11 oct

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@@ -1,11 +1,12 @@
-L'étendue des techniques applicables aux images numériques est aujourd'hui si vaste qu'il serait illusoire de chercher à les décrire ici. Ce chapitre présente plus spécifiquement les algorithmes utilisés en présence d'images (fortement) bruitées. Le bruit rend potentiellement délicate l'extraction des informations utiles contenues dans les images pertubées ou en complique l'interpretation, qu'elle soit automatique ou confiée à la vision humaine. 
-L'intuition nous incite donc à chercher des méthodes efficaces de prétraitement pour réduire la puissance du bruit afin de permettre aux traitements de plus haut niveau comme la segmentation, d'opérer ensuite dans de meilleures conditions.           
+L'étendue des techniques applicables aux images numériques est aujourd'hui si vaste qu'il serait illusoire de chercher à les décrire ici. Ce chapitre présente plus spécifiquement les algorithmes utilisés en présence d'images (fortement) bruitées. Le bruit rend potentiellement délicate l'extraction des informations utiles contenues dans les images perturbées ou en complique l'interprétation, qu'elle soit automatique ou confiée à la vision humaine. 
+L'intuition nous incite donc à chercher des méthodes efficaces de pré-traitement pour réduire la puissance du bruit afin de permettre aux traitements de plus haut niveau comme la segmentation, d'opérer ensuite dans de meilleures conditions.           
 
 Toutefois, il faut également considérer que les opérations préalables de réduction de bruit apportent des modifications statistiques aux images et influent donc potentiellement sur les caractéristiques que l'on cherche à mettre en évidence grâce au traitement principal. En ce sens, il peut-être préférable de chercher à employer des algorithmes de haut niveau travaillant directement sur les images bruitées pour minimiser les effets des altérations apportées par les filtres débruiteurs et conserver toute l'information contenue dans les images perturbées.
-%TODO
-% dire aussi que le prétraitement, ça prend du temps. C'est évident mais c'est mieux en le disant
- Les images auxquelles nous nous intéressons sont généralement les images numériques allant des images naturelles telles que définies par Caselles \cite{Caselles99topographicmaps} aux images d'amplitude isues de l'imagerie radar à ouverture synthétique (ROS ou en anglais SAR) \cite{cutrona1990synthetic}, de l'imagerie médicale à ultrasons (echographie) ou encore biologique dans le cas de la microscopie électronique. 
-Ces dispositifs d'acquisition sont naturellement, et par essence, générateurs de bruits divers, inhérents aux thechnologies mises en \oe uvre au sein de ces systèmes et qui viennent dégrader l'image idéale de la scène que l'on cherche à représenter ou analyser. On sait aujourd'hui caractériser de manière assez précise ces bruits et la section \ref{sec_bruits} en détaille les  origines physiques ainsi que  les propriétés statistiques qui en découlent.
+
+Enfin, toute opération aussi basique soit elle, prend un certain temps qui réduit potentiellement celui disponible pour le traitement de haut niveau. Lorsque les images à analyser sont de grande taille,  procéder à un débruitage préalable peut s'avérer incompatible avec les contraintes de débit et les temps requis par le traitement de haut niveau.
+
+ Les images auxquelles nous nous intéressons sont généralement les images numériques allant des images naturelles telles que définies par Caselles \cite{Caselles99topographicmaps} aux images d'amplitude issues de l'imagerie radar à ouverture synthétique (ROS ou en anglais SAR) \cite{cutrona1990synthetic}, de l'imagerie médicale à ultrasons (échographie) ou encore biologique dans le cas de la microscopie électronique. 
+Ces dispositifs d'acquisition sont naturellement, et par essence, générateurs de bruits divers, inhérents aux technologies mises en \oe uvre au sein de ces systèmes et qui viennent dégrader l'image idéale de la scène que l'on cherche à représenter ou analyser. On sait aujourd'hui caractériser de manière assez précise ces bruits et la section \ref{sec_bruits} en détaille les  origines physiques ainsi que  les propriétés statistiques qui en découlent.
 On peut dores et déjà avancer que la connaissance de l'origine d'une image et donc des propriétés des bruits associés qui en corrompent les informations, est un atout permettant de concevoir des techniques de filtrage adaptées à chaque situation. Toutefois, la recherche d'un filtre universel, bien qu'encore illusoire, n'est pas abandonnée, tant les besoins sont nombreux, divers et souvent complexes.    
        
 \section{Modèle d'image bruitée}
@@ -64,7 +65,7 @@ Le bruit de grenaille est de type multiplicatif et suit une loi de Poisson. La P
 La très grande majorité des algorithmes de réduction de bruit fait l'hypothèse que la perturbation est de type gaussien, même si le développement des systèmes d'imagerie radar et médicale a favorisé l'étude des bruits multiplicatifs du type \textit{speckle} ou \textit{Poisson}.
 Un très grand nombre de travaux proposant des méthodes de réduction de ces bruits ont été menés, ainsi que beaucoup d'états de l'art et d'études comparatives de ces diverses techniques, que nous n'avons pas l'ambition d'égaler.
 
-Nous nous focaliserons sur les techniques en lien avec les travaux que nous avons menés et qui ont donné lieu à des implémentations efficaces  susceptibles de fournir des éléments opérationnels rapides pour le prétraitement des images. 
+Nous nous focaliserons sur les techniques en lien avec les travaux que nous avons menés et qui ont donné lieu à des implémentations efficaces  susceptibles de fournir des éléments opérationnels rapides pour le pré-traitement des images. 
 
 La figure \ref{fig-ny-noises} montre une image de synthèse issue de la base de test COIL \cite{coil}, supposée sans bruit et qui sera considérée comme référence, ainsi que deux versions bruitées, respectivement avec un bruit gaussien d'écart type 25 et un bruit impulsionnel affectant 25\% des pixels. 
 L'indice de qualité le plus employé pour mesurer la similarité entre deux images est le PSNR (pour Peak Signal to Noise Ratio). Il est exprimé en décibels (dB) et se calcule en appliquant la formule  
@@ -202,9 +203,17 @@ On connait peu de versions GPU du filtre médian, peut-être en raison des impl
 Sur architecture GT200 (GTX260), les performances maximales de ces deux versions sont obtenues pour un masque de 3$\times$3 pixels avec respectivement 175~MP/s pour libJacket et 60~MP/s pour PCMF. 
 Une précédente implémentation avait été réalisée, basée sur l'algorithme BVM décrit dans \cite{5402362}. Elle prouve son efficacité dans l'élimination des artefacts générés par les dispositifs d'imagerie médicale magnétique en 3D \cite{chen09}, mais ne permet pas d'exploiter véritablement le parallélisme des GPU en filtrage d'image en 2D.
 
-La figure \ref{fig-compare-jacket-pcmf}, tirée de \cite{5402362}, compare ces trois implémentations et montre que le débit permis par la libJacket décroit très vite avec la taille du masque pour passer à 30~MP/s dès la taille 5$\times$5, alors que le PCMF décroit linéairement jusqu'à la taille 11$\times$11 où il permet encore de traiter quelque 40~MP/s. Ceci s'explique simplement par le fait que libJacket utilise un tri simple pour la sélection de la valeur médiane alors que le PCMF exploite les propriétés des histogrammes cumulés et n'est ainsi que très peu dépendant de la taille du masque.
+\begin{figure}
+  \centering
+  \subfigure[Sur GPU GTX260. Courbe tirée de \cite{5402362}]{\label{fig-compare-jacket-pcmf1}\includegraphics[width=7cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter2/img/compar-median1.png}}\quad
+  \subfigure[Sur GPU C2075. Courbe tirée de \cite{sanchez2013highly}]{\label{fig-compare-jacket-pcmf2}\includegraphics[width=7cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter2/img/compar-median2.png}}
+\caption{Performances relatives des filtres médians implémentés sur GPU dans libJacket/ArrayFire, PCMF et BVM et exécutés sur deux modèle de générations différentes.}
+\label{fig-compare-jacket-pcmf}
+\end{figure}
+
+La figure \ref{fig-compare-jacket-pcmf1}, tirée de \cite{5402362}, compare ces trois implémentations et montre que le débit permis par la libJacket décroit très vite avec la taille du masque pour passer à 30~MP/s dès la taille 5$\times$5, alors que le PCMF décroit linéairement jusqu'à la taille 11$\times$11 où il permet encore de traiter quelque 40~MP/s. Ceci s'explique simplement par le fait que libJacket utilise un tri simple pour la sélection de la valeur médiane alors que le PCMF exploite les propriétés des histogrammes cumulés et n'est ainsi que très peu dépendant de la taille du masque.
  
-Plus récemment, Sanchez \textit{et al.} ont actualisé leurs mesures sur architecture Fermi (GPU C2075) en comparant leur PCMF à la version ré-écrite en C de libJacket, nommée ArrayFire. Les courbes sont celles de la figure \ref{fig-compare-arrayfire-pcmf}, où l'on constate que les variations selon la taille du masque demeurent comparables, avec toutefois des valeurs de débit augmentées, avec près de 185~MP/s pour ArrayFire et 82~MP/s pour PCMF. 
+Plus récemment, Sanchez \textit{et al.} ont actualisé dans \cite{sanchez2013highly} leurs mesures sur architecture Fermi (GPU C2075) en comparant leur PCMF à la version ré-écrite en C de libJacket, nommée ArrayFire. Les courbes sont celles de la figure \ref{fig-compare-jacket-pcmf2}, où l'on constate que les variations selon la taille du masque demeurent comparables, avec toutefois des valeurs de débit augmentées, avec près de 185~MP/s pour ArrayFire et 82~MP/s pour PCMF. 
 
 Parallèlement, on trouve aussi des implémentations de filtre médian dans des traitements plus complexes comme dans \cite{aldinucci2012parallel} où les auteurs décrivent la plus récente évolution de leur technique itérative de réduction de bruit impulsionnel, sans qu'il soit possible d'évaluer le débit du médian seul. 
 
@@ -410,7 +419,7 @@ On voit que la convergence est assez rapide mais que le contour ainsi détérmin
 \subfigure[L'état  du contour après la septième itération]{\includegraphics[width=3cm]{/home/zulu/Documents/these_gilles/THESE/codes/snake/cochon128_tradi_snake_it7.png}}
 \subfigure[L'état du contour après la dixième itération]{\includegraphics[width=3cm]{/home/zulu/Documents/these_gilles/THESE/codes/snake/cochon128_tradi_snake_it10.png}}
 \subfigure[L'état du contour après la centième itération. C'est le contour final.]{\includegraphics[width=3cm]{/home/zulu/Documents/these_gilles/THESE/codes/snake/cochon128_tradi_snake_result.png}}   
-\caption{Segmentation d'une image en niveaux de gris de 128 $\times$ 128 pixels par algorithme dit du \textit{snake}, dans sa version originale. Les paramètres d'élastictié, de raideur et d'attraction ont été fixés respectivement aux valeurs 5, 0.1 et 5. }
+\caption{Segmentation d'une image en niveaux de gris de 128 $\times$ 128 pixels par algorithme dit du \textit{snake}, dans sa version originale. Les paramètres d'élasticité, de raideur et d'attraction ont été fixés respectivement aux valeurs 5, 0.1 et 5. }
 \label{fig-snake-tradi-cochon}
 \end{figure} 
 
@@ -525,6 +534,14 @@ Les algorithmes de type \textit{snake}, très coûteux en temps de calcul, pouva
 Parmi les premières solutions décrites, \cite{snakegvf06} propose une implémentation réalisée en openGL, où les données de gradient sont compactées en texture RVBA de manière à s'affranchir du format 16 bits de la représentation : les deux premiers canaux R et V contiennent les valeursreprésentant respectivement le gradients selon $dx$ et $dy$ sous une forme codée par la valeurs des 2 autres canaux. 
 Par ailleurs, une approximation du système linéaire à résoudre est proposée afin de donner une structure bande symétrique à la matrice à inverser, ce qui améliore considérablement l'efficacité des accès aux données au travers du cache.
 
+\begin{figure}
+  \centering
+  \subfigure[Contour initial]{\label{fig-epaule-init}\includegraphics[height=4cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter2/img/snake-epaule-init.png}}\quad
+  \subfigure[Contour final]{\label{fig-epaule-fin}\includegraphics[height=4cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter2/img/snake-epaule-fin.png}}
+\caption{Segmentation d'une image d'épaule en 1024$^2$ pixels issue d'un examen IRM par l'implémentation du snake GVF de \cite{snakegvf06}. Le contour est représenté en rougeet le contour final est obtenu en 11~s. }
+\label{fig-snakegvf}
+\end{figure}
+
 Les performances annoncées montrent tout d'abord que l'approximation adoptée n'a qu'un impact extrêmement limité sur le résulat de la segmentation avec un écart radial maximal inférieur à 1.3 pixel par rapport au calcul exact effectué sur CPU. Enfin, la segmentation de l'image d'exemple en 1024$^2$ pixels s'effectue en un total de 11~s après l'initialisation manuelle reproduite à la figure \ref{fig-snakegvf}. Cela est annoncé comme presque 30 fois plus rapide que l'implémentation CPU de référence, mais demeure beaucoup trop lent pour un usage interactif.
 
 Une solution directe employant la transformée de fourier pour inverser le système à résoudre a été décrite récemment dans  \cite{zheng2012fast}et programmée en employant la bibliothèque openGL. Les exemples fournis montrent des objets segmentés dans des images d'environ 10000 pixels en une durée de l'ordre de la demi seconde.   
@@ -545,39 +562,18 @@ La figure \ref{fig-gPb} présente quelques résultats d'extraction de contours.
 \label{fig-gPb}
 \end{figure}
 
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+\section{Conclusion}
+La présentation que nous venons de faire des principales techniques de filtrage et de ségmentation ainsi que des implémentations sur GPU qui leur ont été consacrées nous ont permis de mettre une évidence en lumière : malgré leur orientation grand public et les langages de huat niveau permettant d'accéder rapidement à la programmation GPU, la parallélisation efficace d'un algorithme séquentiel destiné à s'exécuter sur ces processeurs n'est pas triviale. Le modèle et les contraintes de programmation leur sont spécifiques et obtenir un code rapide découle nécessairement d'un compromis qui peut parfois être complexe à affiner. 
 
+Ajoutons que les générations de GPU qui se succèdent conservent certes des caractéristiques communes mais diffèrent suffisemment quant-à la distribution des ressources, rendant toute généralité vaine et faisant qu'un code optimisé pour un modèle donné peut devenir moins rapide avec un modèle plus récent. Prenons l'exemple du nombre maximal de registres utilisables par thread ; il est de 128 sur GPU C1060 contre seulement de 63 pour un C2070. Un code faisant un usage optimisé des registres sur C1060 pourra s'exécuter plus lentement sur C2070. C'est un cas de figure sur lequel nous reviendrons plus en détail dans le chapitre consacré au filtre médian.
 
+Cet état de fait rend les résultats publiés par les chercheurs souvent délicats à intérpréter et plus encore à reproduire lorsque l'on souhaite comparer les performances de nos propres codes avec les références du moment, sauf à disposer d'un panel de cartes GPU représentant toutes les évolutions de l'architecture et ce pour au moins les deux grands fabricants de GPUs que sont ATI et Nvidia.   
 
+Pour aider les développeurs à allouer les ressources de manière optimale, ou tout du moins estimer le dégré d'optimisation de leur code à l'aune de la vitesse d'exécution, Nvidia fournit une feuille de calcul appelée \textit{occupancy calculator} dans laquelle ont peut entrer les paramètres d'exécution d'un \textit{kernel} parallèle : nmobre de registres utilisés par chaque thread, quantité de mémoire partagée, modèle de GPU, dimensions de la grille. Le tableur retourne alors l'indice de charge (l'occupancy) qui traduit le rapport, à chaque instant, entre le nombre de warps actifs et le nombre maximal de warps par processeur (SM = Streaming Multiprocessor). L'occupancy se traduit donc par un indice compris entre 0 et 100\% et la recherche de performance semble devoir être la recherche de l'occupancy maximale.
 
+Toutefois, comme l'a clairement demontré Volkov dans \cite{volkov2010better}, ce paradigme peut aisément être remis en cause et Volkov parvient effectivement à améliorer les peformances d'un certain nombre d'exemples génériques dans des conditions de faible valeur d'occupancy. 
+Enfin, nous avons pu constater deux grands modèles d'accès aux données : les algorithmes de filtrage usent quasiment tous de la mémoire partagée comme tampon d'accès aux données de l'image en mémoire globale (ou texture) alors que les algorithmes de segmentation performants s'en affranchissent. La raison en est clairement des motifs d'accès très irréguliers et non contigus pour ces derniers, rendant la gestion efficace de la mémoire partagée délicate et potentiellement si coûteuse qu'elle en devienne sans intérêt.
+Les chapitres suivants présentant nos contributions reviendront sur ces aspects en proposant des solutions pour accroître la performance des algorithmes parallélisés.
 
 
 

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@@ -0,0 +1,437 @@
+\section{Présentation de l'algorithme}
+La principale difficulté soulevée par l'emploi d'algorithmes de type \textit{snake} orientés contour est le choix de la fonction d'énergie externe et la détermination de la nature des images auxquelles elle convient. 
+Dans l'approche orientée régions, les deux régions que sont l'extérieur et l'intérieur du contour (cas mono cible) sont prises en compte dans l'estimation de la forme du contour ;  cela permet d'extraire des formes dans des images où les contours de la cible sont mal définis, en raison d'un fort niveau de bruit par exemple.
+Les algorithmes découlant de cette approche n'ont fait l'objet, à notre connaissance, d'aucune parallèlisation sur GPU, malgré le grand intérêt qu'elles revêtent dans l'interprétation d'images fortement bruitées ( RADAR, médicales,\dots ) et le besoin d'en réduire les temps d'exécution suffisamment  pour permettre l'interactivité. 
+Nous proposons dans la suite de ce chapitre de détailler tout d'abord l'algorithme séquentiel que nous avons pris comme référence, puis d'en présenter la version parallèle pour GPU que nous en avons conçu.
+L'algorithme a été décrit et proposé initialement en 1999 par Chesnaud \textit{et al.} dans \cite{ChesnaudRB99}. L'implémentation que les auteurs ont développé a continué d'être améliorée jusqu'à aujourd'hui et est employée comme brique élémentaire dans des algorithmes plus complexes. La version qui sert de référence ici est une implémentation séquentielle optimisée qui met aussi à profit les capacités de parallélisme des CPU actuels en employant le jeu d'instruction SSE2 des microprocesseurs. La description que nous en faisons dans les lignes qui suivent est très largement inspirée de \cite{ChesnaudRB99} à la différence que nous n'implémentons pas les critères de régularisation du contour ni de minimisation de la longueur de description pour nous focaliser sur la déformation du contour et sa convergence. 
+
+\subsection{Formulation}
+À l'intérieur de l'image observée $\bar{v}$, soient $\bar{t}$ le vecteur composé par les niveaux de gris des $N_t$ pixels de la région cible $\Omega_t$ et $\bar{b}$ celui des $N_b$ pixels du fond $\Omega_b$. Les vecteurs $\bar{t}$ et $\bar{b}$ sont supposés non corrélés et sont caractérisés par leurs densités de probabilité (PDF) respectives $p^{\Theta_t}$ et $p^{\Theta_b}$ ; $\Theta_t$ et $\Theta_b$ étant les vecteurs des paramètres de leurs PDF. Dans le cas gaussien que nous supposerons ici, $\Theta = (\mu, \sigma)$ où $\mu$ est la moyenne et $\sigma^2$ est la variance.
+On note $\Gamma$ le contour de la région cible ($\Gamma \in \Omega_t$), que l'on suppose continu en connexité à 8 voisins. 
+
+Le but de la segmentation est alors de déterminer la géométrie de $\Gamma$ qui maximise un critère de vraisemblance généralisée (GL).
+La vraisemblance sur l'ensemble de l'image, \textit{ie.} la région $\Omega$ est donnée par
+
+\begin{equation}
+P\left(\bar{v} | \Omega_t, \Omega_b, \Theta_t, \Theta_b\right) = P\left(\bar{v} | \Omega_t, \Theta_t\right)P\left(\bar{v}|\Omega_b, \Theta_b\right)   
+\end{equation}
+soit en développant 
+\begin{equation}
+P\left(\bar{v} | \Omega_t, \Omega_b, \Theta_t, \Theta_b\right) = \prod_{x_k\in\Omega_t}p^{\Theta_t}\left(v_k, \Theta_t\right)\prod_{x_k\in\Omega_b}p^{\Theta_b}\left(v_k, \Theta_b\right)  
+\label{eq-lhprod1}
+\end{equation}
+
+Dans le cas gaussien, la PDF étant de la forme 
+\begin{equation}
+p(\alpha) = \frac{1}{\sigma\sqrt{2\pi}}\mathrm{e}^{-\frac{\left(\alpha - \mu\right)^2}{2\sigma^2}}  
+\label{eq-pdfgauss}
+\end{equation}
+
+La substitution de \eqref{eq-pdfgauss} dans \eqref{eq-lhprod1}, suivie du logarithme, permet d'obtenir l'expression de la \textit{log-vraisemblance}
+\begin{equation}
+  \label{eq-gl1}
+  -N_tln\left(\sqrt{2\pi}\right)-N_tln\left(\sigma_t\right)-\frac{1}{2\sigma_t^2}\sum_{x_k\in\Omega_t}\left(v_k-\mu_t\right)^2
+-N_bln\left(\sqrt{2\pi}\right)-N_bln\left(\sigma_b\right)-\frac{1}{2\sigma_b^2}\sum_{x_k\in\Omega_b}\left(v_k-\mu_b\right)^2
+\end{equation}
+dans laquelle les vecteurs $\Theta_t$ et $\Theta_b$ sont estimés suivant la méthode du  maximum de vraisemblance, qui donne l'expression générique suivante pour l'estimée de $\Theta_t$, notée $\widehat{\Theta_t}$ et transposable à l'identique pour $\Theta_b$ 
+\begin{equation}
+  \label{eq-teta}
+  \widehat{\Theta_t} \left(
+\begin{array}{l}
+\widehat{\mu_t} = \frac{1}{N_t} \displaystyle\sum_{x_k\in \Omega_t} v_k \\
+\widehat{\sigma^2_t} = \frac{1}{N_t} \displaystyle\sum_{x_k\in \Omega_t} \left(v_k - \widehat{\mu_t}\right)^2 \\
+  \end{array}
+\right.
+\end{equation}
+
+En intégrant \eqref{eq-teta} dans \eqref{eq-gl1}, il reste, à une constante près, le critère de vraisemblance généralisée suivant, noté GL, que l'on cherche à  optimiser en déterminant la géométrie de $\Gamma$ qui en maximise la valeur et épousera alors au mieux la forme du contour de la cible.
+\begin{equation}
+  \label{eq-gl}
+  GL=\frac{1}{2}\left(N_tln\left(\widehat{\sigma_t^2}\right)+N_bln\left(\widehat{\sigma_b^2}\right)\right)
+\end{equation}
+
+\subsection{Optimisation des calculs}\label{snake-formulation}
+La maximisation de GL est effectuée en employant une technique itérative où sa valeur doit être calculée à chaque déformation du contour $\Gamma$.
+Si l'on se reporte à l'équation \eqref{eq-teta}, on voit que l'obtention de la valeur de GL nécessite, à chaque évaluation d'une géométrie donnée de $\Gamma$, le calcul des sommes  
+\begin{align}
+\label{eq-sommes1}
+  S_v(\Omega_t) &= \sum_{x_k\in \Omega_t} v_k & S_{v^2}(\Omega_t) &= \displaystyle\sum_{x_k\in \Omega_t} v_k^2 \nonumber\\
+  S_v(\Omega_b) &= \sum_{x_k\in \Omega_b} v_k & S_{v^2}(\Omega_b) &= \displaystyle\sum_{x_k\in \Omega_b} v_k^2 \nonumber\\
+\end{align}
+
+Considèrons la région cible $\Omega_t$, les pixels de coordonées $(i,j)$ qui la composent, et généralisons l'écriture des sommes de \eqref{eq-sommes1} en 
+\begin{equation}
+  \label{eq-sommes-gene}
+  S_f(\Omega_t) = \sum_{i=i_{min}}^{i=i_{max}}\sum_{j=j_{min}(i)}^{j=j_{max}(i)}g\left(v(i,j)\right)
+\end{equation}
+où $f$ représente la fonction de valeurs de niveaux de gris à sommer.
+
+En posant 
+\begin{equation}
+  \label{eq-cumuls1}
+  T_g(y,\tau) = \sum_{j=0}^{\tau}g\left(v(y,j)\right)
+\end{equation}
+L'équation \eqref{eq-sommes-gene} devient 
+\begin{equation}
+  \label{eq-somme-cumuls2}
+  S_f(\Omega_t) = \sum_{i=i_{min}}^{i=i_{max}}\left[T_g(i,j_{max}(i))-T_g(i,j_{min}(i)-1)\right]
+\end{equation}
+qui représente  une sommation sur le contour $\Gamma$ que l'on peut écrire 
+\begin{equation}
+  \label{eq-somme-contour}
+  S_f(\Omega_t) = \sum_{(i,j)\in \Gamma}C(i,j)\gamma(i,j)
+\end{equation}
+où $C(i,j)$ est un coefficient lié à la direction du contour au point $(i,j)$ et $\gamma(i,j)$ prend sa valeur selon les règles suivantes
+
+\begin{equation}
+  \label{eq-coefC}
+  \gamma(i,j)=
+  \begin{cases}
+    T(i,j)  & \text{si $C(i,j)= 1 $}\\
+    T(i,j-1)& \text{si $C(i,j)= -1$}\\
+    0       & \text{si $C(i,j)= 0$}\\
+  \end{cases}
+\end{equation}
+
+La valeur de $C(i,j)$ est déterminée pour chaque pixel d'indice $l$ du contour en considérant les pixels d'indices $l-1$ et $l+1$ qui définissent les deux vecteurs $f_{in}$ et $f_{out}$ et leur code selon le codage de Freeman, comme l'illustre la figure \ref{fig-freeman}. La table \ref{tab-freeman} donne les valeurs de $C(i,j)$ selon les valeurs des codes de Freeman des vecteurs $f_{in}$ et $f_{out}$.
+\begin{figure}[htb]
+  \centering
+  \includegraphics[height=3cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/codage-freeman.png}
+  \caption{À gauche : détermination des vecteurs $f_{in}$ et $f_{out}$. À droite : code de Freeman d'un vecteur en fonction de sa direction, l'origine étant supposée au pixel central, en noir. }
+  \label{fig-freeman}
+\end{figure}
+
+\begin{table}[htb]
+  \centering
+\begin{tabular}[htb]{ccccccccc}
+      \toprule
+      &\multicolumn{8}{c}{$f_{out}$}\\
+      \cmidrule(r){2-9}
+      $f_{in}$& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
+      \midrule
+      0     &0&0&0&0&0&-1&-1&-1 \\
+      1     &1&1&1&1&1&0&0&0\\
+      2     &1&1&1&1&1&0&0&0\\
+      3     &1&1&1&1&1&0&0&0\\
+      4     &0&0&0&0&0&-1&-1&-1\\
+      5     &0&0&0&0&0&-1&-1&-1\\
+      6     &0&0&0&0&0&-1&-1&-1\\
+      7     &0&0&0&0&0&-1&-1&-1\\
+      \bottomrule
+\end{tabular}
+   \caption{Valeur du coefficient $C(i,j)$ en fonction des valeurs des codes de Freeman des vecteurs $f_{in}$ et $f_{out}$.}
+      \label{tab-freeman}
+\end{table}
+
+L'intérêt de cette transformation est majeur :
+\begin{itemize}
+\item La sommation en deux dimensions sur la région $\Omega_t$ est ainsi réduite à une sommation à une dimension sur le contour $\Gamma$.
+\item Les valeurs $T_g(i,j)$ peuvent être calculées préalablement à la phase de segmentation proprement dite. Pour le cas gaussien qui nous intéresse, cela revient à pré-calculer les trois images \textit{cumulées} $S_1$, $S_x$ et $S_{x^2}$ définies par
+  \begin{alignat}{4}
+    \label{eq-img-cumul}
+    S_1(i,j) &= \sum_{x=0}^jx & \quad \text{,}\quad S_x(i,j) &= \sum_{x=0}^jv(i,x) & \quad \text{et}&\quad & S_{x^2}(i,j) &= \sum_{x=0}^jv(i,x)^2 
+  \end{alignat}
+\item Les valeurs du coefficient $C(i,j)$ se calculent très facilement durant la génération du contour $\Gamma$.
+\end{itemize}
+
+
+Par ailleurs, le choix d'un contour polygonal permet également d'améliorer l'efficacité de l'algorithme car lors de la phase de segmentation, le déplacement d'un sommet du polygone n'influe que sur les pixels des deux segments qui s'y rapportent, réduisant ainsi la quantité de calculs à effectuer à chaque nouvelle déformation du contour.  
+
+\begin{upminfo}
+  L'approche décrite dans ce chapitre n'est valide que si les segments formant le polygone du contour ne se croisent pas. Il est donc nécessaire, lors de la convergence de la segmentation, d'empêcher les croisements de segments. Une solution simple a été proposée dans \cite{ChesnaudRB99} et nous l'avons parallélisée dans le cadre des travaux présentés ici. 
+\end{upminfo}
+
+\subsection{Implémentation séquentielle}\label{snake-cpu-impl}
+Un des inconvénients des algorithmes de type \textit{snake} est l'influence du contour initial sur la convergence de la segmentation. Pour pallier simplement ce défaut, une technique progressive est adoptée, en initialisant le contour avec peu de sommets (4) puis en augmentant le nombre au fur et à mesure de la convergence. L'algorithme \ref{algo-snake-cpu1} décrit macroscopiquement la solution mise \oe uvre tandis que l'algorithme \ref{algo-snake-cpu2} en présente les détails.
+
+\begin{algorithm}
+\label{algo-snake-cpu}
+\caption{Principe mis en \oe uvre pour la convergence du snake polygonal}
+  Calculer les images cumulées\;
+  Initialiser le contour avec 4 sommets\;
+  \Repeter(\tcc*[f]{niveau contour}){aucun sommet ne peut être ajouté}{
+      \Repeter(\tcc*[f]{niveau sommet}){aucun sommet ne peut être déplacé}{
+        Déplacer chaque sommet autour de sa position actuelle.\;
+       Déplacer le sommet vers la position induisant le meilleur GL\;
+     }
+      Ajouter un sommet au milieu de chaque \textit{grand} segment\;
+  }
+\end{algorithm}
+
+\begin{algorithm}[h]
+\caption{Détail de l'implémentation du snake polygonal} 
+\label{cpualgo}
+   Lire l'image $\bar{v}$\;
+   Calculer les images cumulées $S_1$, $S_x$ $S_{x^2}$ \nllabel{li-img-cumul}\tcc*[r]{en parallèle via SSE2} 
+   $n \leftarrow 0$ \tcc*[r]{indice de boucle niveau contour}
+   $N_n \leftarrow 4$ \tcc*[r]{nombre de n\oe uds}
+   $\Gamma \leftarrow \{\Gamma_0,\Gamma_1,\Gamma_2,\Gamma_3\} $\;
+   $d \leftarrow d_{max}$ \tcc*[r]{pas de déplacement des n\oe uds}
+   $l_{min} = 32$ \tcc*[r]{longueur mini des segments sécables}
+   $\Gamma_i \leftarrow \Gamma_0$ \tcc*[r]{sommet courant}
+   $GL_{ref} \leftarrow GL(\Gamma, N_n, \bar{v}, S_y, S_{y^2})$ \tcc*[r]{voir à partir de ligne 18 pour le détail}
+  \Repeter(\tcc*[f]{niveau contour}){$N_{add}=0$}{
+    $N_{add}\leftarrow 0$\;
+    \Repeter(\tcc*[f]{niveau n\oe ud}){$N_{move}=0$}{
+      $N_{move}\leftarrow 0$\;
+      \Pour{$i=0$ à $i=N_n-1$}{
+       Calculer les positions $\{\Gamma_i^0, \dots, \Gamma_i^7\}$ \tcc*[r]{les 8 voisins de $\Gamma_i$ }
+       \Pour{$w=0$ à $w=7$}{
+          Soustraire à $GL_{ref}$ la contribution des segments $\Gamma_{i-1}\Gamma_i$ et $\Gamma_{i}\Gamma_{i+1}$\;
+         Discrétiser les segments $\Gamma_{i-1}\Gamma_i^w$ et $\Gamma_{i}^w\Gamma_{i+1}$\nllabel{li-bresen}\;
+          Lire dans $S_1$, $S_x$ et $S_{x^2}$ les contributions des pixels de $\Gamma_{i-1}\Gamma_i^w$ et $\Gamma_{i}^w\Gamma_{i+1}$\nllabel{li-contrib-seg-deb}\;
+          Calculer les directions et lire les codes de Freeman \;
+          Calculer $GL_w$ incluant les contributions de $\Gamma_{i-1}\Gamma_i^w$ et $\Gamma_{i}^w\Gamma_{i+1}$ \nllabel{li-contrib-seg-fin}\;
+         \lSi{$GL_w > GL_{ref}$}{
+           $GL_{ref} \leftarrow GL_w$\;
+           $\Gamma_i \leftarrow \Gamma_i^w$\;
+            $N_{move} \leftarrow N_{move}+1$\;
+         }
+       }
+      }
+     $l \leftarrow l+1$\;
+   }
+   \PourCh{segment $\Gamma_i\Gamma_{i+1}$}{
+     \Si{$\|\Gamma_i\Gamma_{i+1}\| > l_{min}$}{
+       Ajouter un n\oe ud au milieu de $\Gamma_i$ et $\Gamma_{i+1}$\;
+       $N_{add} \leftarrow N_{add}+1$\;
+     }
+   }
+   $N_n \leftarrow N_n + N_{add}$\;
+   \lSi{$d > 1$}{ $d \leftarrow d/2$ } \lSinon{ $d \leftarrow 1$ }\;
+   $GL_{ref} \leftarrow GL(\Gamma, N_n, \bar{v}, S_y, S_{y^2})$ \;
+  }
+\end{algorithm}
+
+Les différentes sommations nécessaires au calcul de la valeur du critère $GL$ sont effectuées en parallèle à l'aide du jeu d'instructions SSE2. La taille des registres utilisables est de 128 bits et permet ainsi de traiter des images de 4096$\times$4096 pixels dont les niveaux de gris sont codés sur 16 bits. Cela ne laisse toutefois que 12 bits pour le codage des segments du contour et limite ainsi leur longueur à 4096 pixels. L'organisation d'un registre SSE 128 bits est donc la suivante :
+\begin{itemize}
+\item 24 bits pour les sommes à opérandes dans $S_1$
+\item 24+16 = 40 bits pour les sommes à opérandes dans $S_x$
+\item 24+32 = 60 bits pour les sommes à opérandes dans $S_x^2$
+\end{itemize}
+Soit un total de 124 bits, qui peuvent donc être représentés par un registre SSE2.
+
+\subsection{Performances}
+Les images de 1024$^2$ pixels de la figure \ref{fig-snakecpu-cochon512} montrent l'évolution du contour lors de la segmentation d'une image photographique prise en faible éclairement et bruitée artificiellement par un bruit gaussien d'écart type 25. Les paramètres de la séquence sont fixés empiriquement aux valeurs $d_{max}=16, l_{min}=8$.
+Les temps d'exécution indiqués sont mesurés sur Intel Xeon E5530-2.4GHz with 12Go RAM et sont les valeurs moyennes obtenues pour 10 exécutions.
+\begin{figure}
+  \centering
+  \subfigure[Initialisation : 4 n\oe uds]{\includegraphics[height=3.5cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/im000.png}}
+  \subfigure[Itération 1 : 8 n\oe uds 3~ms]{\includegraphics[height=3.5cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/im001.png}}
+  \subfigure[Itération 2 : 16 n\oe uds 1~ms]{\includegraphics[height=3.5cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/im002.png}}
+  \subfigure[Itération 3, 32 n\oe uds 1~ms]{\includegraphics[height=3.5cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/im003.png}}\\
+  \subfigure[Itération 7 : 223 n\oe uds 3~ms]{\includegraphics[height=3.5cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/im007.png}}
+  \subfigure[Itération 10 : 244 n\oe uds 3~ms]{\includegraphics[height=3.5cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/im010.png}}
+  \subfigure[Itération 13 : 256 n\oe uds 3~ms]{\includegraphics[height=3.5cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/im013.png}}
+  \subfigure[Itération 14 : 256 n\oe uds 3~ms]{\includegraphics[height=3.5cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/im014.png}}
+  \caption{Évolution du contour lors de la segmentation d'une image de 512$^2$ pixels. La convergence est obtenue à l'itération 14 après 44~ms pour un total de  256 n\oe uds.}
+ \label{fig-snakecpu-cochon512}
+\end{figure}
+
+La dépendance vis à vis du contour initial qui est un des principaux soucis liés au snake est ici fortement relativisée. La figure \ref{fig-snakecpu-compinit} montre le contour final segmentant l'image de test de la figure \ref{fig-snakecpu-cochon512} à partir d'un état initial très éloigné du précédent et \textit{a priori} très défavorable compte tenu du fait qu'il est loin de la cible et sans intersection avec elle. Toutefois, le contour final obtenu est très proche de celui obtenu à partir d'un état initial englobant la cible, malgré un n\oe ud qui s'est ``accroché'' au bord de l'image. La convergence est également plus longue à obtenir dans ce cas avec Un total de 17 itérations en 87~ms et 273 n\oe uds. 
+
+\begin{figure}
+  \centering
+  \subfigure[Initialisation 2 ]{\includegraphics[height=4.5cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/snakecpu-init2.png}}
+  \subfigure[Contour final 2 : 273 n\oe uds 87~ms]{\includegraphics[height=4.5cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/snakecpu-compinit2.png}}
+  \subfigure[Contour final 1 : 256 n\oe uds 44~ms]{\includegraphics[height=4.5cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/im014.png}}
+  \caption{Influence du contour initial sur la segmentation. Le contour final 1 est celui de la figure \ref{fig-snakecpu-cochon512}.}
+  \label{fig-snakecpu-compinit}
+\end{figure}
+
+La dimension de l'image à traiter a également un effet sur le résultat et naturellement sur le temps de calcul. Si l'on conserve les mêmes paramètres d'optimisation que pour la segmentation de l'image 512$^2$ pixels et un contour initial dont les cotés sont à une distance des bords équivalente à 10\% des cotés de l'image, le résultat sur une image identique de 4000$^2$ pixels est  obtenu en 1.3~s avec 1246 n\oe uds ; il est reproduit  à la figure \ref{fig-snakecpu-cochon4ka}. Le nombre de pixels appartenant à la région cible est tel que l'amplitude des déplacements autorisés pour chaque n\oe ud ($d$) peut se révéler trop faible vis à vis du seuil d'acceptation des mouvements. On observe que les zones à gradient élevé ne posent pas de problème et sont détourées de la même manière, alors que dans le bas de l'image où figure une zone de gradient faible (ombre), la cible se trouve maintenant quelque peu surévaluée en surface là ou elle était plutôt sous évaluée dans l'image en 512$^2$ pixels. 
+On parvient à un résultat très proche beaucoup plus rapidement en adaptant les paramètres à la taille de l'image, comme le montre par exemple la segmentation de la figure \ref{fig-snakecpu-cochon4kb}, effectuée avec $d_{max}=128$ et $l_{min}=32$ et qui converge vers un contour de 447 n\oe uds en moins de 0.7~s.
+Au delà des 16 millions de pixels (4000$^2$ pixels), l'implémentation séquentielle est toujours possible mais doit se priver des instructions SSE. Nous avons, avec leur accord, adapté le code des auteurs en ce sens et réalisé les mesures pour des tailles allant jusqu'à 150~MP. La table \ref{tab-snakecpu-speed-size} en synthétise les résultats en distinguant chaque fois le temps pris par les pré-calculs et celui nécessaire à la convergence de la segmentation. 
+
+\begin{figure}
+  \centering
+  \subfigure[$d_{max}=16$ et $l_{min}=8$, 1246 n\oe uds en 1.3~s]{\label{fig-snakecpu-cochon4ka}\includegraphics[height=5cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/snakecpu-cochon4k.png}}
+  \subfigure[$d_{max}=128$ et $l_{min}=32$, 447 n\oe uds en 0.7~s]{\label{fig-snakecpu-cochon4kb}\includegraphics[height=5cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/snakecpu-cochon4k-128-32.png}}
+  \caption{Segmentation de l'image de test en 4000$^2$ pixels.}
+  \label{fig-snakecpu-cochon4k}
+\end{figure}
+
+
+\begin{table}[h]
+  \centering
+  \begin{tabular}{rrrr}
+      \toprule
+      &\multicolumn{3}{c}{Taille de l'image}\\
+      &\multicolumn{3}{c}{(millions de pixels)}\\
+      \cmidrule(r){2-4}
+      & 15 & 100 & 150 \\
+      \midrule
+      Pré-calculs &0,13&0,91&1,4\\
+      Segmentation&0,46&3,17&4,3\\
+      {\bf Total} &0,51&4,08&5,7\\
+      \bottomrule
+\end{tabular}
+   \caption{Performances (en secondes) de la segmentation par snake polygonal sur CPU en fonction de la taille de l'image à traiter. Le temps sont obtenus avec la même image de test dilatée et bruitée et un contour initial carré dont la distance aux bords est proportionnelle à la taille de l'image. Seule l'image en 15~MP a pu être traitée par une implémentation utilisant SSE2.}
+      \label{tab-snakecpu-speed-size}
+\end{table}
+
+
+Enfin, il faut aussi considérer les tailles relatives de la cible et de l'image. Ainsi, si on fait l'hypothèse d'une cible de petite taille ``noyée'' dans une image de grandes dimensions, les résultats de la segmentation seront impactés en raison, cette fois, d'une moindre adaptation à la cible lors des toutes premières itérations, les plus grossières, où le nombre de n\oe uds et réduit et le pas de déplacement potentiellement grand vis à vis de la cible. Ce cas de figure est illustré par la segmentation reproduite à la figure \ref{fig-snakecpu-cochon4kc3} et qui met en évidence une qualité moindre par la confusion des zones les plus sombres de la cible avec le fond.
+
+\begin{figure}[h]
+  \centering
+  \includegraphics[height=5cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/cochon4kc-128-8.png}
+  \caption{Segmentation de l'image de test en 4000$^2$ pixels avec une cible de petite taille. Le contour initial est celui utilisé à la figure \ref{fig-snakecpu-cochon4k}.}
+  \label{fig-snakecpu-cochon4kc3}
+\end{figure}
+ 
+
+
+\section{Implémentation parallèle GPU du snake polygonal}
+L'analyse de l'exécution du programme séquentiel révèle la prépondérance des blocs fonctionnels suivants, dans l'ordre d'importance, qui occupent à eux seuls plus de 80\% du temps total d'exécution :
+\begin{itemize}
+ \item Le calcul de la contribution des segments (lignes \ref{li-contrib-seg-deb} à \ref{li-contrib-seg-fin} dans l'algorithme \ref{cpualgo}) 
+  \item La génération des trois images cumulées, avant le début des itérations (ligne \ref{li-img-cumul}).
+  \item La discrétisation des segments définis par les coordonnées de leurs extrémités (ligne \ref{li-bresen}).
+\end{itemize}
+Cette proportion est globalement conservée lorsque la taille de l'image à traiter varie, comme le montre le graphique de la figure \ref{fig-snakecpu-chronos1} 
+
+\begin{figure}
+  \centering
+  \includegraphics[width=0.5\linewidth, height=0.25\linewidth]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/cpu_profil.png}
+\label{fig-snakecpu-chronos1}
+  \caption{Évolution du coût relatif des trois fonctions les plus consommatrices en temps de calcul en fonction de la taille de l'image à traiter.}
+\end{figure} 
+
+Si l'effort de parallélisation porte essentiellement sur ces fonctions coûteuses, l'ensemble du traitement est réalisé sur le GPU afin de réduire autant que possible les transferts entre le GPU et le système hôte qui, selon le volume concerné, sont susceptibles de grever considérablement la performance globale. L'hôte ne conserve que l'initiative du transfert initial et le contrôle de la boucle principale, ne nécessitant l'échange que d'un seul octet à chaque itération (le nombre de nouveau n\oe uds $N_{add}$).
+
+Les traitements étant totalement indépendants, nos traitons séparément la parallélisation des pré-calculs et celle de la segmentation.
+
+\subsection{pré-calculs des images cumulées}   
+Pour réduire la quantité de mémoire requise, nous avons choisi de ne pas générer l'image $S_1$ mais plutôt d'en calculer les valeurs à la volée. L'expression en est simple et le temps pris par les opération élémentaires qu'elle met en jeu est largement compensé par le gain obtenu en économisant les accès mémoire qui auraient été nécessaires, ce qui n'est pas le cas des deux autres images $S_x$ et $S_x^2$ dont le calcul est quant à lui réalisé en appliquant la méthode des \textit{prefixsums} décrite dans \cite{BlellochTR90}.
+
+Les sommations se font au niveau de chaque ligne de l'image, que l'on décompose en $n$ blocs de $bs$ pixels où $bs$ correspond aussi au nombre de threads exécutés par chaque bloc de la grille de calcul. La valeur $bs$ étant obligatoirement une puissance de 2 supérieure à 32, le bloc de pixels d'indice $n-1$ doit éventuellment être complété par des valeurs nulles. Chaque bloc de thread réalise son traitement indépendemment des autres, mais l'ensemble des sommes de bloc étant requise pour le calcul des sommes globales, une synchronisation est nécessaire à deux endroits du calcul. Nous avons choisi d'assurer ces synchronisations en découpant le traitement en trois \textit{kernels} distincts, rendant par la même occasion le code plus concis :
+\begin{itemize}
+\item \texttt{compute\_block\_prefixes()} est le \textit{kernel} effectuant, en mémoire partagée, les prefixsums inclusif de chaque bloc, puis mémorisant les sommes des blocs, \textit{i.e} le dernier élément du prefixsum associé, dans deux vecteurs $V_x$ et $V_x^2$ en mémoire globale. L'ensemble des prefixsums est également mémorisé en mémoire globale. La largeur de l'image n'étant pas nécessairement une puissance de 2, il est nécessaire de faire du remplissage avec des valeurs nulles dans le dernier bloc (indice $n-1$). 
+\item \texttt{scan\_blocksums()} est le \textit{kernel} effectuant les prefixsum exclusifs des vecteurs $V_x$ et $V_x^2$. Les résultat demeurent respectivement dans $V_x$ et $V_x^2$.
+\item \texttt{add\_sums2prefixes()} est le \textit{kernel} effectuant les additions de chaque élément d'indice $i$ des vecteurs $V_x$ (respectivement $V_x^2$ avec tous les éléments du prefixsum du bloc de même indice $i$ associé. 
+\end{itemize}
+
+Les diagrammes de la figure \ref{fig-calcul-cumuls} donnent le détail des opérations effectuées par ces trois \textit{kernels} pour l'image cumulée $S_x$. La seconde image cumulée $S_x^2$ est obtenues exactement de la même manière en sommant non plus les valeurs $v_k$ mais $v^2_k$.
+
+\begin{figure}
+  \centering
+  \subfigure[Détail des opérations effectuées par le \textit{kernel} \texttt{compute\_block\_prefixes()}. La valeur $bs$ correspond au nombre de pixels de chaque bloc, qui est aussi le nombre de threads exécuté par chaque bloc de la grille de calcul.]{\resizebox{0.9\linewidth}{!}{ \input{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/GPUcumuls.pdf_t}}}\vspace{1cm}
+\subfigure[Détail des opérations effectuées par le \textit{kernel} \texttt{scan\_blocksums()}.]{\resizebox{0.9\linewidth}{!}{ \input{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/GPUscansomblocs.pdf_t}}}\vspace{1cm}
+\subfigure[Détail des opérations effectuées par le \textit{kernel} \texttt{add\_sums2prefixes()}.]{\resizebox{0.9\linewidth}{!}{ \input{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/GPUaddsoms2cumuls.pdf_t}}}
+\label{fig-calcul-cumuls}
+  \caption{Calcul des images cumulées $S_x$ et $S_x^2$ en trois étapes successives. a) cumul partiel bloc par bloc et mémorisation de la somme de chaque bloc. b) cumul sur le  vecteur des sommes partielles. c) ajout des sommes partielles à chaque élément des blocs cumulés.}
+\end{figure}
+
+Les gains de performance de cette implémentation GPU comparée à l'implémentation CPU/SSE2 sont ceux de la table \ref{tab-speedup-cumuls}, soit un GPU environ 7 fois plus rapide pour des images de taille 15 à 150 millions de pixels. L'influence de la taille d'image sur le gain est faible, mais on peut toutefois noter que plus l'image est grande plus le gain est important.
+On obtient des accélérations supérieures en rendant le calcul moins générique et en développant des versions spécifiques des trois \textit{kernels}, dédiées par exemple au traitement des images dont largeur est multiple de 256 pixels.
+\begin{table}
+  \centering
+  \begin{tabular}{rrrr}
+      \toprule
+      &\multicolumn{3}{c}{Taille de l'image}\\
+      &\multicolumn{3}{c}{(millions de pixels)}\\
+      \cmidrule(r){2-4}
+      & 15 & 100 & 150 \\
+      \midrule
+      temps CPU (s)      &0,13&0,91&1,4\\
+      temps GPU (s)      &0,02&0,13&0,2\\
+      {\bf Accélération} &{\bf 6,5} &{\bf 6,9} &{\bf 7,0}\\
+      \bottomrule
+\end{tabular}
+   \caption{Accélération constatée, pour le calcul des images cumulées, de l'implémentation GPU par rapport à l'implémentation CPU de référence.}
+      \label{tab-speedup-cumuls}
+\end{table}
+
+
+\subsection{Calcul des contribution des segments}
+
+Le déplacement d'un des $N_n$ n\oe uds du contour $\Gamma$ vers l'une des 8 positions voisines permises, impose d'évaluer les contributions des 8 paires de segments associées, soit $16N_n$ segments pour la totalité du contour, que nous évaluons en parallèle au sein du \textit{kernel} \texttt{GPU\_compute\_segments\_contribs()}. Pour ce faire, chaque segment doit tout d'abord être discrêtisé en une suite de pixels puis, en conservant la règle \textit{1 pixel par thread} la contribution de chaque pixel est déterminée avant de toutes les additionner pour obtenir la contribution du segment. 
+Les pixels représentant les n\oe uds font l'objet d'un traitement spécifique impliquant les codes de Freeman, pour ne pas fausser les contributions globales (voir paragraphe \ref{snake-cpu-impl}).  
+
+Pour optimiser l'exécution de ce kernel et réduire l'effet de la disparité des longueurs des segments, nous créons un motif régulier en mémoire, basé sur la longueur $npix_{max}$ du plus grand segment et moyennant le recours au remplissage par des valeurs neutres là où c'est nécessaire pour les autres segments.
+
+Si $bs_{max}$ est la taille de bloc maximale admissible par le GPU, la taille $bs$ des blocs de threads/pixels employée pour le calcul des contributions des segments est alors déterminée de la façon suivante :
+\[
+bs=
+\begin{cases}
+2^p     & \text{ avec $2^{p-1} < npix_{max} \leq 2^p$ si $npix_{max}\in[33; bs_{max}]$}\\
+32      & \text{ si $npix_{max} \leq 32$}\\
+bs_{max} & \text{ si $npix_{max} > bs_{max}$}
+\end{cases} 
+\]
+ 
+Dans notre implémentation, les calculs sont faits en mémoire partagée et la quantité nécessaire limite la taille de bloc admissible. Nous limitons celle-ci à 256 sur C1060 et 512 sur C2050. Toutefois, les tests ont montré que sur ces deux versions de l'architecture, La taille maximale conduisant aux meilleures performances est de 256 threads par bloc. 
+
+Le \textit{kernel} \texttt{GPU\_compute\_segments\_contribs()} calcule alors en paralèle pour tous les segments les coordonnées de tous les pixels qui les composent. Nous mettons pour cela en \oe uvre l'algorithme de Bresenham, \textit{i.e} la méthode du segment semi-ouvert, en distinguant les cas où
+\begin{itemize}
+\item la valeur absolue de la pente $k$ du segment à discrêtiser est supérieure à $1$; on applique alors la méthode au segment \textit{horizontal} semi-ouvert et on obtient un pixel par ligne.
+\item la valeur absolue de la pente $k$ du segment à discrêtiser est inférieure ou égale à  $1$; on applique alors la méthode au segment \textit{vertical} semi-ouvert et on obtient un pixel par colonne.
+\end{itemize}
+Cette distinction nous permet de conserver la règle \textit{1 pixel par thread} importante pour la régularité des motifs d'accès en mémoire et aussi pour \textit{charger} au maximum le GPU.
+
+La figure \ref{fig-structure-segment} représente la structure décrite ci-dessus pour la représentation en mémoire des segments à évaluer. La première ligne montre le détail du premier segment, avec la correspondance \textit{1 pixel par thread} et le découpage en un nombre de blocs suffisant pour permettre de décrire le plus long des segments. 
+
+La seconde ligne présente l'ordre dans lequel sont concaténés les 16 groupes de blocs-segment associés au déplacement d'un n\oe ud particulier. 
+
+\begin{figure}
+  \centering
+  \resizebox{0.8\linewidth}{8cm}{ \input{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/contribs_segments.pdf_t}}
+\label{fig-structure-segment}
+  \caption{Structuration des données en mémoire du GPU pour l'évaluation en parallèle de l'ensemble des évolutions possibles du contour.}
+\end{figure}
+
+Aux deux dernières lignes est décrite la concaténation des ensembles de 16 blocs-segment, avec la particularité de séparer la description des n\oe uds d'indices pairs et ceux d'indices impairs. Cela permet de moins s'écarter de l'heuristique d'optimisation en vigueur dans la version séquentielle : les statistiques globales comme la valeur de critère $GL$ y sont recalculées après chaque déplacement, alors que dans le cas d'un traitement parallèle, les statistiques de référence ne sont calculées qu'après le déplacement simultané des $N_n$ n\oe uds; les résultats et la convergence en sont potentiellement affectés, comme l'illustrent les situations présentées à la figure \ref{fig-cycle-contribs-segments}, car les segments composant véritablement le contour après déplacement des n\oe uds n'ont pas nécessairement été évalués (segments en rouge dans la figure \ref{fig-cycle-contribs-segments}). 
+  
+\begin{figure}
+  \centering
+  \subfigure[Contour de référence.]{\includegraphics[height=3cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/cycle-contribs-segments-cpu1.png}}\quad
+\subfigure[Déplacement du n\oe ud $N_1$. Le critère est amélioré.]{\includegraphics[height=3cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/cycle-contribs-segments-cpu2.png}}\quad
+\subfigure[Déplacement du n\oe ud $N_2$. Le critère est amélioré.]{\includegraphics[height=3cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/cycle-contribs-segments-cpu3.png}}\\
+\subfigure[Déplacement en parallèle de tous les n\oe uds. Les segments du contour n'ont pas été évalués. On ne peut pas dire, a priori si le critère est amélioré.]{\includegraphics[height=3cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/cycle-contribs-segments-gpu1.png}}\quad
+\subfigure[Déplacement en parallèle des n\oe uds impairs. Le critère est amélioré.]{\includegraphics[height=3cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/cycle-contribs-segments-gpu2.png}}\quad
+\subfigure[Déplacement en parallèle des n\oe uds pairs. Un seul segment n'a pas été évalué.]{\includegraphics[height=3cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/cycle-contribs-segments-gpu3.png}}
+\label{fig-cycle-contribs-segments}
+  \caption{Comparaison des cycles de déplacement des n\oe uds. Ligne du haut : version séquentielle. Ligne du bas : version parallèle. Les segments en rouge sont des segments du contour non évalués, alors que ceux en pointillés sont les paires ayant reçu les meilleures évaluations parmi les 8 déplacements possibles des n\oe uds correspondant.}
+\end{figure}
+
+La représentation en mémoire des segments conduit à avoir un certain nombre non prévisible de threads inactifs dans la grille, sans que cela soit préjudiciable aux performances car cela n'engendre pas de branches divergentes. 
+
+Les calculs liés à l'évaluation des contributions des pixels sont réalisés en mémoire partagée. Seule une très petite quantité de données doit être stockée en mémoire globale. Il s'agit, pour chaque segment (pas pixel) :
+\begin{itemize}
+\item des coordonnées de son milieu. Cela permet l'ajout efficace quand c'est nécessaire.
+\item les coordonnées des deux derniers pixels de chaque extrémité. Ils sont nécessaires pour calculer la dérivée aux extrémités et ainsi déterminer le code de Freeman des n\oe uds.      
+\end{itemize}
+
+Pour obtenir les contributions des segments, \textit{i.e} les sommes des contributions des leurs pixels, une première phase de réduction partielle est effectuée au niveau de chaque bloc.
+
+Une synchronisation est alors nécessaire avant d'effectuer les sommes de l'ensemble des contributions partielles qui fournissent les contributions globales des segments. Le contour modifié est alors construit comme la suite des meilleures positions déterminées pour chaque n\oe ud.
+Un calcul des statistiques globales du nouveau contour ainsi que du critère $GL$ est alors nécessaire et applique à nouveau les techniques décrites dans ce paragraphe.  
+Enfin l'ajout des nouveaux n\oe uds se fait simplement pour les segments suffisamment grands, en utilisant les coordonnées des pixels milieux mémorisées lors de la discrêtisation des segments. 
+
+
+\subsubsection{Cas particulier des segments dont la pente $k$ vérifie $|k|\leq 1$}
+Comme nous venons de le voir, les segments dont la pente $k$ vérifie $|k|\leq 1$ sont discrêtisés à raison de \textit{1 pixel par colonne} et comportent donc le plus souvent plusieurs pixels sur une ligne donnée, comme le montrent les schémas de la figure \ref{fig-segment-k<1}. 
+D'après la formulation générale du snake faite au paragraphe \ref{snake-formulation}, le coefficient $C(i,j)$ est à appliquer en chaque point du contour. La technique de discrêtisation employée conduit à des coefficients $C(i,j)$ constants sur l'ensemble des pixels des segments dont la pente $k$ vérifie  $|k|> 1$, mais ce n'est pas le cas pour ceux dont la pente $k$ est inférieure ou égale à $1$. Les quatre cas, un par quadrant, qui peuvent se présenter sont représentés à la figure \ref{fig-segment-k<1}. On y constate en se reportant à la table \ref{tab-freeman} que tout pixel dont les voisins immédiats sont sur la même ligne à un coefficient $C(i,j)=0$ ($F_{in}=f_{out}=0$). Les deux pixels des extrémités, n'ayant quant à eux qu'un voisin, ont un coefficient qui dépend du quadrant :
+\begin{itemize}
+\item dans les quandrant  1 et 2
+  \begin{itemize}
+  \item le premier pixel d'une ligne a un coefficient $C(i,j)=1$.
+    \item le dernier pixel d'une ligne a un coefficient $C(i,j)=0$
+  \end{itemize}
+ \item dans les quandrant  3 et 4
+   \begin{itemize}
+   \item le dernier pixel d'une ligne à un coefficient $C(i,j)=-1$.
+     \item le premier pixel d'une ligne a un coefficient $C(i,j)=0$.
+   \end{itemize}
+\end{itemize}
+
+Les accès en mémoire aux contributions de ces pixels dans les images cumulées sont évités et une contribution nulle leur est automatiquement attribuée dès l'étape de discrêtisation au sein du kernel \texttt{GPU\_compute\_segments\_contribs()}.
+
+
+
+
+
+
+
+

diff --git a/THESE/Chapters/chapter3/chapter3.tex~ b/THESE/Chapters/chapter3/chapter3.tex~
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+\section{Présentation de l'algorithme}
+La principale difficulté soulevée par l'emploi d'algorithmes de type \textit{snake} orientés contour est le choix de la fonction d'énergie externe et la détermination de la nature des images auxquelles elle convient. 
+Dans l'approche orientée régions, les deux régions que sont l'extérieur et l'intérieur du contour (cas mono cible) sont prises en compte dans l'estimation de la forme du contour ;  cela permet d'extraire des formes dans des images où les contours de la cible sont mal définis, en raison d'un fort niveau de bruit par exemple.
+Les algorithmes découlant de cette approche n'ont fait l'objet, à notre connaissance, d'aucune parallèlisation sur GPU, malgré le grand intérêt qu'elles revêtent dans l'interprétation d'images fortement bruitées ( RADAR, médicales,\dots ) et le besoin d'en réduire les temps d'exécution suffisamment  pour permettre l'interactivité. 
+Nous proposons dans la suite de ce chapitre de détailler tout d'abord l'algorithme séquentiel que nous avons pris comme référence, puis d'en présenter la version parallèle pour GPU que nous en avons conçu.
+L'algorithme a été décrit et proposé initialement en 1999 par Chesnaud \textit{et al.} dans \cite{ChesnaudRB99}. L'implémentation que les auteurs ont développé a continué d'être améliorée jusqu'à aujourd'hui et est employée comme brique élémentaire dans des algorithmes plus complexes. La version qui sert de référence ici est une implémentation séquentielle optimisée qui met aussi à profit les capacités de parallélisme des CPU actuels en employant le jeu d'instruction SSE2 des microprocesseurs. La description que nous en faisons dans les lignes qui suivent est très largement inspirée de \cite{ChesnaudRB99} à la différence que nous n'implémentons pas les critères de régularisation du contour ni de minimisation de la longueur de description pour nous focaliser sur la déformation du contour et sa convergence. 
+
+\subsection{Formulation}
+À l'intérieur de l'image observée $\bar{v}$, soient $\bar{t}$ le vecteur composé par les niveaux de gris des $N_t$ pixels de la région cible $\Omega_t$ et $\bar{b}$ celui des $N_b$ pixels du fond $\Omega_b$. Les vecteurs $\bar{t}$ et $\bar{b}$ sont supposés non corrélés et sont caractérisés par leurs densités de probabilité (PDF) respectives $p^{\Theta_t}$ et $p^{\Theta_b}$ ; $\Theta_t$ et $\Theta_b$ étant les vecteurs des paramètres de leurs PDF. Dans le cas gaussien que nous supposerons ici, $\Theta = (\mu, \sigma)$ où $\mu$ est la moyenne et $\sigma^2$ est la variance.
+On note $\Gamma$ le contour de la région cible ($\Gamma \in \Omega_t$), que l'on suppose continu en connexité à 8 voisins. 
+
+Le but de la segmentation est alors de déterminer la géométrie de $\Gamma$ qui maximise un critère de vraisemblance généralisée (GL).
+La vraisemblance sur l'ensemble de l'image, \textit{ie.} la région $\Omega$ est donnée par
+
+\begin{equation}
+P\left(\bar{v} | \Omega_t, \Omega_b, \Theta_t, \Theta_b\right) = P\left(\bar{v} | \Omega_t, \Theta_t\right)P\left(\bar{v}|\Omega_b, \Theta_b\right)   
+\end{equation}
+soit en développant 
+\begin{equation}
+P\left(\bar{v} | \Omega_t, \Omega_b, \Theta_t, \Theta_b\right) = \prod_{x_k\in\Omega_t}p^{\Theta_t}\left(v_k, \Theta_t\right)\prod_{x_k\in\Omega_b}p^{\Theta_b}\left(v_k, \Theta_b\right)  
+\label{eq-lhprod1}
+\end{equation}
+
+Dans le cas gaussien, la PDF étant de la forme 
+\begin{equation}
+p(\alpha) = \frac{1}{\sigma\sqrt{2\pi}}\mathrm{e}^{-\frac{\left(\alpha - \mu\right)^2}{2\sigma^2}}  
+\label{eq-pdfgauss}
+\end{equation}
+
+La substitution de \eqref{eq-pdfgauss} dans \eqref{eq-lhprod1}, suivie du logarithme, permet d'obtenir l'expression de la \textit{log-vraisemblance}
+\begin{equation}
+  \label{eq-gl1}
+  -N_tln\left(\sqrt{2\pi}\right)-N_tln\left(\sigma_t\right)-\frac{1}{2\sigma_t^2}\sum_{x_k\in\Omega_t}\left(v_k-\mu_t\right)^2
+-N_bln\left(\sqrt{2\pi}\right)-N_bln\left(\sigma_b\right)-\frac{1}{2\sigma_b^2}\sum_{x_k\in\Omega_b}\left(v_k-\mu_b\right)^2
+\end{equation}
+dans laquelle les vecteurs $\Theta_t$ et $\Theta_b$ sont estimés suivant la méthode du  maximum de vraisemblance, qui donne l'expression générique suivante pour l'estimée de $\Theta_t$, notée $\widehat{\Theta_t}$ et transposable à l'identique pour $\Theta_b$ 
+\begin{equation}
+  \label{eq-teta}
+  \widehat{\Theta_t} \left(
+\begin{array}{l}
+\widehat{\mu_t} = \frac{1}{N_t} \displaystyle\sum_{x_k\in \Omega_t} v_k \\
+\widehat{\sigma^2_t} = \frac{1}{N_t} \displaystyle\sum_{x_k\in \Omega_t} \left(v_k - \widehat{\mu_t}\right)^2 \\
+  \end{array}
+\right.
+\end{equation}
+
+En intégrant \eqref{eq-teta} dans \eqref{eq-gl1}, il reste, à une constante près, le critère de vraisemblance généralisée suivant, noté GL, que l'on cherche à  optimiser en déterminant la géométrie de $\Gamma$ qui en maximise la valeur et épousera alors au mieux la forme du contour de la cible.
+\begin{equation}
+  \label{eq-gl}
+  GL=\frac{1}{2}\left(N_tln\left(\widehat{\sigma_t^2}\right)+N_bln\left(\widehat{\sigma_b^2}\right)\right)
+\end{equation}
+
+\subsection{Optimisation des calculs}
+La maximisation de GL est effectuée en employant une technique itérative où sa valeur doit être calculée à chaque déformation du contour $\Gamma$.
+Si l'on se reporte à l'équation \eqref{eq-teta}, on voit que l'obtention de la valeur de GL nécessite, à chaque évaluation d'une géométrie donnée de $\Gamma$, le calcul des sommes  
+\begin{align}
+\label{eq-sommes1}
+  S_v(\Omega_t) &= \sum_{x_k\in \Omega_t} v_k & S_{v^2}(\Omega_t) &= \displaystyle\sum_{x_k\in \Omega_t} v_k^2 \nonumber\\
+  S_v(\Omega_b) &= \sum_{x_k\in \Omega_b} v_k & S_{v^2}(\Omega_b) &= \displaystyle\sum_{x_k\in \Omega_b} v_k^2 \nonumber\\
+\end{align}
+
+Considèrons la région cible $\Omega_t$, les pixels de coordonées $(i,j)$ qui la composent, et généralisons l'écriture des sommes de \eqref{eq-sommes1} en 
+\begin{equation}
+  \label{eq-sommes-gene}
+  S_f(\Omega_t) = \sum_{i=i_{min}}^{i=i_{max}}\sum_{j=j_{min}(i)}^{j=j_{max}(i)}g\left(v(i,j)\right)
+\end{equation}
+où $f$ représente la fonction de valeurs de niveaux de gris à sommer.
+
+En posant 
+\begin{equation}
+  \label{eq-cumuls1}
+  T_g(y,\tau) = \sum_{j=0}^{\tau}g\left(v(y,j)\right)
+\end{equation}
+L'équation \eqref{eq-sommes-gene} devient 
+\begin{equation}
+  \label{eq-somme-cumuls2}
+  S_f(\Omega_t) = \sum_{i=i_{min}}^{i=i_{max}}\left[T_g(i,j_{max}(i))-T_g(i,j_{min}(i)-1)\right]
+\end{equation}
+qui représente  une sommation sur le contour $\Gamma$ que l'on peut écrire 
+\begin{equation}
+  \label{eq-somme-contour}
+  S_f(\Omega_t) = \sum_{(i,j)\in \Gamma}C(i,j)\gamma(i,j)
+\end{equation}
+où $C(i,j)$ est un coefficient lié à la direction du contour au point $(i,j)$ et $\gamma(i,j)$ prend sa valeur selon les règles suivantes
+
+\begin{equation}
+  \label{eq-coefC}
+  \gamma(i,j)=
+  \begin{cases}
+    T(i,j)  & \text{si $C(i,j)= 1 $}\\
+    T(i,j-1)& \text{si $C(i,j)= -1$}\\
+    0       & \text{si $C(i,j)= 0$}\\
+  \end{cases}
+\end{equation}
+
+La valeur de $C(i,j)$ est déterminée pour chaque pixel d'indice $l$ du contour en considérant les pixels d'indices $l-1$ et $l+1$ qui définissent les deux vecteurs $f_{in}$ et $f_{out}$ et leur code selon le codage de Freeman, comme l'illustre la figure \ref{fig-freeman}. La table \ref{tab-freeman} donne les valeurs de $C(i,j)$ selon les valeurs des codes de Freeman des vecteurs $f_{in}$ et $f_{out}$.
+\begin{figure}[htb]
+  \centering
+  \includegraphics[height=3cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/codage-freeman.png}
+  \caption{À gauche : détermination des vecteurs $f_{in}$ et $f_{out}$. À droite : code de Freeman d'un vecteur en fonction de sa direction, l'origine étant supposée au pixel central, en noir. }
+  \label{fig-freeman}
+\end{figure}
+
+\begin{table}[htb]
+  \centering
+\begin{tabular}[htb]{ccccccccc}
+      \toprule
+      &\multicolumn{8}{c}{$f_{out}$}\\
+      \cmidrule(r){2-9}
+      $f_{in}$& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
+      \midrule
+      0     &0&0&0&0&0&-1&-1&-1 \\
+      1     &1&1&1&1&1&0&0&0\\
+      2     &1&1&1&1&1&0&0&0\\
+      3     &1&1&1&1&1&0&0&0\\
+      4     &0&0&0&0&0&-1&-1&-1\\
+      5     &0&0&0&0&0&-1&-1&-1\\
+      6     &0&0&0&0&0&-1&-1&-1\\
+      7     &0&0&0&0&0&-1&-1&-1\\
+      \bottomrule
+\end{tabular}
+   \caption{Valeur du coefficient $C(i,j)$ en fonction des valeurs des codes de Freeman des vecteurs $f_{in}$ et $f_{out}$.}
+      \label{tab-freeman}
+\end{table}
+
+L'intérêt de cette transformation est majeur :
+\begin{itemize}
+\item La sommation en deux dimensions sur la région $\Omega_t$ est ainsi réduite à une sommation à une dimension sur le contour $\Gamma$.
+\item Les valeurs $T_g(i,j)$ peuvent être calculées préalablement à la phase de segmentation proprement dite. Pour le cas gaussien qui nous intéresse, cela revient à pré-calculer les trois images \textit{cumulées} $S_1$, $S_x$ et $S_{x^2}$ définies par
+  \begin{alignat}{4}
+    \label{eq-img-cumul}
+    S_1(i,j) &= \sum_{x=0}^jx & \quad \text{,}\quad S_x(i,j) &= \sum_{x=0}^jv(i,x) & \quad \text{et}&\quad & S_{x^2}(i,j) &= \sum_{x=0}^jv(i,x)^2 
+  \end{alignat}
+\item Les valeurs du coefficient $C(i,j)$ se calculent très facilement durant la génération du contour $\Gamma$.
+\end{itemize}
+
+
+Par ailleurs, le choix d'un contour polygonal permet également d'améliorer l'efficacité de l'algorithme car lors de la phase de segmentation, le déplacement d'un sommet du polygone n'influe que sur les pixels des deux segments qui s'y rapportent, réduisant ainsi la quantité de calculs à effectuer à chaque nouvelle déformation du contour.  
+
+\begin{upminfo}
+  L'approche décrite dans ce chapitre n'est valide que si les segments formant le polygone du contour ne se croisent pas. Il est donc nécessaire, lors de la convergence de la segmentation, d'empêcher les croisements de segments. Une solution simple a été proposée dans \cite{ChesnaudRB99} et nous l'avons parallélisée dans le cadre des travaux présentés ici. 
+\end{upminfo}
+
+\subsection{Implémentation séquentielle}
+Un des inconvénients des algorithmes de type \textit{snake} est l'influence du contour initial sur la convergence de la segmentation. Pour pallier simplement ce défaut, une technique progressive est adoptée, en initialisant le contour avec peu de sommets (4) puis en augmentant le nombre au fur et à mesure de la convergence. L'algorithme \ref{algo-snake-cpu1} décrit macroscopiquement la solution mise \oe uvre tandis que l'algorithme \ref{algo-snake-cpu2} en présente les détails.
+
+\begin{algorithm}
+\label{algo-snake-cpu}
+\caption{Principe mis en \oe uvre pour la convergence du snake polygonal}
+  Calculer les images cumulées\;
+  Initialiser le contour avec 4 sommets\;
+  \Repeter(\tcc*[f]{niveau contour}){aucun sommet ne peut être ajouté}{
+      \Repeter(\tcc*[f]{niveau sommet}){aucun sommet ne peut être déplacé}{
+        Déplacer chaque sommet autour de sa position actuelle.\;
+       Déplacer le sommet vers la position induisant le meilleur GL\;
+     }
+      Ajouter un sommet au milieu de chaque \textit{grand} segment\;
+  }
+\end{algorithm}
+
+\begin{algorithm}[h]
+\caption{Détail de l'implémentation du snake polygonal} 
+\label{cpualgo}
+   Lire l'image $\bar{v}$\;
+   Calculer les images cumulées $S_1$, $S_x$ $S_{x^2}$ \nllabel{li-img-cumul}\tcc*[r]{en parallèle via SSE2} 
+   $n \leftarrow 0$ \tcc*[r]{indice de boucle niveau contour}
+   $N_n \leftarrow 4$ \tcc*[r]{nombre de n\oe uds}
+   $\Gamma \leftarrow \{\Gamma_0,\Gamma_1,\Gamma_2,\Gamma_3\} $\;
+   $d \leftarrow d_{max}$ \tcc*[r]{pas de déplacement des n\oe uds}
+   $l_{min} = 32$ \tcc*[r]{longueur mini des segments sécables}
+   $\Gamma_i \leftarrow \Gamma_0$ \tcc*[r]{sommet courant}
+   $GL_{ref} \leftarrow GL(\Gamma, N_n, \bar{v}, S_y, S_{y^2})$ \tcc*[r]{voir à partir de ligne 18 pour le détail}
+  \Repeter(\tcc*[f]{niveau contour}){$N_{add}=0$}{
+    $N_{add}\leftarrow 0$\;
+    \Repeter(\tcc*[f]{niveau n\oe ud}){$N_{move}=0$}{
+      $N_{move}\leftarrow 0$\;
+      \Pour{$i=0$ à $i=N_n-1$}{
+       Calculer les positions $\{\Gamma_i^0, \dots, \Gamma_i^7\}$ \tcc*[r]{les 8 voisins de $\Gamma_i$ }
+       \Pour{$w=0$ à $w=7$}{
+          Soustraire à $GL_{ref}$ la contribution des segments $\Gamma_{i-1}\Gamma_i$ et $\Gamma_{i}\Gamma_{i+1}$\;
+         Discrétiser les segments $\Gamma_{i-1}\Gamma_i^w$ et $\Gamma_{i}^w\Gamma_{i+1}$\nllabel{li-bresen}\;
+          Lire dans $S_1$, $S_x$ et $S_{x^2}$ les contributions des pixels de $\Gamma_{i-1}\Gamma_i^w$ et $\Gamma_{i}^w\Gamma_{i+1}$\nllabel{li-contrib-seg-deb}\;
+          Calculer les directions et lire les codes de Freeman \;
+          Calculer $GL_w$ incluant les contributions de $\Gamma_{i-1}\Gamma_i^w$ et $\Gamma_{i}^w\Gamma_{i+1}$ \nllabel{li-contrib-seg-fin}\;
+         \lSi{$GL_w > GL_{ref}$}{
+           $GL_{ref} \leftarrow GL_w$\;
+           $\Gamma_i \leftarrow \Gamma_i^w$\;
+            $N_{move} \leftarrow N_{move}+1$\;
+         }
+       }
+      }
+     $l \leftarrow l+1$\;
+   }
+   \PourCh{segment $\Gamma_i\Gamma_{i+1}$}{
+     \Si{$\|\Gamma_i\Gamma_{i+1}\| > l_{min}$}{
+       Ajouter un n\oe ud au milieu de $\Gamma_i$ et $\Gamma_{i+1}$\;
+       $N_{add} \leftarrow N_{add}+1$\;
+     }
+   }
+   $N_n \leftarrow N_n + N_{add}$\;
+   \lSi{$d > 1$}{ $d \leftarrow d/2$ } \lSinon{ $d \leftarrow 1$ }\;
+   $GL_{ref} \leftarrow GL(\Gamma, N_n, \bar{v}, S_y, S_{y^2})$ \;
+  }
+\end{algorithm}
+
+Les différentes sommations nécessaires au calcul de la valeur du critère $GL$ sont effectuées en parallèle à l'aide du jeu d'instructions SSE2. La taille des registres utilisables est de 128 bits et permet ainsi de traiter des images de 4096$\times$4096 pixels dont les niveaux de gris sont codés sur 16 bits. Cela ne laisse toutefois que 12 bits pour le codage des segments du contour et limite ainsi leur longueur à 4096 pixels. L'organisation d'un registre SSE 128 bits est donc la suivante :
+\begin{itemize}
+\item 24 bits pour les sommes à opérandes dans $S_1$
+\item 24+16 = 40 bits pour les sommes à opérandes dans $S_x$
+\item 24+32 = 60 bits pour les sommes à opérandes dans $S_x^2$
+\end{itemize}
+Soit un total de 124 bits, qui peuvent donc être représentés par un registre SSE2.
+
+\subsection{Performances}
+Les images de 1024$^2$ pixels de la figure \ref{fig-snakecpu-cochon512} montrent l'évolution du contour lors de la segmentation d'une image photographique prise en faible éclairement et bruitée artificiellement par un bruit gaussien d'écart type 25. Les paramètres de la séquence sont fixés empiriquement aux valeurs $d_{max}=16, l_{min}=8$.
+Les temps d'exécution indiqués sont mesurés sur Intel Xeon E5530-2.4GHz with 12Go RAM et sont les valeurs moyennes obtenues pour 10 exécutions.
+\begin{figure}
+  \centering
+  \subfigure[Initialisation : 4 n\oe uds]{\includegraphics[height=3.5cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/im000.png}}
+  \subfigure[Itération 1 : 8 n\oe uds 3~ms]{\includegraphics[height=3.5cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/im001.png}}
+  \subfigure[Itération 2 : 16 n\oe uds 1~ms]{\includegraphics[height=3.5cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/im002.png}}
+  \subfigure[Itération 3, 32 n\oe uds 1~ms]{\includegraphics[height=3.5cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/im003.png}}\\
+  \subfigure[Itération 7 : 223 n\oe uds 3~ms]{\includegraphics[height=3.5cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/im007.png}}
+  \subfigure[Itération 10 : 244 n\oe uds 3~ms]{\includegraphics[height=3.5cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/im010.png}}
+  \subfigure[Itération 13 : 256 n\oe uds 3~ms]{\includegraphics[height=3.5cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/im013.png}}
+  \subfigure[Itération 14 : 256 n\oe uds 3~ms]{\includegraphics[height=3.5cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/im014.png}}
+  \caption{Évolution du contour lors de la segmentation d'une image de 512$^2$ pixels. La convergence est obtenue à l'itération 14 après 44~ms pour un total de  256 n\oe uds.}
+ \label{fig-snakecpu-cochon512}
+\end{figure}
+
+La dépendance vis à vis du contour initial qui est un des principaux soucis liés au snake est ici fortement relativisée. La figure \ref{fig-snakecpu-compinit} montre le contour final segmentant l'image de test de la figure \ref{fig-snakecpu-cochon512} à partir d'un état initial très éloigné du précédent et \textit{a priori} très défavorable compte tenu du fait qu'il est loin de la cible et sans intersection avec elle. Toutefois, le contour final obtenu est très proche de celui obtenu à partir d'un état initial englobant la cible, malgré un n\oe ud qui s'est ``accroché'' au bord de l'image. La convergence est également plus longue à obtenir dans ce cas avec Un total de 17 itérations en 87~ms et 273 n\oe uds. 
+
+\begin{figure}
+  \centering
+  \subfigure[Initialisation 2 ]{\includegraphics[height=4.5cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/snakecpu-init2.png}}
+  \subfigure[Contour final 2 : 273 n\oe uds 87~ms]{\includegraphics[height=4.5cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/snakecpu-compinit2.png}}
+  \subfigure[Contour final 1 : 256 n\oe uds 44~ms]{\includegraphics[height=4.5cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/im014.png}}
+  \caption{Influence du contour initial sur la segmentation. Le contour final 1 est celui de la figure \ref{fig-snakecpu-cochon512}.}
+  \label{fig-snakecpu-compinit}
+\end{figure}
+
+La dimension de l'image à traiter a également un effet sur le résultat et naturellement sur le temps de calcul. Si l'on conserve les mêmes paramètres d'optimisation que pour la segmentation de l'image 512$^2$ pixels et un contour initial dont les cotés sont à une distance des bords équivalente à 10\% des cotés de l'image, le résultat sur une image identique de 4000$^2$ pixels est  obtenu en 1.3~s avec 1246 n\oe uds ; il est reproduit  à la figure \ref{fig-snakecpu-cochon4ka}. Le nombre de pixels appartenant à la région cible est tel que l'amplitude des déplacements autorisés pour chaque n\oe ud ($d$) peut se révéler trop faible vis à vis du seuil d'acceptation des mouvements. On observe que les zones à gradient élevé ne posent pas de problème et sont détourées de la même manière, alors que dans le bas de l'image où figure une zone de gradient faible (ombre), la cible se trouve maintenant quelque peu surévaluée en surface là ou elle était plutôt sous évaluée dans l'image en 512$^2$ pixels. 
+On parvient à un résultat très proche beaucoup plus rapidement en adaptant les paramètres à la taille de l'image, comme le montre par exemple la segmentation de la figure \ref{fig-snakecpu-cochon4kb}, effectuée avec $d_{max}=128$ et $l_{min}=32$ et qui converge vers un contour de 447 n\oe uds en moins de 0.7~s.
+Au delà des 16 millions de pixels (4000$^2$ pixels), l'implémentation séquentielle est toujours possible mais doit se priver des instructions SSE. Nous avons, avec leur accord, adapté le code des auteurs en ce sens et réalisé les mesures pour des tailles allant jusqu'à 150~MP. La table \ref{tab-snakecpu-speed-size} en synthétise les résultats en distinguant chaque fois le temps pris par les pré-calculs et celui nécessaire à la convergence de la segmentation. 
+
+\begin{figure}
+  \centering
+  \subfigure[$d_{max}=16$ et $l_{min}=8$, 1246 n\oe uds en 1.3~s]{\label{fig-snakecpu-cochon4ka}\includegraphics[height=5cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/snakecpu-cochon4k.png}}
+  \subfigure[$d_{max}=128$ et $l_{min}=32$, 447 n\oe uds en 0.7~s]{\label{fig-snakecpu-cochon4kb}\includegraphics[height=5cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/snakecpu-cochon4k-128-32.png}}
+  \caption{Segmentation de l'image de test en 4000$^2$ pixels.}
+  \label{fig-snakecpu-cochon4k}
+\end{figure}
+
+
+\begin{table}
+  \centering
+  \begin{tabular}{rrrr}
+      \toprule
+      &\multicolumn{3}{c}{Taille de l'image}\\
+      &\multicolumn{3}{c}{(millions de pixels)}\\
+      \cmidrule(r){2-4}
+      & 15 & 100 & 150 \\
+      \midrule
+      Pré-calculs &0,13&0,91&1,4\\
+      Segmentation&0,46&3,17&4,3\\
+      {\bf Total} &0,51&4,08&5,7\\
+      \bottomrule
+\end{tabular}
+   \caption{Performances (en secondes) de la segmentation par snake polygonal sur CPU en fonction de la taille de l'image à traiter. Le temps sont obtenus avec la même image de test dilatée et bruitée et un contour initial carré dont la distance aux bords est proportionnelle à la taille de l'image.}
+      \label{tab-snakecpu-speed-size}
+\end{table}
+
+
+Enfin, il faut aussi considérer les tailles relatives de la cible et de l'image. Ainsi, si on fait l'hypothèse d'une cible de petite taille ``noyée'' dans une image de grandes dimensions, les résultats de la segmentation seront impactés en raison, cette fois, d'une moindre adaptation à la cible lors des toutes premières itérations, les plus grossières, où le nombre de n\oe uds et réduit et le pas de déplacement potentiellement grand vis à vis de la cible. Ce cas de figure est illustré par la segmentation reproduite à la figure \ref{fig-snakecpu-cochon4kc3} et qui met en évidence une qualité moindre par la confusion des zones les plus sombres de la cible avec le fond.
+
+\begin{figure}[h]
+  \centering
+  \includegraphics[height=5cm]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/cochon4kc-128-8.png}
+  \caption{Segmentation de l'image de test en 4000$^2$ pixels avec une cible de petite taille. Le contour initial est celui utilisé à la figure \ref{fig-snakecpu-cochon4k}.}
+  \label{fig-snakecpu-cochon4kc3}
+\end{figure}
+ 
+
+
+\section{Implémentation parallèle GPU du snake polygonal}
+L'analyse de l'exécution du programme séquentiel révèle la prépondérance des blocs fonctionnels suivants, dans l'ordre d'importance, qui occupent à eux seuls plus de 80\% du temps total d'exécution :
+\begin{itemize}
+ \item Le calcul de la contribution des segments (lignes \ref{li-contrib-seg-deb} à \ref{li-contrib-seg-fin} dans l'algorithme \ref{cpualgo}) 
+  \item La génération des trois images cumulées, avant le début des itérations (ligne \ref{li-img-cumul}).
+  \item La discrétisation des segments définis par les coordonnées de leurs extrémités (ligne \ref{li-bresen}).
+\end{itemize}
+Cette proportion est globalement conservée lorsque la taille de l'image à traiter varie, comme le montre le graphique de la figure \ref{fig-snakecpu-chronos1} 
+
+\begin{figure}
+  \centering
+  \includegraphics[width=0.5\linewidth, height=0.25\linewidth]{/home/zulu/Documents/these_gilles/THESE/Chapters/chapter3/img/cpu_profil.png}
+\label{fig-snakecpu-chronos1}
+  \caption{Évolution du coût relatif des trois fonctions les plus consommatrices en temps de calcul en fonction de la taille de l'image à traiter.}
+\end{figure} 
+
+Si l'effort de parallélisation porte essentiellement sur ces fonctions coûteuses, l'ensemble du traitement est réalisé sur le GPU afin de réduire autant que possible les transferts entre le GPU et le système hôte qui, selon le volume concerné, sont susceptibles de grever considérablement la performance globale. L'hôte ne conserve que l'initiative du transfert initial et le contrôle de la boucle principale, ne nécessitant l'échange que d'un seul octet à chaque itération (le nombre de nouveau n\oe uds $N_{add}$).
+
+Les traitements étant totalement indépendants, nos traitons séparément la parallélisation des pré-calculs et celle de la segmentation.
+
+\subsection{pré-calculs des images cumulées}   
+Pour réduire la quantité de mémoire requise, nous avons choisi de ne pas générer l'image $S_1$ mais plutôt d'en calculer les valeurs à la volée. L'expression en est simple et le temps pris par les opération élémentaires qu'elle met en jeu est largement compensé par le gain obtenu en économisant les accès mémoire qui auraient été nécessaires, ce qui n'est pas le cas des deux autres images $S_x$ et $S_x^2$ dont le calcul est quant à lui réalisé en appliquant la méthode des \textit{prefixsums} inclusifs décrite dans \cite{BlellochTR90}. 
+
+La prise en compte de grandes images nécessite cependant de les traiter par tranches successives calibrées en fonction de la contrainte limitative qui peut-être :
+\begin{itemize}
+\item le nombre maximal de threads dans une grille, pour le cas où l'on adopte la règle ``1 pixel par thread''.
+\item la quantité de mémoire partagée par blocs de threads, si l'on traite plus d'un pixel par thread.
+\end{itemize}
+
+La solution qui apporte les meilleures performances est celle du traitement 1 pixel par thread, associé à un découpage de l'image en tranche équilibrées. Le seuil au delà duquel ce découpage devient nécessaire se situe à 33 millions de pixels pour C1060 et 66 millions pour C2050.
+ 
+
+
+
+
+
+
+
+
+

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new file mode 100644 (file)
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+4 0 0 50 -1 4 12 0.0000 2 195 4335 810 2160 $N_{TB}$ blocks of $bs$ threads for one segment\001
+4 2 -1 50 -1 4 12 0.0000 2 150 630 2565 1980 block 0\001
+4 2 -1 50 -1 4 12 0.0000 2 195 1515 6930 1980 block $N_{TB}-1$\001
+4 2 -1 50 -1 4 12 0.0000 2 150 630 4410 1980 block 1\001
+4 0 0 50 -1 4 12 0.0000 2 195 2625 810 3780 16 segments around one node\001
+4 2 0 50 -1 4 12 0.0000 2 195 2730 1800 3555 $\\overrightarrow{P_{i-1}T_{i,0}}$\001
+4 2 0 50 -1 4 12 0.0000 2 195 2730 3825 3555 $\\overrightarrow{P_{i-1}T_{i,7}}$\001
+4 2 0 50 -1 4 12 0.0000 2 195 2790 4860 3555 $\\overrightarrow{T_{i,0}P_{i+1}}$\001
+4 2 0 50 -1 4 12 0.0000 2 195 2790 6885 3555 $\\overrightarrow{T_{i,7}P_{i+1}}$\001
+4 2 0 50 -1 4 12 0.0000 2 195 2280 6930 5085 $P_{2(N_n/2+N_n\\%2-1)}$\001
+4 2 0 50 -1 4 12 0.0000 2 195 675 4005 5085 $P_{2}$\001
+4 2 0 50 -1 4 12 0.0000 2 195 675 2385 5085 $P_{0}$\001
+4 1 0 50 -1 4 12 0.0000 2 180 2715 4005 5310 $(N_n/2+N_n\\%2)$ even nodes\001
+4 2 0 50 -1 4 12 0.0000 2 195 675 2385 6255 $P_{1}$\001
+4 2 0 50 -1 4 12 0.0000 2 195 675 4005 6255 $P_{3}$\001
+4 2 0 50 -1 4 12 0.0000 2 195 1695 6930 6255 $P_{2(N_n/2-1)+1}$\001
+4 1 0 50 -1 4 12 0.0000 2 180 1800 4005 6480 $(N_n/2)$ odd nodes\001

diff --git a/THESE/Chapters/chapter3/img/contribs_segments.fig.bak b/THESE/Chapters/chapter3/img/contribs_segments.fig.bak
new file mode 100644 (file)
index 0000000..549b078
--- /dev/null
+++ b/THESE/Chapters/chapter3/img/contribs_segments.fig.bak
@@ -0,0 +1,1189 @@
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+4 2 0 50 -1 4 12 0.0000 2 195 2790 6885 3555 $\\overrightarrow{T_{i,7}P_{i+1}}$\001
+4 2 0 50 -1 4 12 0.0000 2 195 2280 6930 5085 $P_{2(N_n/2+N_n\\%2-1)}$\001
+4 2 0 50 -1 4 12 0.0000 2 195 675 4005 5085 $P_{2}$\001
+4 2 0 50 -1 4 12 0.0000 2 195 675 2385 5085 $P_{0}$\001
+4 1 0 50 -1 4 12 0.0000 2 180 2715 4005 5310 $(N_n/2+N_n\\%2)$ even nodes\001
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+4 2 0 50 -1 4 12 0.0000 2 195 1695 6930 6255 $P_{2(N_n/2-1)+1}$\001
+4 1 0 50 -1 4 12 0.0000 2 180 1800 4005 6480 $(N_n/2)$ odd nodes\001

diff --git a/THESE/Chapters/chapter3/img/contribs_segments.pdf b/THESE/Chapters/chapter3/img/contribs_segments.pdf
new file mode 100644 (file)
index 0000000..ad12216
Binary files /dev/null and b/THESE/Chapters/chapter3/img/contribs_segments.pdf differ

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+      <!-- License: creative commons attribution -->
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diff --git a/THESE/biblio.bib b/THESE/biblio.bib
index 92f642f2b1cc1113d3fb01a94a0ec6d12242d009..e9628e940e8ee2e4e12acf13c1f7732a71d5a5e1 100755 (executable)
--- a/THESE/biblio.bib
+++ b/THESE/biblio.bib
@@ -503,6 +503,21 @@ number={3},
 pages={493-504},  
 doi={10.1016/j.bpj.2009.10.037}, 
 }
+@article{sanchez2013highly,
+year={2013},
+issn={1939-8018},
+journal={Journal of Signal Processing Systems},
+volume={71},
+number={3},
+doi={10.1007/s11265-012-0715-1},
+title={Highly Parallelable Bidimensional Median Filter for Modern Parallel Programming Models},
+url={http://dx.doi.org/10.1007/s11265-012-0715-1},
+publisher={Springer US},
+keywords={Nonlinear filters; Parallel algorithms; Image processing},
+author={Sánchez, RicardoM. and Rodríguez, PaulA.},
+pages={221-235},
+language={English}
+}
 @article{Sanchez-2-2012,
 year={2012},
 issn={1939-8018},
@@ -516,6 +531,7 @@ author={Sánchez, RicardoM. and Rodríguez, PaulA.},
 pages={1-15},
 language={English}
 }
+
 @article{cutrona1990synthetic,
   title={Synthetic aperture radar},
   author={Cutrona, LJ},
@@ -1295,4 +1311,27 @@ ISSN={1550-5499},}
   pages={416--423},
   year={2001},
   organization={IEEE}
+}
+@article{volkov2010better,
+  title={Better performance at lower occupancy},
+  author={Volkov, Vasily},
+  journal={Proceedings of the GPU Technology Conference, GTC},
+  volume={10},
+  year={2010}
+}
+@article{perrot2013fine,
+  title={Fine-tuned High-speed Implementation of a GPU-based Median Filter},
+  author={Perrot, Gilles and Domas, St{\'e}phane and Couturier, Rapha{\"e}l},
+  journal={Journal of Signal Processing Systems},
+  pages={1--6},
+  year={2013},
+  publisher={Springer}
+}
+@inproceedings{sanchez2012bidimensional,
+  title={Bidimensional median filter for parallel computing architectures},
+  author={Sanchez, Ricardo M and Rodriguez, Paul A},
+  booktitle={Acoustics, Speech and Signal Processing (ICASSP), 2012 IEEE International Conference on},
+  pages={1549--1552},
+  year={2012},
+  organization={IEEE}
 }
\ No newline at end of file

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+\@writefile{toc}{\contentsline {subsubsection}{\numberline {3.2.2.1}Cas particulier des segments dont la pente $k$ v\IeC {\'e}rifie $|k|\leq 1$}{53}{subsubsection.3.2.2.1}}
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+\newlabel{cpualgo}{{3}{54}{Implémentation séquentielle\relax }{algocfline.3}{}}
+\@writefile{lot}{\contentsline {table}{\numberline {3.2}{\ignorespaces Performances (en secondes) de la segmentation par snake polygonal sur CPU en fonction de la taille de l'image \IeC {\`a} traiter. Le temps sont obtenus avec la m\IeC {\^e}me image de test dilat\IeC {\'e}e et bruit\IeC {\'e}e et un contour initial carr\IeC {\'e} dont la distance aux bords est proportionnelle \IeC {\`a} la taille de l'image. Seule l'image en 15~MP a pu \IeC {\^e}tre trait\IeC {\'e}e par une impl\IeC {\'e}mentation utilisant SSE2.}}{55}{table.3.2}}
+\newlabel{tab-snakecpu-speed-size}{{3.2}{55}{Performances (en secondes) de la segmentation par snake polygonal sur CPU en fonction de la taille de l'image à traiter. Le temps sont obtenus avec la même image de test dilatée et bruitée et un contour initial carré dont la distance aux bords est proportionnelle à la taille de l'image. Seule l'image en 15~MP a pu être traitée par une implémentation utilisant SSE2}{table.3.2}{}}
+\@writefile{lot}{\contentsline {table}{\numberline {3.3}{\ignorespaces Acc\IeC {\'e}l\IeC {\'e}ration constat\IeC {\'e}e, pour le calcul des images cumul\IeC {\'e}es, de l'impl\IeC {\'e}mentation GPU par rapport \IeC {\`a} l'impl\IeC {\'e}mentation CPU de r\IeC {\'e}f\IeC {\'e}rence.}}{55}{table.3.3}}
+\newlabel{tab-speedup-cumuls}{{3.3}{55}{Accélération constatée, pour le calcul des images cumulées, de l'implémentation GPU par rapport à l'implémentation CPU de référence}{table.3.3}{}}
+\@writefile{toc}{\contentsline {chapter}{\numberline {4}Le filtrage des images sur GPU}{57}{chapter.4}}
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+\@writefile{toc}{\contentsline {section}{\numberline {4.2}Filtre m\IeC {\'e}dian}{57}{section.4.2}}
+\@writefile{toc}{\contentsline {section}{\numberline {4.3}Filtres de convolution}{57}{section.4.3}}
 \bibstyle{plain}
 \bibdata{biblio}
-\@writefile{toc}{\contentsline {chapter}{\numberline {5}Conclusion g\IeC {\'e}n\IeC {\'e}rale}{43}{chapter.5}}
+\@writefile{toc}{\contentsline {chapter}{\numberline {5}Conclusion g\IeC {\'e}n\IeC {\'e}rale}{59}{chapter.5}}
 \@writefile{lof}{\addvspace {10\p@ }}
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 \bibcite{kddcup99}{1}
@@ -312,93 +418,97 @@
 \bibcite{arora1998approximation}{7}
 \bibcite{bauer2009segmentation}{8}
 \bibcite{bertaux2004speckle}{9}
-\bibcite{boykov2004experimental}{10}
-\bibcite{bresenham1965algorithm}{11}
-\bibcite{1467423}{12}
-\bibcite{BuadesCM06}{13}
-\bibcite{Caselles99topographicmaps}{14}
-\bibcite{5459410}{15}
-\bibcite{cates2004gist}{16}
-\bibcite{chandran2009computational}{17}
-\bibcite{che2008performance}{18}
-\bibcite{chen09}{19}
-\bibcite{1093941}{20}
-\bibcite{cheng1995mean}{21}
-\bibcite{cherkassky1997implementing}{22}
-\bibcite{ChesnaudRB99}{23}
-\bibcite{cohen1993surface}{24}
-\bibcite{comaniciu1999mean}{25}
-\bibcite{comaniciu2002mean}{26}
-\bibcite{cutrona1990synthetic}{27}
-\bibcite{Dabov06imagedenoising}{28}
-\bibcite{Dabov09bm3dimage}{29}
-\bibcite{Daubechies:1992:TLW:130655}{30}
-\bibcite{dixit2005gpu}{31}
-\bibcite{elad2006image}{32}
-\bibcite{felzenszwalb2004efficient}{33}
-\bibcite{fluck2006gpu}{34}
-\bibcite{foley1994introduction}{35}
-\bibcite{ford1955simple}{36}
-\bibcite{fukunaga1975estimation}{37}
-\bibcite{fulkerson2012really}{38}
-\bibcite{GallandBR03}{39}
-\bibcite{GermainR01}{40}
-\bibcite{nlmeansgpubelge}{41}
-\bibcite{snakegvf06}{42}
-\bibcite{healey1994radiometric}{43}
-\bibcite{hochbaum2013simplifications}{44}
-\bibcite{5170921}{45}
-\bibcite{humphrey1924psychology}{46}
-\bibcite{jeong2009scalable}{47}
-\bibcite{5402362}{48}
-\bibcite{KassWT88}{49}
-\bibcite{keselman1998extraction}{50}
-\bibcite{cmla2009Kes}{51}
-\bibcite{kohli2007dynamic}{52}
-\bibcite{lefohn2003inter}{53}
-\bibcite{lefohn2003interactive}{54}
-\bibcite{lefohn2005streaming}{55}
-\bibcite{li2009mean}{56}
-\bibcite{li2011robust}{57}
-\bibcite{macqueen1967some}{58}
-\bibcite{Mallat:2008:WTS:1525499}{59}
-\bibcite{mancuso2001introduction}{60}
-\bibcite{martin2001database}{61}
-\bibcite{coil}{62}
-\bibcite{osher1988fronts}{63}
-\bibcite{4310076}{64}
-\bibcite{PALHANOXAVIERDEFONTES}{65}
-\bibcite{pelleg2000x}{66}
-\bibcite{4287006}{67}
-\bibcite{1521458}{68}
-\bibcite{4587843}{69}
-\bibcite{Roberts:2010:WGA:1921479.1921499}{70}
-\bibcite{ronfard1994region}{71}
-\bibcite{rumpf2001level}{72}
-\bibcite{6288187}{73}
-\bibcite{sethian1996fast}{74}
-\bibcite{kmeansgpuopengl}{75}
-\bibcite{shi2000normalized}{76}
-\bibcite{snakegvfopencl12}{77}
-\bibcite{convolutionsoup}{78}
-\bibcite{graphcutscuda}{79}
-\bibcite{strang1999discrete}{80}
-\bibcite{theuwissen2001ccd}{81}
-\bibcite{710815}{82}
-\bibcite{tukey77}{83}
-\bibcite{vedaldi2008quick}{84}
-\bibcite{4563095}{85}
-\bibcite{Vineet:2009:FMS:1572769.1572796}{86}
-\bibcite{wang2001image}{87}
-\bibcite{wang2003image}{88}
-\bibcite{Wang04imagequality}{89}
-\bibcite{wu1993optimal}{90}
-\bibcite{xiao2010efficient}{91}
-\bibcite{5206542}{92}
-\bibcite{Zahn:1971:GMD:1309266.1309359}{93}
-\bibcite{zheng2011performance}{94}
-\bibcite{zheng2012fast}{95}
+\bibcite{BlellochTR90}{10}
+\bibcite{boykov2004experimental}{11}
+\bibcite{bresenham1965algorithm}{12}
+\bibcite{1467423}{13}
+\bibcite{BuadesCM06}{14}
+\bibcite{Caselles99topographicmaps}{15}
+\bibcite{5459410}{16}
+\bibcite{cates2004gist}{17}
+\bibcite{chandran2009computational}{18}
+\bibcite{che2008performance}{19}
+\bibcite{chen09}{20}
+\bibcite{1093941}{21}
+\bibcite{cheng1995mean}{22}
+\bibcite{cherkassky1997implementing}{23}
+\bibcite{ChesnaudRB99}{24}
+\bibcite{cohen1993surface}{25}
+\bibcite{comaniciu1999mean}{26}
+\bibcite{comaniciu2002mean}{27}
+\bibcite{cutrona1990synthetic}{28}
+\bibcite{Dabov06imagedenoising}{29}
+\bibcite{Dabov09bm3dimage}{30}
+\bibcite{Daubechies:1992:TLW:130655}{31}
+\bibcite{dixit2005gpu}{32}
+\bibcite{elad2006image}{33}
+\bibcite{felzenszwalb2004efficient}{34}
+\bibcite{fluck2006gpu}{35}
+\bibcite{foley1994introduction}{36}
+\bibcite{ford1955simple}{37}
+\bibcite{fukunaga1975estimation}{38}
+\bibcite{fulkerson2012really}{39}
+\bibcite{GallandBR03}{40}
+\bibcite{GermainR01}{41}
+\bibcite{nlmeansgpubelge}{42}
+\bibcite{snakegvf06}{43}
+\bibcite{healey1994radiometric}{44}
+\bibcite{hochbaum2013simplifications}{45}
+\bibcite{5170921}{46}
+\bibcite{humphrey1924psychology}{47}
+\bibcite{jeong2009scalable}{48}
+\bibcite{5402362}{49}
+\bibcite{KassWT88}{50}
+\bibcite{keselman1998extraction}{51}
+\bibcite{cmla2009Kes}{52}
+\bibcite{kohli2007dynamic}{53}
+\bibcite{lefohn2003inter}{54}
+\bibcite{lefohn2003interactive}{55}
+\bibcite{lefohn2005streaming}{56}
+\bibcite{li2009mean}{57}
+\bibcite{li2011robust}{58}
+\bibcite{macqueen1967some}{59}
+\bibcite{Mallat:2008:WTS:1525499}{60}
+\bibcite{mancuso2001introduction}{61}
+\bibcite{martin2001database}{62}
+\bibcite{coil}{63}
+\bibcite{osher1988fronts}{64}
+\bibcite{4310076}{65}
+\bibcite{PALHANOXAVIERDEFONTES}{66}
+\bibcite{pelleg2000x}{67}
+\bibcite{4287006}{68}
+\bibcite{1521458}{69}
+\bibcite{4587843}{70}
+\bibcite{Roberts:2010:WGA:1921479.1921499}{71}
+\bibcite{ronfard1994region}{72}
+\bibcite{rumpf2001level}{73}
+\bibcite{6288187}{74}
+\bibcite{sethian1996fast}{75}
+\bibcite{kmeansgpuopengl}{76}
+\bibcite{shi2000normalized}{77}
+\bibcite{snakegvfopencl12}{78}
+\bibcite{convolutionsoup}{79}
+\bibcite{graphcutscuda}{80}
+\bibcite{strang1999discrete}{81}
+\bibcite{sanchez2013highly}{82}
+\bibcite{theuwissen2001ccd}{83}
+\bibcite{710815}{84}
+\bibcite{tukey77}{85}
+\bibcite{vedaldi2008quick}{86}
+\bibcite{4563095}{87}
+\bibcite{Vineet:2009:FMS:1572769.1572796}{88}
+\bibcite{volkov2010better}{89}
+\bibcite{wang2001image}{90}
+\bibcite{wang2003image}{91}
+\bibcite{Wang04imagequality}{92}
+\bibcite{wu1993optimal}{93}
+\bibcite{xiao2010efficient}{94}
+\bibcite{5206542}{95}
+\bibcite{Zahn:1971:GMD:1309266.1309359}{96}
+\bibcite{zheng2011performance}{97}
+\bibcite{zheng2012fast}{98}
 \citation{zheng2011performance}
 \citation{graphcutscuda}
 \citation{fulkerson2012really}
+\citation{snakegvf06}
 \citation{martin2001database}

diff --git a/THESE/these.bbl b/THESE/these.bbl
index 1dd0d5a101f625ae4e6b0606dd3cfb470ede6d2a..4245ef5d547ae2090646a1d8f44511928a238666 100644 (file)
--- a/THESE/these.bbl
+++ b/THESE/these.bbl
@@ -47,6 +47,12 @@ Nicolas Bertaux, Yann Frauel, Philippe R{\'e}fr{\'e}gier, and Bahram Javidi.
   gray-level regularization.
 \newblock {\em JOSA A}, 21(12):2283--2291, 2004.
 
+\bibitem{BlellochTR90}
+Guy~E. Blelloch.
+\newblock Prefix sums and their applications.
+\newblock Technical Report CMU-CS-90-190, School of Computer Science, Carnegie
+  Mellon University, November 1990.
+
 \bibitem{boykov2004experimental}
 Yuri Boykov and Vladimir Kolmogorov.
 \newblock An experimental comparison of min-cut/max-flow algorithms for energy
@@ -478,6 +484,12 @@ Gilbert Strang.
 \newblock The discrete cosine transform.
 \newblock {\em SIAM review}, 41(1):135--147, 1999.
 
+\bibitem{sanchez2013highly}
+RicardoM. Sánchez and PaulA. Rodríguez.
+\newblock Highly parallelable bidimensional median filter for modern parallel
+  programming models.
+\newblock {\em Journal of Signal Processing Systems}, 71(3):221--235, 2013.
+
 \bibitem{theuwissen2001ccd}
 Albert~JP Theuwissen.
 \newblock Ccd or cmos image sensors for consumer digital still photography?
@@ -513,6 +525,11 @@ Vibhav Vineet, Pawan Harish, Suryakant Patidar, and P.~J. Narayanan.
 \newblock In {\em Proceedings of the Conference on High Performance Graphics
   2009}, HPG '09, pages 167--171, New York, NY, USA, 2009. ACM.
 
+\bibitem{volkov2010better}
+Vasily Volkov.
+\newblock Better performance at lower occupancy.
+\newblock {\em Proceedings of the GPU Technology Conference, GTC}, 10, 2010.
+
 \bibitem{wang2001image}
 Song Wang and Jeffrey~Mark Siskind.
 \newblock Image segmentation with minimum mean cut.

diff --git a/THESE/these.blg b/THESE/these.blg
index db7083f2b904f196010b9540d62a55b0c6872e01..c0d2699609784cfbcddbdc57ddc7fe4e0f0dba7e 100644 (file)
--- a/THESE/these.blg
+++ b/THESE/these.blg
@@ -10,45 +10,45 @@ Warning--empty journal in cohen1993surface
 Warning--empty journal in dixit2005gpu
 Warning--can't use both volume and number fields in macqueen1967some
 Warning--can't use both volume and number fields in xiao2010efficient
-You've used 95 entries,
+You've used 98 entries,
             2118 wiz_defined-function locations,
-            1037 strings with 20359 characters,
-and the built_in function-call counts, 36042 in all, are:
-= -- 3622
-> -- 1535
-< -- 50
-+ -- 620
-- -- 502
-* -- 2473
-:= -- 5637
-add.period$ -- 304
-call.type$ -- 95
-change.case$ -- 519
+            1054 strings with 20816 characters,
+and the built_in function-call counts, 36777 in all, are:
+= -- 3696
+> -- 1562
+< -- 51
++ -- 631
+- -- 510
+* -- 2518
+:= -- 5771
+add.period$ -- 313
+call.type$ -- 98
+change.case$ -- 532
 chr.to.int$ -- 0
-cite$ -- 103
-duplicate$ -- 1479
-empty$ -- 2910
-format.name$ -- 502
-if$ -- 7833
+cite$ -- 106
+duplicate$ -- 1509
+empty$ -- 2969
+format.name$ -- 510
+if$ -- 7983
 int.to.chr$ -- 0
-int.to.str$ -- 95
-missing$ -- 95
-newline$ -- 475
-num.names$ -- 192
-pop$ -- 582
+int.to.str$ -- 98
+missing$ -- 97
+newline$ -- 490
+num.names$ -- 198
+pop$ -- 590
 preamble$ -- 1
-purify$ -- 428
+purify$ -- 438
 quote$ -- 0
-skip$ -- 1122
+skip$ -- 1143
 stack$ -- 0
-substring$ -- 2471
-swap$ -- 461
-text.length$ -- 50
+substring$ -- 2507
+swap$ -- 464
+text.length$ -- 51
 text.prefix$ -- 0
 top$ -- 0
-type$ -- 370
+type$ -- 382
 warning$ -- 8
-while$ -- 349
-width$ -- 97
-write$ -- 1062
+while$ -- 356
+width$ -- 100
+write$ -- 1095
 (There were 8 warnings)

diff --git a/THESE/these.lof b/THESE/these.lof
index 5e7bab4b603472ea5b5db3722c932c0eb5d327cc..6eae3f910aabe5bb87c671ff06d533b2759ae436 100644 (file)
--- a/THESE/these.lof
+++ b/THESE/these.lof
@@ -33,48 +33,85 @@
 \contentsline {subfigure}{\numberline {(c)}{\ignorespaces {$f=5$ et $t=2$, PSNR=29.0~dB MSSIM=0.39}}}{19}{figure.2.6}
 \contentsline {subfigure}{\numberline {(d)}{\ignorespaces {$f=5$ et $t=5$, PSNR=29.0~dB MSSIM=0.40}}}{19}{figure.2.6}
 \contentsline {figure}{\numberline {2.7}{\ignorespaces Filtrage par BM3D, PSNR=29.3~dB MSSIM=0.41}}{19}{figure.2.7}
-\contentsline {figure}{\numberline {2.8}{\ignorespaces Illustration pr\IeC {\'e}-chargement en m\IeC {\'e}moire partag\IeC {\'e}e mise en \oe uvre dans \cite {zheng2011performance} pour l'impl\IeC {\'e}mentation, entre autres, du filtre bilat\IeC {\'e}ral. a) en vert le bloc de threads associ\IeC {\'e} aux pixels centraux. b-e) les blocs de pixels successivement pr\IeC {\'e}-charg\IeC {\'e}s en m\IeC {\'e}moire partag\IeC {\'e}e. f) la configuration finale de la ROI en m\IeC {\'e}moire partag\IeC {\'e}e.}}{22}{figure.2.8}
-\contentsline {figure}{\numberline {2.9}{\ignorespaces Segmentation d'une image en niveaux de gris de 128 $\times $ 128 pixels par analyse simple d'histogramme. Colonne de gauche : image d'entr\IeC {\'e}e. Colonne centrale : histogramme des niveaux de gris. Colonne de droite : r\IeC {\'e}sultat de la segmentation.}}{24}{figure.2.9}
-\contentsline {subfigure}{\numberline {(a)}{\ignorespaces {Image initiale comportant deux zones : le fond et le cochon (la cible)}}}{24}{figure.2.9}
-\contentsline {subfigure}{\numberline {(b)}{\ignorespaces {Histogramme des niveaux de gris}}}{24}{figure.2.9}
-\contentsline {subfigure}{\numberline {(c)}{\ignorespaces {Image binaire repr\IeC {\'e}sentant la segmentation. Seuil estim\IeC {\'e} \IeC {\`a} 101 apr\IeC {\`e}s 4 it\IeC {\'e}rations.}}}{24}{figure.2.9}
-\contentsline {subfigure}{\numberline {(d)}{\ignorespaces {Image initiale bruit\IeC {\'e}e}}}{24}{figure.2.9}
-\contentsline {subfigure}{\numberline {(e)}{\ignorespaces {Histogramme des niveaux de gris}}}{24}{figure.2.9}
-\contentsline {subfigure}{\numberline {(f)}{\ignorespaces {Image binaire repr\IeC {\'e}sentant la segmentation. Seuil estim\IeC {\'e} \IeC {\`a} 99 apr\IeC {\`e}s 5 it\IeC {\'e}rations.}}}{24}{figure.2.9}
-\contentsline {figure}{\numberline {2.10}{\ignorespaces Segmentation d'une image en niveaux de gris de 128 $\times $ 128 pixels par simplification de graphe de type \textit {Normalized cut} pour un nombre $s$ de segments variant de 2 \IeC {\`a} 5.}}{26}{figure.2.10}
-\contentsline {subfigure}{\numberline {(a)}{\ignorespaces {$s = 2$}}}{26}{figure.2.10}
-\contentsline {subfigure}{\numberline {(b)}{\ignorespaces {$s = 3$}}}{26}{figure.2.10}
-\contentsline {subfigure}{\numberline {(c)}{\ignorespaces {$s = 4$}}}{26}{figure.2.10}
-\contentsline {subfigure}{\numberline {(d)}{\ignorespaces {$s = 5$}}}{26}{figure.2.10}
-\contentsline {figure}{\numberline {2.11}{\ignorespaces Segmentation d'une image en niveaux de gris de 128 $\times $ 128 pixels par algorithme \textit {k-means} pour un nombre $s$ de segments variant de 2 \IeC {\`a} 5. Chaque couleur est associ\IeC {\'e}e \IeC {\`a} un segment. Les couleurs sont choisies pour une meilleure visualisation des diff\IeC {\'e}rents segments.}}{27}{figure.2.11}
+\contentsline {figure}{\numberline {2.8}{\ignorespaces Performances relatives des filtres m\IeC {\'e}dians impl\IeC {\'e}ment\IeC {\'e}s sur GPU dans libJacket/ArrayFire, PCMF et BVM et ex\IeC {\'e}cut\IeC {\'e}s sur deux mod\IeC {\`e}le de g\IeC {\'e}n\IeC {\'e}rations diff\IeC {\'e}rentes.}}{21}{figure.2.8}
+\contentsline {subfigure}{\numberline {(a)}{\ignorespaces {Sur GPU GTX260. Courbe tir\IeC {\'e}e de \cite {5402362}}}}{21}{figure.2.8}
+\contentsline {subfigure}{\numberline {(b)}{\ignorespaces {Sur GPU C2075. Courbe tir\IeC {\'e}e de \cite {sanchez2013highly}}}}{21}{figure.2.8}
+\contentsline {figure}{\numberline {2.9}{\ignorespaces Illustration pr\IeC {\'e}-chargement en m\IeC {\'e}moire partag\IeC {\'e}e mise en \oe uvre dans \cite {zheng2011performance} pour l'impl\IeC {\'e}mentation, entre autres, du filtre bilat\IeC {\'e}ral. a) en vert le bloc de threads associ\IeC {\'e} aux pixels centraux. b-e) les blocs de pixels successivement pr\IeC {\'e}-charg\IeC {\'e}s en m\IeC {\'e}moire partag\IeC {\'e}e. f) la configuration finale de la ROI en m\IeC {\'e}moire partag\IeC {\'e}e.}}{22}{figure.2.9}
+\contentsline {figure}{\numberline {2.10}{\ignorespaces Segmentation d'une image en niveaux de gris de 128 $\times $ 128 pixels par analyse simple d'histogramme. Colonne de gauche : image d'entr\IeC {\'e}e. Colonne centrale : histogramme des niveaux de gris. Colonne de droite : r\IeC {\'e}sultat de la segmentation.}}{25}{figure.2.10}
+\contentsline {subfigure}{\numberline {(a)}{\ignorespaces {Image initiale comportant deux zones : le fond et le cochon (la cible)}}}{25}{figure.2.10}
+\contentsline {subfigure}{\numberline {(b)}{\ignorespaces {Histogramme des niveaux de gris}}}{25}{figure.2.10}
+\contentsline {subfigure}{\numberline {(c)}{\ignorespaces {Image binaire repr\IeC {\'e}sentant la segmentation. Seuil estim\IeC {\'e} \IeC {\`a} 101 apr\IeC {\`e}s 4 it\IeC {\'e}rations.}}}{25}{figure.2.10}
+\contentsline {subfigure}{\numberline {(d)}{\ignorespaces {Image initiale bruit\IeC {\'e}e}}}{25}{figure.2.10}
+\contentsline {subfigure}{\numberline {(e)}{\ignorespaces {Histogramme des niveaux de gris}}}{25}{figure.2.10}
+\contentsline {subfigure}{\numberline {(f)}{\ignorespaces {Image binaire repr\IeC {\'e}sentant la segmentation. Seuil estim\IeC {\'e} \IeC {\`a} 99 apr\IeC {\`e}s 5 it\IeC {\'e}rations.}}}{25}{figure.2.10}
+\contentsline {figure}{\numberline {2.11}{\ignorespaces Segmentation d'une image en niveaux de gris de 128 $\times $ 128 pixels par simplification de graphe de type \textit {Normalized cut} pour un nombre $s$ de segments variant de 2 \IeC {\`a} 5.}}{27}{figure.2.11}
 \contentsline {subfigure}{\numberline {(a)}{\ignorespaces {$s = 2$}}}{27}{figure.2.11}
 \contentsline {subfigure}{\numberline {(b)}{\ignorespaces {$s = 3$}}}{27}{figure.2.11}
 \contentsline {subfigure}{\numberline {(c)}{\ignorespaces {$s = 4$}}}{27}{figure.2.11}
 \contentsline {subfigure}{\numberline {(d)}{\ignorespaces {$s = 5$}}}{27}{figure.2.11}
-\contentsline {figure}{\numberline {2.12}{\ignorespaces Segmentation d'une image en niveaux de gris de 128 $\times $ 128 pixels par algorithme \textit {mean-shift} pour un rayon de voisinage $r$ de 100, 50, 35 et 25 pixels permettant d'obtenir un nombre $s$ de segments variant respectivement de 2 \IeC {\`a} 5. Le volume minimal admis pour un segment est fix\IeC {\'e} \IeC {\`a} 100 pixels. Chaque couleur est associ\IeC {\'e}e \IeC {\`a} un segment. Les couleurs sont choisies pour une meilleure visualisation des diff\IeC {\'e}rents segments.}}{28}{figure.2.12}
-\contentsline {subfigure}{\numberline {(a)}{\ignorespaces {$r=100 \Rightarrow s = 2$}}}{28}{figure.2.12}
-\contentsline {subfigure}{\numberline {(b)}{\ignorespaces {$r=50 \Rightarrow s = 3$}}}{28}{figure.2.12}
-\contentsline {subfigure}{\numberline {(c)}{\ignorespaces {$r=35 \Rightarrow s = 4$}}}{28}{figure.2.12}
-\contentsline {subfigure}{\numberline {(d)}{\ignorespaces {$r=25 \Rightarrow s = 5$}}}{28}{figure.2.12}
-\contentsline {figure}{\numberline {2.13}{\ignorespaces Segmentation d'une image en niveaux de gris de 128 $\times $ 128 pixels par algorithme dit du \textit {snake}, dans sa version originale. Les param\IeC {\`e}tres d'\IeC {\'e}lasticti\IeC {\'e}, de raideur et d'attraction ont \IeC {\'e}t\IeC {\'e} fix\IeC {\'e}s respectivement aux valeurs 5, 0.1 et 5. }}{30}{figure.2.13}
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-\contentsline {figure}{\numberline {2.14}{\ignorespaces \IeC {\'E}volution du nombre de pixels actifs pour les it\IeC {\'e}ration successives de l'impl\IeC {\'e}mentation de l'algorithme push-relabel de \cite {graphcutscuda}. Les petites images montrent la localisation des pixels actifs apr\IeC {\`e}s chaque it\IeC {\'e}ration, en blanc.}}{33}{figure.2.14}
-\contentsline {figure}{\numberline {2.15}{\ignorespaces Segmentation d'une image couleur de 512$\times $512 pixels par l'impl\IeC {\'e}mentation GPU quick-shift de \cite {fulkerson2012really}.}}{34}{figure.2.15}
-\contentsline {subfigure}{\numberline {(a)}{\ignorespaces {Image originale}}}{34}{figure.2.15}
-\contentsline {subfigure}{\numberline {(b)}{\ignorespaces {$\tau =10$ et $\sigma =2$}}}{34}{figure.2.15}
-\contentsline {subfigure}{\numberline {(c)}{\ignorespaces {$\tau =10$ et $\sigma =10$}}}{34}{figure.2.15}
-\contentsline {subfigure}{\numberline {(d)}{\ignorespaces {$\tau =20$ et $\sigma =10$}}}{34}{figure.2.15}
-\contentsline {figure}{\numberline {2.16}{\ignorespaces Segmentation d'une image couleur de 2256$\times $3008 pixels.}}{35}{figure.2.16}
+\contentsline {figure}{\numberline {2.12}{\ignorespaces Segmentation d'une image en niveaux de gris de 128 $\times $ 128 pixels par algorithme \textit {k-means} pour un nombre $s$ de segments variant de 2 \IeC {\`a} 5. Chaque couleur est associ\IeC {\'e}e \IeC {\`a} un segment. Les couleurs sont choisies pour une meilleure visualisation des diff\IeC {\'e}rents segments.}}{28}{figure.2.12}
+\contentsline {subfigure}{\numberline {(a)}{\ignorespaces {$s = 2$}}}{28}{figure.2.12}
+\contentsline {subfigure}{\numberline {(b)}{\ignorespaces {$s = 3$}}}{28}{figure.2.12}
+\contentsline {subfigure}{\numberline {(c)}{\ignorespaces {$s = 4$}}}{28}{figure.2.12}
+\contentsline {subfigure}{\numberline {(d)}{\ignorespaces {$s = 5$}}}{28}{figure.2.12}
+\contentsline {figure}{\numberline {2.13}{\ignorespaces Segmentation d'une image en niveaux de gris de 128 $\times $ 128 pixels par algorithme \textit {mean-shift} pour un rayon de voisinage $r$ de 100, 50, 35 et 25 pixels permettant d'obtenir un nombre $s$ de segments variant respectivement de 2 \IeC {\`a} 5. Le volume minimal admis pour un segment est fix\IeC {\'e} \IeC {\`a} 100 pixels. Chaque couleur est associ\IeC {\'e}e \IeC {\`a} un segment. Les couleurs sont choisies pour une meilleure visualisation des diff\IeC {\'e}rents segments.}}{29}{figure.2.13}
+\contentsline {subfigure}{\numberline {(a)}{\ignorespaces {$r=100 \Rightarrow s = 2$}}}{29}{figure.2.13}
+\contentsline {subfigure}{\numberline {(b)}{\ignorespaces {$r=50 \Rightarrow s = 3$}}}{29}{figure.2.13}
+\contentsline {subfigure}{\numberline {(c)}{\ignorespaces {$r=35 \Rightarrow s = 4$}}}{29}{figure.2.13}
+\contentsline {subfigure}{\numberline {(d)}{\ignorespaces {$r=25 \Rightarrow s = 5$}}}{29}{figure.2.13}
+\contentsline {figure}{\numberline {2.14}{\ignorespaces Segmentation d'une image en niveaux de gris de 128 $\times $ 128 pixels par algorithme dit du \textit {snake}, dans sa version originale. Les param\IeC {\`e}tres d'\IeC {\'e}lasticit\IeC {\'e}, de raideur et d'attraction ont \IeC {\'e}t\IeC {\'e} fix\IeC {\'e}s respectivement aux valeurs 5, 0.1 et 5. }}{30}{figure.2.14}
+\contentsline {subfigure}{\numberline {(a)}{\ignorespaces {Les \IeC {\'e}tats initial et suivant chacune des trois premi\IeC {\`e}res it\IeC {\'e}rations}}}{30}{figure.2.14}
+\contentsline {subfigure}{\numberline {(b)}{\ignorespaces {L'\IeC {\'e}tat du contour apr\IeC {\`e}s la septi\IeC {\`e}me it\IeC {\'e}ration}}}{30}{figure.2.14}
+\contentsline {subfigure}{\numberline {(c)}{\ignorespaces {L'\IeC {\'e}tat du contour apr\IeC {\`e}s la dixi\IeC {\`e}me it\IeC {\'e}ration}}}{30}{figure.2.14}
+\contentsline {subfigure}{\numberline {(d)}{\ignorespaces {L'\IeC {\'e}tat du contour apr\IeC {\`e}s la centi\IeC {\`e}me it\IeC {\'e}ration. C'est le contour final.}}}{30}{figure.2.14}
+\contentsline {figure}{\numberline {2.15}{\ignorespaces \IeC {\'E}volution du nombre de pixels actifs pour les it\IeC {\'e}ration successives de l'impl\IeC {\'e}mentation de l'algorithme push-relabel de \cite {graphcutscuda}. Les petites images montrent la localisation des pixels actifs apr\IeC {\`e}s chaque it\IeC {\'e}ration, en blanc.}}{33}{figure.2.15}
+\contentsline {figure}{\numberline {2.16}{\ignorespaces Segmentation d'une image couleur de 512$\times $512 pixels par l'impl\IeC {\'e}mentation GPU quick-shift de \cite {fulkerson2012really}.}}{35}{figure.2.16}
 \contentsline {subfigure}{\numberline {(a)}{\ignorespaces {Image originale}}}{35}{figure.2.16}
-\contentsline {subfigure}{\numberline {(b)}{\ignorespaces {Image segment\IeC {\'e}e par mean-shift standard}}}{35}{figure.2.16}
-\contentsline {subfigure}{\numberline {(c)}{\ignorespaces {Image segment\IeC {\'e}e par mean-shift kd-tree}}}{35}{figure.2.16}
-\contentsline {figure}{\numberline {2.17}{\ignorespaces Segmentation d'images issues d'examens IRM par la m\IeC {\'e}thode des level set \IeC {\`a} bande \IeC {\'e}troite.}}{36}{figure.2.17}
-\contentsline {subfigure}{\numberline {(a)}{\ignorespaces {Cerveau 256$\times $256$\times $256 en 7~s}}}{36}{figure.2.17}
-\contentsline {subfigure}{\numberline {(b)}{\ignorespaces {Reins et aorte, 256$\times $256$\times $272 en 16~s}}}{36}{figure.2.17}
-\contentsline {figure}{\numberline {2.18}{\ignorespaces Extraction de contour par la version GPU de l'algorithme gPb. Les images sont issues de la base BSDS \cite {martin2001database}}}{38}{figure.2.18}
+\contentsline {subfigure}{\numberline {(b)}{\ignorespaces {$\tau =10$ et $\sigma =2$}}}{35}{figure.2.16}
+\contentsline {subfigure}{\numberline {(c)}{\ignorespaces {$\tau =10$ et $\sigma =10$}}}{35}{figure.2.16}
+\contentsline {subfigure}{\numberline {(d)}{\ignorespaces {$\tau =20$ et $\sigma =10$}}}{35}{figure.2.16}
+\contentsline {figure}{\numberline {2.17}{\ignorespaces Segmentation d'une image couleur de 2256$\times $3008 pixels.}}{35}{figure.2.17}
+\contentsline {subfigure}{\numberline {(a)}{\ignorespaces {Image originale}}}{35}{figure.2.17}
+\contentsline {subfigure}{\numberline {(b)}{\ignorespaces {Image segment\IeC {\'e}e par mean-shift standard}}}{35}{figure.2.17}
+\contentsline {subfigure}{\numberline {(c)}{\ignorespaces {Image segment\IeC {\'e}e par mean-shift kd-tree}}}{35}{figure.2.17}
+\contentsline {figure}{\numberline {2.18}{\ignorespaces Segmentation d'images issues d'examens IRM par la m\IeC {\'e}thode des level set \IeC {\`a} bande \IeC {\'e}troite.}}{37}{figure.2.18}
+\contentsline {subfigure}{\numberline {(a)}{\ignorespaces {Cerveau 256$\times $256$\times $256 en 7~s}}}{37}{figure.2.18}
+\contentsline {subfigure}{\numberline {(b)}{\ignorespaces {Reins et aorte, 256$\times $256$\times $272 en 16~s}}}{37}{figure.2.18}
+\contentsline {figure}{\numberline {2.19}{\ignorespaces Segmentation d'une image d'\IeC {\'e}paule en 1024$^2$ pixels issue d'un examen IRM par l'impl\IeC {\'e}mentation du snake GVF de \cite {snakegvf06}. Le contour est repr\IeC {\'e}sent\IeC {\'e} en rougeet le contour final est obtenu en 11~s. }}{37}{figure.2.19}
+\contentsline {subfigure}{\numberline {(a)}{\ignorespaces {Contour initial}}}{37}{figure.2.19}
+\contentsline {subfigure}{\numberline {(b)}{\ignorespaces {Contour final}}}{37}{figure.2.19}
+\contentsline {figure}{\numberline {2.20}{\ignorespaces Extraction de contour par la version GPU de l'algorithme gPb. Les images sont issues de la base BSDS \cite {martin2001database}}}{38}{figure.2.20}
 \addvspace {10\p@ }
+\contentsline {figure}{\numberline {3.1}{\ignorespaces \IeC {\`A} gauche : d\IeC {\'e}termination des vecteurs $f_{in}$ et $f_{out}$. \IeC {\`A} droite : code de Freeman d'un vecteur en fonction de sa direction, l'origine \IeC {\'e}tant suppos\IeC {\'e}e au pixel central, en noir. }}{43}{figure.3.1}
+\contentsline {figure}{\numberline {3.2}{\ignorespaces \IeC {\'E}volution du contour lors de la segmentation d'une image de 512$^2$ pixels. La convergence est obtenue \IeC {\`a} l'it\IeC {\'e}ration 14 apr\IeC {\`e}s 44~ms pour un total de 256 n\oe uds.}}{46}{figure.3.2}
+\contentsline {subfigure}{\numberline {(a)}{\ignorespaces {Initialisation : 4 n\oe uds}}}{46}{figure.3.2}
+\contentsline {subfigure}{\numberline {(b)}{\ignorespaces {It\IeC {\'e}ration 1 : 8 n\oe uds 3~ms}}}{46}{figure.3.2}
+\contentsline {subfigure}{\numberline {(c)}{\ignorespaces {It\IeC {\'e}ration 2 : 16 n\oe uds 1~ms}}}{46}{figure.3.2}
+\contentsline {subfigure}{\numberline {(d)}{\ignorespaces {It\IeC {\'e}ration 3, 32 n\oe uds 1~ms}}}{46}{figure.3.2}
+\contentsline {subfigure}{\numberline {(e)}{\ignorespaces {It\IeC {\'e}ration 7 : 223 n\oe uds 3~ms}}}{46}{figure.3.2}
+\contentsline {subfigure}{\numberline {(f)}{\ignorespaces {It\IeC {\'e}ration 10 : 244 n\oe uds 3~ms}}}{46}{figure.3.2}
+\contentsline {subfigure}{\numberline {(g)}{\ignorespaces {It\IeC {\'e}ration 13 : 256 n\oe uds 3~ms}}}{46}{figure.3.2}
+\contentsline {subfigure}{\numberline {(h)}{\ignorespaces {It\IeC {\'e}ration 14 : 256 n\oe uds 3~ms}}}{46}{figure.3.2}
+\contentsline {figure}{\numberline {3.3}{\ignorespaces Influence du contour initial sur la segmentation. Le contour final 1 est celui de la figure \ref {fig-snakecpu-cochon512}.}}{46}{figure.3.3}
+\contentsline {subfigure}{\numberline {(a)}{\ignorespaces {Initialisation 2 }}}{46}{figure.3.3}
+\contentsline {subfigure}{\numberline {(b)}{\ignorespaces {Contour final 2 : 273 n\oe uds 87~ms}}}{46}{figure.3.3}
+\contentsline {subfigure}{\numberline {(c)}{\ignorespaces {Contour final 1 : 256 n\oe uds 44~ms}}}{46}{figure.3.3}
+\contentsline {figure}{\numberline {3.4}{\ignorespaces Segmentation de l'image de test en 4000$^2$ pixels.}}{47}{figure.3.4}
+\contentsline {subfigure}{\numberline {(a)}{\ignorespaces {$d_{max}=16$ et $l_{min}=8$, 1246 n\oe uds en 1.3~s}}}{47}{figure.3.4}
+\contentsline {subfigure}{\numberline {(b)}{\ignorespaces {$d_{max}=128$ et $l_{min}=32$, 447 n\oe uds en 0.7~s}}}{47}{figure.3.4}
+\contentsline {figure}{\numberline {3.5}{\ignorespaces Segmentation de l'image de test en 4000$^2$ pixels avec une cible de petite taille. Le contour initial est celui utilis\IeC {\'e} \IeC {\`a} la figure \ref {fig-snakecpu-cochon4k}.}}{47}{figure.3.5}
+\contentsline {figure}{\numberline {3.6}{\ignorespaces \IeC {\'E}volution du co\IeC {\^u}t relatif des trois fonctions les plus consommatrices en temps de calcul en fonction de la taille de l'image \IeC {\`a} traiter.}}{48}{figure.3.6}
+\contentsline {figure}{\numberline {3.7}{\ignorespaces Calcul des images cumul\IeC {\'e}es $S_x$ et $S_x^2$ en trois \IeC {\'e}tapes successives. a) cumul partiel bloc par bloc et m\IeC {\'e}morisation de la somme de chaque bloc. b) cumul sur le vecteur des sommes partielles. c) ajout des sommes partielles \IeC {\`a} chaque \IeC {\'e}l\IeC {\'e}ment des blocs cumul\IeC {\'e}s.}}{50}{figure.3.7}
+\contentsline {subfigure}{\numberline {(a)}{\ignorespaces {D\IeC {\'e}tail des op\IeC {\'e}rations effectu\IeC {\'e}es par le \textit {kernel} \texttt {compute\_block\_prefixes()}. La valeur $bs$ correspond au nombre de pixels de chaque bloc, qui est aussi le nombre de threads ex\IeC {\'e}cut\IeC {\'e} par chaque bloc de la grille de calcul.}}}{50}{figure.3.7}
+\contentsline {subfigure}{\numberline {(b)}{\ignorespaces {D\IeC {\'e}tail des op\IeC {\'e}rations effectu\IeC {\'e}es par le \textit {kernel} \texttt {scan\_blocksums()}.}}}{50}{figure.3.7}
+\contentsline {subfigure}{\numberline {(c)}{\ignorespaces {D\IeC {\'e}tail des op\IeC {\'e}rations effectu\IeC {\'e}es par le \textit {kernel} \texttt {add\_sums2prefixes()}.}}}{50}{figure.3.7}
+\contentsline {figure}{\numberline {3.8}{\ignorespaces Structuration des donn\IeC {\'e}es en m\IeC {\'e}moire du GPU pour l'\IeC {\'e}valuation en parall\IeC {\`e}le de l'ensemble des \IeC {\'e}volutions possibles du contour.}}{51}{figure.3.8}
+\contentsline {figure}{\numberline {3.9}{\ignorespaces Comparaison des cycles de d\IeC {\'e}placement des n\oe uds. Ligne du haut : version s\IeC {\'e}quentielle. Ligne du bas : version parall\IeC {\`e}le. Les segments en rouge sont des segments du contour non \IeC {\'e}valu\IeC {\'e}s, alors que ceux en pointill\IeC {\'e}s sont les paires ayant re\IeC {\c c}u les meilleures \IeC {\'e}valuations parmi les 8 d\IeC {\'e}placements possibles des n\oe uds correspondant.}}{52}{figure.3.9}
+\contentsline {subfigure}{\numberline {(a)}{\ignorespaces {Contour de r\IeC {\'e}f\IeC {\'e}rence.}}}{52}{figure.3.9}
+\contentsline {subfigure}{\numberline {(b)}{\ignorespaces {D\IeC {\'e}placement du n\oe ud $N_1$. Le crit\IeC {\`e}re est am\IeC {\'e}lior\IeC {\'e}.}}}{52}{figure.3.9}
+\contentsline {subfigure}{\numberline {(c)}{\ignorespaces {D\IeC {\'e}placement du n\oe ud $N_2$. Le crit\IeC {\`e}re est am\IeC {\'e}lior\IeC {\'e}.}}}{52}{figure.3.9}
+\contentsline {subfigure}{\numberline {(d)}{\ignorespaces {D\IeC {\'e}placement en parall\IeC {\`e}le de tous les n\oe uds. Les segments du contour n'ont pas \IeC {\'e}t\IeC {\'e} \IeC {\'e}valu\IeC {\'e}s. On ne peut pas dire, a priori si le crit\IeC {\`e}re est am\IeC {\'e}lior\IeC {\'e}.}}}{52}{figure.3.9}
+\contentsline {subfigure}{\numberline {(e)}{\ignorespaces {D\IeC {\'e}placement en parall\IeC {\`e}le des n\oe uds impairs. Le crit\IeC {\`e}re est am\IeC {\'e}lior\IeC {\'e}.}}}{52}{figure.3.9}
+\contentsline {subfigure}{\numberline {(f)}{\ignorespaces {D\IeC {\'e}placement en parall\IeC {\`e}le des n\oe uds pairs. Un seul segment n'a pas \IeC {\'e}t\IeC {\'e} \IeC {\'e}valu\IeC {\'e}.}}}{52}{figure.3.9}
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diff --git a/THESE/these.lot b/THESE/these.lot
index 9eb1143f81a75b5aff110a7f90dc3ad4700921b9..d46ffb54b05842870187b2db57278160a6f31b62 100644 (file)
--- a/THESE/these.lot
+++ b/THESE/these.lot
@@ -2,5 +2,8 @@
 \addvspace {10\p@ }
 \addvspace {10\p@ }
 \addvspace {10\p@ }
+\contentsline {table}{\numberline {3.1}{\ignorespaces Valeur du coefficient $C(i,j)$ en fonction des valeurs des codes de Freeman des vecteurs $f_{in}$ et $f_{out}$.}}{44}{table.3.1}
+\contentsline {table}{\numberline {3.2}{\ignorespaces Performances (en secondes) de la segmentation par snake polygonal sur CPU en fonction de la taille de l'image \IeC {\`a} traiter. Le temps sont obtenus avec la m\IeC {\^e}me image de test dilat\IeC {\'e}e et bruit\IeC {\'e}e et un contour initial carr\IeC {\'e} dont la distance aux bords est proportionnelle \IeC {\`a} la taille de l'image. Seule l'image en 15~MP a pu \IeC {\^e}tre trait\IeC {\'e}e par une impl\IeC {\'e}mentation utilisant SSE2.}}{55}{table.3.2}
+\contentsline {table}{\numberline {3.3}{\ignorespaces Acc\IeC {\'e}l\IeC {\'e}ration constat\IeC {\'e}e, pour le calcul des images cumul\IeC {\'e}es, de l'impl\IeC {\'e}mentation GPU par rapport \IeC {\`a} l'impl\IeC {\'e}mentation CPU de r\IeC {\'e}f\IeC {\'e}rence.}}{55}{table.3.3}
 \addvspace {10\p@ }
 \addvspace {10\p@ }

diff --git a/THESE/these.out b/THESE/these.out
index 69afe0e88aa22308f46e9e041d145aa705f964ed..a1c6d6c92e201f6844e44f91eb94eb25fa14cc02 100644 (file)
--- a/THESE/these.out
+++ b/THESE/these.out
@@ -30,9 +30,17 @@
 \BOOKMARK [2][]{subsection.2.6.3}{2.6.3 K-means, mean-shift et apparent\351s}{section.2.6}
 \BOOKMARK [2][]{subsection.2.6.4}{2.6.4 Snakes et Level set}{section.2.6}
 \BOOKMARK [2][]{subsection.2.6.5}{2.6.5 Algorithmes hybrides}{section.2.6}
-\BOOKMARK [0][]{chapter.3}{3 La segmentation orient\351e r\351gions dans les images bruit\351es}{}
-\BOOKMARK [1][]{section.3.1}{3.1 Pr\351sentation - existant}{chapter.3}
-\BOOKMARK [1][]{section.3.2}{3.2 La parall\350lisation du snake polygonal}{chapter.3}
+\BOOKMARK [1][]{section.2.7}{2.7 Conclusion}{chapter.2}
+\BOOKMARK [0][]{chapter.3}{3 La segmentation par snake polygonal orient\351 r\351gions}{}
+\BOOKMARK [1][]{section.3.1}{3.1 Pr\351sentation de l'algorithme}{chapter.3}
+\BOOKMARK [2][]{subsection.3.1.1}{3.1.1 Formulation}{section.3.1}
+\BOOKMARK [2][]{subsection.3.1.2}{3.1.2 Optimisation des calculs}{section.3.1}
+\BOOKMARK [2][]{subsection.3.1.3}{3.1.3 Impl\351mentation s\351quentielle}{section.3.1}
+\BOOKMARK [2][]{subsection.3.1.4}{3.1.4 Performances}{section.3.1}
+\BOOKMARK [1][]{section.3.2}{3.2 Impl\351mentation parall\350le GPU du snake polygonal}{chapter.3}
+\BOOKMARK [2][]{subsection.3.2.1}{3.2.1 pr\351-calculs des images cumul\351es}{section.3.2}
+\BOOKMARK [2][]{subsection.3.2.2}{3.2.2 Calcul des contribution des segments}{section.3.2}
+\BOOKMARK [3][]{subsubsection.3.2.2.1}{3.2.2.1 Cas particulier des segments dont la pente k v\351rifie |k|1}{subsection.3.2.2}
 \BOOKMARK [0][]{chapter.4}{4 Le filtrage des images sur GPU}{}
 \BOOKMARK [1][]{section.4.1}{4.1 Algorithme de r\351duction de bruit par recherche des lignes de niveaux}{chapter.4}
 \BOOKMARK [1][]{section.4.2}{4.2 Filtre m\351dian}{chapter.4}

diff --git a/THESE/these.pdf b/THESE/these.pdf
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diff --git a/THESE/these.tex b/THESE/these.tex
index c0b8ada819f4d7d53e35139ca8f8c706043d8462..21ae898fb677223d508fd8b5892fb5710bcbf6f6 100644 (file)
--- a/THESE/these.tex
+++ b/THESE/these.tex
@@ -1,10 +1,13 @@
 \documentclass[french]{spimufcphdthesis}
 \usepackage[utf8]{inputenc}
-           
+            
 \usepackage{graphicx}
 \usepackage{color, xcolor}
 \usepackage[ruled,lined,linesnumbered]{algorithm2e}
-   
+  
+\usepackage{amsmath}
+\usepackage{booktabs}   
+
 %%--------------------
 %% Set the title, subtitle, defense date, and
 %% the registration number of the PhD thesis.
@@ -14,7 +17,7 @@
 %% The third mandatory parameter is the reference number given by
 %% the University Library after the PhD defense.
 \declarethesis[Sous-titre]{Titre}{17 septembre 2012}{XXX}
-   
+      
 %%--------------------
 %% Set the author of the PhD thesis
 \addauthor[gilles.perrot@univ-fcomte.fr]{Gilles}{Perrot}
@@ -25,7 +28,7 @@
 \addjury{Incroyable}{Hulk}{Rapporteur}{Professeur à l'Université de Gotham City \\ Commentaire secondaire}
 \addjury{Super}{Man}{Examinateur}{Professeur à l'Université de Gotham City}
 \addjury{Bat}{Man}{Directeur de thèse}{Professeur à l'Université de Gotham City}
- 
+   
 %%--------------------
 %% Set the laboratory where the thesis was made
 %\addlaboratory{Laboratoire Waynes Industry}
@@ -33,7 +36,7 @@
 %%-------------------- 
 %% Set the English abstract
 \thesisabstract[english]{This is the abstract in English}
-      
+       
 %%--------------------   
 %% Set the English keywords. They only appear if
 %% there is an English abstract
@@ -70,11 +73,14 @@
 %\cfoot{} 
 %% Right footer
 %\rfoot{}
- 
+
+
+%\declareupmtheorem{rq}{Remarque}{Liste des remarques}
+   
 \begin{document}
     
 \tableofcontents
-                                                                   
+                                                                         
 %-------------------- 
 % The content of the PhD thesis
 % objectifs, cadre  
@@ -89,14 +95,14 @@
 % segmentation
 % filtrage : réduction de bruit, convolutions 
 % tracking         
-% pattern recognition
-\chapter{La segmentation orientée régions dans les images bruitées}
-\section{Présentation - existant}  
+% pattern recognition 
+\chapter{La segmentation par snake polygonal orienté régions}
+\input{Chapters/chapter3/chapter3.tex}     
 % segmentation, intérêt, domaines
-% notations, formules 
-% snake polygonal
-\section{La parallèlisation du snake polygonal}
-% détails  
+% notations, formules   
+% snake polygonal 
+%                
+% détails        
 % résultats, images
 % --> multitarget
  
@@ -114,7 +120,7 @@
 % intégrer logiciels dans sections
 \chapter{Conclusion générale}
 % et perspectives 
- 
+   
 %--------------------
 % Bibliography, indexes...
 \bibliographystyle{plain}

diff --git a/THESE/these.toc b/THESE/these.toc
index 96de7f57eeb2ba736281fd2e12165a91f4300d44..70c0726308f7855e8f692fe6b713b2df6c567f09 100644 (file)
--- a/THESE/these.toc
+++ b/THESE/these.toc
@@ -1,41 +1,49 @@
 \select@language {french}
 \contentsline {chapter}{\numberline {1}Introduction}{7}{chapter.1}
 \contentsline {chapter}{\numberline {2}Le traitement des images bruit\IeC {\'e}es}{11}{chapter.2}
-\contentsline {section}{\numberline {2.1}Mod\IeC {\`e}le d'image bruit\IeC {\'e}e}{11}{section.2.1}
+\contentsline {section}{\numberline {2.1}Mod\IeC {\`e}le d'image bruit\IeC {\'e}e}{12}{section.2.1}
 \contentsline {section}{\numberline {2.2}Mod\IeC {\`e}les de bruit}{12}{section.2.2}
 \contentsline {subsection}{\numberline {2.2.1}Le bruit gaussien}{12}{subsection.2.2.1}
 \contentsline {subsection}{\numberline {2.2.2}Le speckle}{12}{subsection.2.2.2}
 \contentsline {subsection}{\numberline {2.2.3}Le bruit ``sel et poivre''}{13}{subsection.2.2.3}
 \contentsline {subsection}{\numberline {2.2.4}Le bruit de Poisson}{13}{subsection.2.2.4}
 \contentsline {section}{\numberline {2.3}Les techniques de r\IeC {\'e}duction de bruit}{13}{section.2.3}
-\contentsline {subsection}{\numberline {2.3.1}Les op\IeC {\'e}rateurs de base}{14}{subsection.2.3.1}
-\contentsline {subsubsection}{\numberline {2.3.1.1}Le filtre de convolution}{14}{subsubsection.2.3.1.1}
+\contentsline {subsection}{\numberline {2.3.1}Les op\IeC {\'e}rateurs de base}{15}{subsection.2.3.1}
+\contentsline {subsubsection}{\numberline {2.3.1.1}Le filtre de convolution}{15}{subsubsection.2.3.1.1}
 \contentsline {subsubsection}{\numberline {2.3.1.2}Le filtre m\IeC {\'e}dian}{15}{subsubsection.2.3.1.2}
 \contentsline {subsubsection}{\numberline {2.3.1.3}Le filtre bilat\IeC {\'e}ral}{16}{subsubsection.2.3.1.3}
 \contentsline {subsubsection}{\numberline {2.3.1.4}Les algorithmes de filtrage par dictionnaire}{18}{subsubsection.2.3.1.4}
 \contentsline {subsection}{\numberline {2.3.2}Les algorithmes de filtrage par patches}{18}{subsection.2.3.2}
-\contentsline {section}{\numberline {2.4}Les impl\IeC {\'e}mentations GPU des algorithmes de filtrage}{19}{section.2.4}
+\contentsline {section}{\numberline {2.4}Les impl\IeC {\'e}mentations GPU des algorithmes de filtrage}{20}{section.2.4}
 \contentsline {subsection}{\numberline {2.4.1}Le filtrage par convolution}{20}{subsection.2.4.1}
 \contentsline {subsection}{\numberline {2.4.2}Le filtre m\IeC {\'e}dian}{20}{subsection.2.4.2}
 \contentsline {subsection}{\numberline {2.4.3}Le filtre bilat\IeC {\'e}ral}{21}{subsection.2.4.3}
-\contentsline {subsection}{\numberline {2.4.4}Les filtres par patches}{22}{subsection.2.4.4}
+\contentsline {subsection}{\numberline {2.4.4}Les filtres par patches}{23}{subsection.2.4.4}
 \contentsline {section}{\numberline {2.5}Les techniques de segmentation}{23}{section.2.5}
-\contentsline {subsection}{\numberline {2.5.1}Analyse d'histogramme}{23}{subsection.2.5.1}
-\contentsline {subsection}{\numberline {2.5.2}Partitionnement de graphe}{25}{subsection.2.5.2}
-\contentsline {subsection}{\numberline {2.5.3}kernel-means, mean-shift et apparent\IeC {\'e}s}{26}{subsection.2.5.3}
-\contentsline {subsection}{\numberline {2.5.4}Les contours actifs, ou \textit {snakes}}{28}{subsection.2.5.4}
-\contentsline {subsection}{\numberline {2.5.5}M\IeC {\'e}thodes hybrides}{30}{subsection.2.5.5}
+\contentsline {subsection}{\numberline {2.5.1}Analyse d'histogramme}{24}{subsection.2.5.1}
+\contentsline {subsection}{\numberline {2.5.2}Partitionnement de graphe}{26}{subsection.2.5.2}
+\contentsline {subsection}{\numberline {2.5.3}kernel-means, mean-shift et apparent\IeC {\'e}s}{27}{subsection.2.5.3}
+\contentsline {subsection}{\numberline {2.5.4}Les contours actifs, ou \textit {snakes}}{29}{subsection.2.5.4}
+\contentsline {subsection}{\numberline {2.5.5}M\IeC {\'e}thodes hybrides}{31}{subsection.2.5.5}
 \contentsline {section}{\numberline {2.6}Les impl\IeC {\'e}mentations GPU des techniques de segmentation}{31}{section.2.6}
-\contentsline {subsection}{\numberline {2.6.1}Calcul d'histogramme}{31}{subsection.2.6.1}
+\contentsline {subsection}{\numberline {2.6.1}Calcul d'histogramme}{32}{subsection.2.6.1}
 \contentsline {subsection}{\numberline {2.6.2}Partitionnement de graphe}{32}{subsection.2.6.2}
-\contentsline {subsection}{\numberline {2.6.3}K-means, mean-shift et apparent\IeC {\'e}s}{32}{subsection.2.6.3}
-\contentsline {subsection}{\numberline {2.6.4}Snakes et Level set}{34}{subsection.2.6.4}
-\contentsline {subsection}{\numberline {2.6.5}Algorithmes hybrides}{37}{subsection.2.6.5}
-\contentsline {chapter}{\numberline {3}La segmentation orient\IeC {\'e}e r\IeC {\'e}gions dans les images bruit\IeC {\'e}es}{39}{chapter.3}
-\contentsline {section}{\numberline {3.1}Pr\IeC {\'e}sentation - existant}{39}{section.3.1}
-\contentsline {section}{\numberline {3.2}La parall\IeC {\`e}lisation du snake polygonal}{39}{section.3.2}
-\contentsline {chapter}{\numberline {4}Le filtrage des images sur GPU}{41}{chapter.4}
-\contentsline {section}{\numberline {4.1}Algorithme de r\IeC {\'e}duction de bruit par recherche des lignes de niveaux}{41}{section.4.1}
-\contentsline {section}{\numberline {4.2}Filtre m\IeC {\'e}dian}{41}{section.4.2}
-\contentsline {section}{\numberline {4.3}Filtres de convolution}{41}{section.4.3}
-\contentsline {chapter}{\numberline {5}Conclusion g\IeC {\'e}n\IeC {\'e}rale}{43}{chapter.5}
+\contentsline {subsection}{\numberline {2.6.3}K-means, mean-shift et apparent\IeC {\'e}s}{33}{subsection.2.6.3}
+\contentsline {subsection}{\numberline {2.6.4}Snakes et Level set}{35}{subsection.2.6.4}
+\contentsline {subsection}{\numberline {2.6.5}Algorithmes hybrides}{38}{subsection.2.6.5}
+\contentsline {section}{\numberline {2.7}Conclusion}{39}{section.2.7}
+\contentsline {chapter}{\numberline {3}La segmentation par snake polygonal orient\IeC {\'e} r\IeC {\'e}gions}{41}{chapter.3}
+\contentsline {section}{\numberline {3.1}Pr\IeC {\'e}sentation de l'algorithme}{41}{section.3.1}
+\contentsline {subsection}{\numberline {3.1.1}Formulation}{41}{subsection.3.1.1}
+\contentsline {subsection}{\numberline {3.1.2}Optimisation des calculs}{42}{subsection.3.1.2}
+\contentsline {subsection}{\numberline {3.1.3}Impl\IeC {\'e}mentation s\IeC {\'e}quentielle}{44}{subsection.3.1.3}
+\contentsline {subsection}{\numberline {3.1.4}Performances}{45}{subsection.3.1.4}
+\contentsline {section}{\numberline {3.2}Impl\IeC {\'e}mentation parall\IeC {\`e}le GPU du snake polygonal}{48}{section.3.2}
+\contentsline {subsection}{\numberline {3.2.1}pr\IeC {\'e}-calculs des images cumul\IeC {\'e}es}{48}{subsection.3.2.1}
+\contentsline {subsection}{\numberline {3.2.2}Calcul des contribution des segments}{49}{subsection.3.2.2}
+\contentsline {subsubsection}{\numberline {3.2.2.1}Cas particulier des segments dont la pente $k$ v\IeC {\'e}rifie $|k|\leq 1$}{53}{subsubsection.3.2.2.1}
+\contentsline {chapter}{\numberline {4}Le filtrage des images sur GPU}{57}{chapter.4}
+\contentsline {section}{\numberline {4.1}Algorithme de r\IeC {\'e}duction de bruit par recherche des lignes de niveaux}{57}{section.4.1}
+\contentsline {section}{\numberline {4.2}Filtre m\IeC {\'e}dian}{57}{section.4.2}
+\contentsline {section}{\numberline {4.3}Filtres de convolution}{57}{section.4.3}
+\contentsline {chapter}{\numberline {5}Conclusion g\IeC {\'e}n\IeC {\'e}rale}{59}{chapter.5}