
              Format IMG. Version 2.0, july 1991.

 'Region-Growing' method starting on a grid made of 1x1 pixels cells .
                    2 region channels are used.


Estimation for memory occupation:

     Model           :       12.46 Mbytes.
     System(approx.) :        0.75 Mbytes.
              Total  :       13.22 Mbytes.


Initialisation
   ...of the regions,   initial number of regions    65536
   ...of the boundary,  initial number of frontiers  130560
   ...of tops,          initial number of tops       66045
 finished !

Initial state of the segmentation:
Elastic energie = 0 , boundary length= 130560.
------> Scale parameter= 2.
Total energy decrease 50796.
Elastic energie = 8435 , boundary length = 100944.
Number of regions: 36551. 
\end{verbatim}
$\vdots$
\begin{verbatim}
Total energy decrease 23.
Elastic energie = 1.908e+04 , boundary length = 70999.
Number of regions: 23318. 

Total energy decrease 0.
Elastic energie = 1.908e+04 , boundary length = 70999.
Number of regions: 23318. 

------> New scale parameter= 4.
Total energy decrease 12901.
Elastic energie = 4.03e+04 , boundary length = 62468.
Number of regions: 18205. 
\end{verbatim}
$\vdots$
\begin{verbatim}
------> New scale parameter= 2304.
Total energy decrease 37022.
Elastic energie = 1.046e+07 , boundary length = 9001.
Number of regions: 275. 

Total energy decrease 1703.
Elastic energie = 1.046e+07 , boundary length = 9000.
Number of regions: 274. 

Total energy decrease 0.
Elastic energie = 1.046e+07 , boundary length = 9000.
Number of regions: 274. 

 Writting the boundaries.

 Image saved in file SegCt1_B_2304_bureau.img.

 Writting the dashed boundaries.

 Image saved in file SegCt1_oB_2304_bureau.img.

 Writting the boundaries.

 Constructing u.

 Image saved in file SegCt1_u_2304_bureau.img.

 Construction of the global view.

 Image saved in file SegCt1_g_2304_bureau.img.
\end{verbatim}
\medskip

Les trois fichiers r\'esultat :\\
 \verb-SegCt1_B_2304_bureau.img- pour les bords, \\
\verb-SegCt1_oB_2304_bureau.img- pour l'original avec les bords, \\
\verb-SegCt1_u_2304_bureau.img- pour l'approximation constante par morceaux \\
et le fichier global, \verb-SegCt1_g_2304_bureau.img-, \\
ont \'et\'e cr\'ees en 80 secondes. L'image \verb-SegCt1_g_2304_bureau.img- est visible sur la figure~\ref{fig_Exemple}.

\section{Exp\'eriences avec le logiciel de segmentation.}

\subsection{Probl\`emes rencontr\'es.}

Les ``probl\`emes'' sont tout simplement des mauvaises segmentations, c'est-\`a-dire des segmentations propos\'ees par l'ordinateur mais que l'observateur humain n'accepte pas. Les causes sont diverses: on a des ``erreurs'' dues \`a la m\'ethode et il y a l'impl\'ementation sur ordinateur qui fausse le r\'esultat. L'ordre de pr\'esentation des divers probl\`emes rencontr\'es correspond plus ou moins \`a leur apparition lors du d\'eveloppement et raffinement du programme.
%-----Buro-----
%\begin{figure}[htb]
%  \insertps{figures/BurO.PS}{80mm}{-1mm}
%  \caption{Le bureau de l'INRIA.}
%  \label{fig_Buro}
%\end{figure}
%------------------
%-----BordD1-----
\begin{figure}[htb]
  \insertps{figures/BordD1.PS}{75mm}{8mm}
  \caption{A gauche l'original, \`a droite les bords, en pointill\'e, sur l'image (grille-$4\!\times\!4$).}
  \label{fig_BordD1}
\end{figure}
%------------------

\clearpage
\bigskip
\begin{flushleft}{\bf a) Bords doubles.}
\newline
\end{flushleft}

Les bords doubles apparaissent pour deux raisons diff\'erentes. La premi\`ere est la pr\'ecision de la grille initiale. En effet si l'on prend comme segmentation initiale une grille de mailles $4\times4$ pixels on ne pourra pas, sauf par hasard, obtenir les bords exacts (voir figure~\ref{fig_BordD1}). Les mailles contenues enti\`erement dans une r\'egion homog\`ene vont s'agglutiner pour remplir cette r\'egion, tandis que les mailles chevauchant deux r\'egions ou plus auront en g\'en\'eral des moyennes de gris diff\'erentes (resp. interm\'ediaires) de celles des zones homog\`enes et de ce fait ne se ``colleront'' \`a aucune o\`u, si un recollement se fait, d\'eplacent les ``vraies'' fronti\`eres. La fa\c{c}on \'evidente de contourner ce probl\`eme est d'utiliser la segmentation triviale au d\'epart. Ainsi on travaille \`a la pr\'ecision du pixel et c'est le mieux que l'on puisse faire sur une image digitalis\'ee. Mais cette solution n\'ecessite une m\'emoire centrale assez grande; pour une image de $256\times 256$ pixels on a avec une grille-$4\!\times\!4$ seulement 4096 r\'egions, par contre avec une grille-$1\!\times\!1$ on a 65536 r\'egions initiales, d'o\`u un accroissement important de la taille des donn\'ees (on passe de 1.5 Moctets \`a 13.2 Moctets).
%-----BordD2-----
\begin{figure}[htb]
  \insertps{figures/BordD2.PS}{70mm}{5mm}
  \caption{A gauche l'image, \`a droite les bords trouv\'es (d\'etail de l'image de bureau).}
  \label{fig_BordD2}
\end{figure}
%------------------

Le deuxi\`eme cas d'apparition de bords doubles se produit en utilisant une grille-$1\!\times\!1$. Comme l'on peut voir sur la figure~\ref{fig_BordD2} (extrait du bureau de l'INRIA) il y a les bords fins, larges d'un seul pixel et ceux trac\'es plus gras, larges de quelques pixels. L'origine des bords ``gras'' est la suivante : le passage d'une zone homog\`ene \`a une autre, voisine, se fait tr\`es souvent en passant par un pixel de niveau de gris interm\'ediaire. Suivant la valeur du param\`etre d'\'echelle il se forme alors une sorte de ``r\'egion-bord'' qui est tr\`es plate mais qui suit le bord sur une longueur importante.

Un ``bon'' r\'eglage du param\`etre d'\'echelle pourra faire dispara\^{\i}tre ces regions. Par ailleurs si, pour un $\lambda$ fix\'e, on arrive \`a obtenir une segmentation 2-normale proche d'un minimum, on \'elimine automatiquement les r\'egions ``plates'' (voir remarque~\ref{reg.rem_eltr}).

\clearpage
\bigskip
\begin{flushleft}{\bf b) Bords oscillants.}
\newline
\end{flushleft}

Si, d\`es le d\'epart, on utilise un param\`etre d'\'echelle $\lambda$ grand dans l'algorithme de croissance de r\'egions, on risque d'obtenir des agglutinations de zones qui ne sont pas homog\`enes, le terme de longueur multipli\'e par $\lambda$ l'emportant dans le crit\`ere. On risque d'obtenir des segmentations quelconques qui n'ont rien \`a voir avec une ``bonne'' segmentation de l'image (voir figure~\ref{fig_BordO}).

%-----BordO--------
\begin{figure}[htb]
  \insertps{figures/BordO.PS}{80mm}{-1mm}
  \caption{A gauche bords avec croissance du param\`etre et \`a droite sans croissance (m\^eme valeur pour $\lambda=2304$, grille $1\times1$).}
  \label{fig_BordO}
\end{figure}
%------------------ 

Ces effets peuvent \^etre \'evit\'es en ordonnant les couples de r\'egions par le gain d'\'energie qu'ils fournissent (cf.\cite{CohVSG}). Notre algorithme n'utilise pas directement cette technique. En effet, comme il est multi\'echelle, on commence tout naturellement par les petites \'echelles, donc des valeurs petites pour $\lambda$ et on cr\'ee ainsi un ordre partiel de ``merging'' proche de celui mentionn\'e plus haut.

On remarquera que l'energie finale $E$ d'une segmentation obtenue gr\^ace \`a l'algorithme de segmentation multi\'echelle est inf\'erieure \`a celle obtenue par un algorithme sans croissance de param\`etre ($E_{2304}=31,128\cdot 10^6$ pour la segmentation de gauche de la figure~\ref{fig_BordO} et $E_{2304}=31,771\cdot 10^6$ pour celle de droite).
\clearpage

\bigskip
\begin{flushleft}{\bf c) Influence de l'impl\'ementation sur ordinateur.}
\newline
\end{flushleft}

\label{sec_InfImpl}L'algorithme de segmentation est a priori d\'ependant de la repr\'esentation des donn\'ees dans la m\'emoire de l'ordinateur (voir aussi la remarque de Pavlidis sur l'utilisation des ``quad-trees'' \cite{Pav3}). Notre programme est \'ecrit de fa\c{c}on \`a donner la plus grand libert\'e au regroupement de r\'egions.

%-----BordHV-----
\begin{figure}[htb]
  \insertps{figures/BordHV.PS}{75mm}{3mm}
  \caption{Segmentations du bureau et de sa rotation de $90^\circ$ \`a $\lambda =2304$ fix\'e.}
  \label{fig_BordHV}
\end{figure}
%------------------

Sur la figure~\ref{fig_BordHV} on peut voir les segmentations trouv\'ees sans faire cro\^\i tre le param\`etre d'\'echelle et en faisant l'initialisation de deux fa\c{c}ons diff\'erentes: parcours ligne par ligne de la grille initiale et parcours colonne par colonne. Ceci revient en fait \`a segmenter l'image et \`a segmenter une rotation de $90^\circ$ de la m\^eme image.

%-----BordR-----
\begin{figure}[htb]
  \insertps{figures/BordR.PS}{70mm}{3mm}
  \caption{Segmentation multi\'echelle du bureau et de sa rotation de $90^\circ$, \`a l'\'echelle $\lambda =2304$.}
  \label{fig_BordR}
\end{figure}
%------------------

Avec l'approche multi\'echelle on a toujours une instabilit\'e par rapport aux rotations mais n\'eanmoins l'effet est moins important. On constate dans la figure~\ref{fig_BordR} ``de visu'' que les bords et r\'egions sont presque les m\^{e}mes. Il y a toutefois des bords qui apparaissent  \`a droite et pas \`a gauche : c'est d\^{u} \`a l'effet de seuillage de la fonctionnelle. Il peut arriver qu'un bord soit ``tout pr\^{e}t'' de dispara\^{\i}tre \`a une certaine valeur de $\lambda$. 
\clearpage

\subsection{Quelques exemples.}

On va pr\'esenter quelques applications du logiciel de segmentation \`a des images d'origines diverses. Dans un premier temps il s'agit de montrer le domaine de validit\'e du mod\`ele a priori (constant par morceaux) et de voir ce que l'on peut attendre d'un outil de segmentation bas\'e sur les seuls niveaux de gris et utilisant un seul param\`etre, l'\'echelle.

On va segmenter une image qui correspond parfaitement au mod\`ele constant par morceaux. Comme l'on peut voir sur la figure~\ref{fig_Synt2}, la reconstruction est parfaite dans les zones constantes et les bords coincident exactement avec les fronti\`eres des objets. La partie affine (en haut \`a droite dans l'image originale) par contre est reconstruite en utilisant une fonction en escalier.

Si par contre on applique l'algorithme \`a une image qui ne correspond pas aussi bien au mod\`ele que l'image du bureau ou l'image de synth\`ese pr\'ec\'edente, on aura des probl\`emes \`a d\'eterminer une partition ``correcte''. Ainsi un  visage de femme ne convient pas \`a une approximation constante par morceaux (voir \ref{fig_LA_WoB} , \ref{fig_LA_Wou}).
\medskip

Un exemple de l'aspect multi\'echelle est donn\'e dans la figure~\ref{fig_muscle_ms}. Les figures \ref{fig_Plans} et \ref{fig_cup} montrent des r\'esultats obtenus avec un mod\`ele affine p.m., $u(x,y)=ax+by+c$ (en collaboration avec Sixto Alvarez). 
\medskip

Le temps de calcul (sur SUN-Sparc station 1+) pour une image $256\times256$ avec $D=1$ ( maille initiale $1\times1$ pixel) est d'environ 80 secondes , pour le mod\`ele constant p.m.; pour le mod\`ele affine p.m. il faut 120-150 secondes (\`a chaque fois on g\'en\`ere des segmentations \`a plusieurs \'echelles).
\medskip

Pour tirer une conclusion sur l'efficacit\'e du programme de segmentation il reste encore des exp\'eriences \`a faire, notamment avec des mod\`eles affines p.m. , quadratiques/cubiques p.m. . Mais \'etant donn\'e que les temps d'ex\'ecution sont tr\`es raisonnables et que le programme peut s'utiliser avec un grand nombre de mod\`eles, on peut affirmer qu'on a un outil tr\`es souple qui permet de tester diverses hypoth\`eses ( voir aussi page \pageref{sec_tex_examples})

%-----Synt-----
\begin{figure}[htb]
  \insertps{figures/Synt2.PS}{140mm}{7mm}
  \caption{Echelle $\lambda=1536$, $D=1$ 44 r\'egions, 64 fronti\`eres et 50 sommets.}
  \label{fig_Synt2}
\end{figure}
%------------------
%-----LA_WoB-----
\begin{figure}[htb]
  \insertps{figures/LA_Wo.PS}{115mm}{20mm}
  \insertps{figures/LA_WoB.PS}{115mm}{20mm}
  \caption{Image originale et ensemble des bords : $\lambda=2048$ , $D=1$ pixel, longueur des bords : 16767.}
  \label{fig_LA_WoB}
\end{figure}
%------------------
%-----LA_Wou-----
\begin{figure}[htb]
  \insertps{figures/LA_Wo.PS}{115mm}{20mm}
  \insertps{figures/LA_Wou.PS}{115mm}{20mm}
  \caption{Image originale et estimation constante p.m. : $\lambda=2048$ , $D=1$ pixel et 88 r\'egions.}
  \label{fig_LA_Wou}
\end{figure}
%------------------
%-----aquitain-----
\begin{figure}[htb]
  \insertps{figures/aquitain.PS}{150mm}{2mm}
  \caption{Echelle $\lambda=2048$, $D=1$ ,  117 r\'egions,  344 fronti\`eres et  longueur des bords 8736. (Image originale : GRECO TdSI)}
  \label{fig_aquitain}
\end{figure}
%------------------
%-----lacornou-----
\begin{figure}[htb]
  \insertps{figures/lacornou.PS}{150mm}{1mm}
  \caption{Echelle $\lambda=1792$, $D=1$. Nombre de r\'egions 264, longueur des bords 9876. (Image originale : GRECO TdSI)}
  \label{fig_lacornou}
\end{figure}
%------------------
%-----satB-----
\begin{figure}[b]
  \insertps{figures/sat.PS}{115mm}{20mm}
  \insertps{figures/satB.PS}{115mm}{20mm}
  \caption{En haut image originale (image spot DRET, \'etang de Montady). En bas carte des bords trouv\'es : $\lambda$ =2304, $D=2$ , nombre de r\'egions : 73, longueur des bords : 13232, \'energie \'elastique : $4.5\cdot10^7$.}
  \label{fig_satB}
\end{figure}
%------------------
%-----satu-----
\begin{figure}[b]
  \insertps{figures/sat.PS}{115mm}{20mm}
  \insertps{figures/satu.PS}{115mm}{20mm}
  \caption{En haut image originale (image spot DRET, \'etang de Montady). En bas reconstruction de $u$ : $\lambda$ =2304, $D=2$ , nombre de r\'egions : 73, longueur des bords : 13232, \'energie \'elastique : $4.5\cdot10^7$.}
  \label{fig_satu}
\end{figure}
%------------------
%-----muscle-----
\begin{figure}[htb]
  \insertps{figures/muscle.PS}{150mm}{1mm}
  \caption{Commentaire associ\'e: "2-normal segmentation (Mumford and Shah model), grey-level. Parameter : 3328, increased with mode (3). Grid of 1*1 pixels. Nb. of regions: 153, frontiers: 358, tips: 289, Elastic energy 2.09e+07, length 6903. Original image, u, B and oB represented." (Image originale : GRECO TdSI)}
  \label{fig_muscle}
\end{figure}
%------------------
%-----muscle_ms-----
\begin{figure}[b]
  \insertps{figures/muscle_ms.PS}{210mm}{6mm}
  \caption{$D=1$ \`a toutes les \'echelles. Premi\`ere ligne : $\lambda$ =1024, nombre de r\'egions : 799, longueur des bords : 12579, \'energie \'elastique : $9.9\cdot10^6$. Deuxi\`eme ligne : $\lambda$ =2816, nombre de r\'egions : 223, longueur des bords : 8142, \'energie \'elastique : $1.7\cdot10^7$. Troisi\`eme ligne : $\lambda$ =4096, nombre de r\'egions : 105, longueur des bords : 6091, \'energie \'elastique : $2.3\cdot10^7$.}
  \label{fig_muscle_ms}
\end{figure}
%------------------
%-----cup-----
\begin{figure}[htb]
  \insertps{figures/Ct_cup.PS}{95mm}{30mm}
  \insertps{figures/Af_cup.PS}{95mm}{30mm}
  \caption{Segmentation d'une image de profondeur (``range-image'') avec une grille $1\times1$ et \`a l'\'echelle $\lambda=1920$. En haut : mod\`ele constant p.m., 113 r\'egions et l'\'energie est $5.36\cdot10^6$. En bas : mod\`ele affine p.m., 111 r\'egions et l'\'energie vaut $5.26\cdot10^6$.}
  \label{fig_cup}
\end{figure}
%------------------
%-----Plans-----
\begin{figure}[htb]
  \insertps{figures/Plans.PS}{70mm}{42mm}
  \insertps{figures/Plans_Ct.PS}{70mm}{7mm}
  \insertps{figures/Plans_Af.PS}{70mm}{7mm}
  \begin{picture}(0,0)
    \put(193,180){\small 0}
    \put(40,190){\small 1}
    \put(48,93){\small 2}
    \put(93,125){\small 3}
    \put(180,64){\small 4}
    \put(35,46){\small 5}
  \end{picture}
  \caption{En haut : image de plans affines, au milieu segmentation constante p.m. \`a l'\'echelle $\lambda=14592$ , en bas segmentation affine p.m. \`a la m\^{e}me \'echelle. Pour}
  \label{fig_Plans}
\end{figure}
%------------------
\clearpage

\noindent le mod\`ele constant p.m. on trouve 15 r\'egions et une \'energie de $4.57\cdot10^7$. Le mod\`ele affine p.m. donne 6 r\'egions et l'\'energie vaut $3.26\cdot10^7$. Comparons pour ce dernier mod\`ele les expressions de $u$ et $g$ :
\medskip

\begin{center}
\begin{tabular}{cl}
R\'egion 0  & $\left\{\begin{array}{l}
                      g(x,y) = 2\cdot x -2 \cdot y -288     \\
                      u(x,y) = 1,1\cdot x -1,8 \cdot y -51,5
                      \end{array}\right.$                           \\
                                                                    \\
R\'egion 1  & $\left\{\begin{array}{l}
                      g(x,y) = 2\cdot x + 2\cdot y + 10    \\
                      u(x,y) = 1,8\cdot x + 1,8\cdot y +4,4
                      \end{array}\right.$                           \\
                                                                    \\
R\'egion 2  & $\left\{\begin{array}{l}
                      g(x,y) = 255     \\
                      u(x,y) = 255
                      \end{array}\right.$                           \\
                                                                    \\
R\'egion 3  & $\left\{\begin{array}{l}
                      g(x,y) =  x +  y -110    \\
                      u(x,y) = 0,9\cdot x + 0,9\cdot y -80,8
                      \end{array}\right.$                           \\
                                                                    \\
R\'egion 4  & $\left\{\begin{array}{l}
                      g(x,y) = -1,5\cdot x -1,5 \cdot y +675 \\
                      u(x,y) = - x -  y +504
                      \end{array}\right.$                           \\
                                                                    \\
R\'egion 5  & $\left\{\begin{array}{l}
                      g(x,y) = -2\cdot x + 2\cdot y -128    \\
                      u(x,y) = -1,8\cdot x + 1,1\cdot y -51,5
                      \end{array}\right.$                           \\
                                                                    \\
\end{tabular}
\end{center}

\noindent {\bf Remarques : }Les r\'egions 3 et 4 ne correspondent pas au domaines des deux plans dans l'image originale. On peut constater que les coefficients d'inclinaison sont d\'etermin\'es de fa\c{c}on plus pr\'ecise que le terme constant, ceci est d\^{u} \`a l'initialisation dans le cas $D=1$. En effet dans la version actuelle du programme on initialise le terme constant avec la valeur de gris du pixel et les deux autres coefficients sont annul\'es, bien qu'ils sont en fait ind\'etermin\'es.
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%%EndProlog
%%BeginSetup
%%Feature: *Resolution 300
TeXDict begin @a4
%%EndSetup
%%Page: 29 1
bop 59 -68 a Fa(Chapitre)16 b(1)1555 b(29)213 1937 y @beginspecial
@setspecial
%%BeginDocument: lopez/Text2b+4.PS
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522 57 8 [522 0 0 -57 0 57] 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1.000000 1.000000 scale
522 57 8 [522 0 0 -57 0 57] 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image
0.000000 -0.087719 translate
1.000000 0.087719 scale
522 5 8 [522 0 0 -5 0 5] <0000ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff000000000000ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff00ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff00000000ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff000000000000ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff00ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff00000000ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff000000000000ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff00ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000> 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%%EndDocument
 @endspecial 59 1997 a Fb(\014g.4)155 b(Segmen)n(tation)17
b(de)h(textures)g(naturelles.)59 2081 y Fc(Auteurs:)161 b(Ko)q(ep\015er)16
b(-)h(Lop)q(ez)g(-)f(Morel.)59 2164 y(A)h(gauc)o(he)h(:)24
b(deux)17 b(textures)g(de)g(Bro)q(datz,)h(\022)-24 b(a)18 b(droite)g(:)24
b(trois)17 b(di\013)o(\023)-23 b(eren)o(tes)17 b(textures)g(de)g(Bro)q(datz)
59 2225 y(\(image)f(256)i(x)f(256\).)26 b(Dessous)18 b(:)24
b(estimation)15 b(des)j(b)q(ords)g(\(Segmen)o(tation)e(d'une)h(image)f
(\014ltr)o(\023)-23 b(ee)59 2285 y(en)21 b(18)g(canaux)h(di\013)o(\023)-23
b(eren)o(ts\).)34 b(Il)20 b(faut)h(noter)g(que)f(la)h(m)o(^)-23
b(eme)18 b(texture)i(est)g(r)o(\023)-23 b(ep)o(\023)g(et)o(\023)g(ee)20
b(deux)g(fois)h(\022)-24 b(a)59 2345 y(gauc)o(he.)21 b(Bien)15
b(que)h(qu'une)f(fron)o(ti)o(\022)-23 b(ere)15 b(soit)h(apparen)o(te,)g
(l'algorithme)d(reconnait)j(l'iden)o(tit)o(\023)-23 b(e)14
b("en)59 2405 y(mo)o(y)o(enne")g(des)j(textures)e(de)h(haut)h(en)f(bas)h(et)f
(segmen)o(te)e(correctemen)o(t.)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF
