X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/hdrcouchot.git/blobdiff_plain/322d0e3286e2bc9cae4178aa7488b5631c57a8ba..d33e664452e3655370cbe069e3f6fbd16842c818:/stegoyousra.tex?ds=inline

diff --git a/stegoyousra.tex b/stegoyousra.tex
index dc17a7a..d931298 100644
--- a/stegoyousra.tex
+++ b/stegoyousra.tex
@@ -1,69 +1,65 @@
 La plupart des schÃ©mas de stÃ©ganographie sont conÃ§us de sorte  Ã  minimiser une 
-fonction de distortion. Dans les exemples du chapitre prÃ©cÃ©dent, 
-ces fonctions de distortion sont construites dans l'obejctif de prÃ©server 
-les caractÃ©ristiques de l'images. 
+fonction de distorsion. Dans les exemples du chapitre prÃ©cÃ©dent, 
+ces fonctions de distorsion sont construites dans l'objectif de prÃ©server 
+les caractÃ©ristiques de l'image. 
 On comprend aisÃ©ment que dans des rÃ©gions uniformes ou sur des bords clairement dÃ©finis,
 une modification mÃªme mineure de l'image est facilement dÃ©tectable.
-Au contraire dans les textures, le bruit ou les rÃ©gions chaotiques 
+Au contraire les textures, le bruit ou les rÃ©gions chaotiques 
 sont  difficiles Ã  modÃ©liser. Les caractÃ©ristiques des images 
 dont ces zones ont Ã©tÃ© modifiÃ©es sont ainsi similaires Ã  celles
 des images initiales.
 
 Ces rÃ©gions sont caractÃ©risÃ©es par des courbes de niveau trÃ¨s perturbÃ©es.
-Ce chapitre prÃ©sente une nouvelle fonction de distortion pour la stÃ©ganographie 
+Ce chapitre prÃ©sente une nouvelle fonction de distorsion pour la stÃ©ganographie 
 qui est basÃ©e sur les dÃ©rivÃ©es du second ordre, l'outil mathÃ©matique usuel 
 pour les courbes de niveau.
 
 Pour peu qu'on sache dÃ©finir une fonction $P$ 
 qui associe Ã  chaque pixel $(x,y)$ sa valeur $P(x,y)$,
 les pixels tels que les dÃ©rivÃ©es secondes de $P$ ont des valeurs Ã©levÃ©es 
-sont des bon candidats pour contenir un bit du message.
+sont des bons candidats pour contenir un bit du message.
 Cependant, une telle fonction $P$ n'est connue que de maniÃ¨re discrÃ¨te,
 \textit{i.e.}, en un nombre fini de points. 
 Les dÃ©rivÃ©es premiÃ¨res et secondes ne peuvent donc pas Ãªtre Ã©valuÃ©es mathÃ©matiquement.
-Au mieux, on peut construire une fonction qui approxime ces $P$ sur cet ensemble 
+Au mieux, on peut construire une fonction qui approxime les 
+dÃ©rivÃ©es de $P$ sur cet ensemble 
 de pixels. Ordonner alors les pixels selon la matrice hessienne 
 (\textit{i.e.}, la matrice des dÃ©rivÃ©es secondes) n'est pas trivial puisque celle-ci 
 contient de nombreuses valeurs pour un seul pixel donnÃ©. 
 
 On verra dans ce chapitre comment des approximations des dÃ©rivÃ©es 
-premiÃ¨res et secondes pour des images numriques (Section~\ref{sec:gradient}) on peu Ãªtre
+premiÃ¨res et secondes pour des images numÃ©riques (Section~\ref{sec:gradient}) 
+ont pu Ãªtre
 obtenues.
 Deux propositions de dÃ©rivÃ©es secondes sont ensuite 
 donnÃ©es et prouvÃ©es (Section~\ref{sec:second} et Section~\ref{sec:poly}).
-Une adaptation d'une fonction de distortion existante est Ã©tudiÃ©e
+Une adaptation d'une fonction de distorsion existante est Ã©tudiÃ©e
 en Section~\ref{sec:distortion} et des expÃ©riences sont prÃ©sentÃ©es 
 en Section~\ref{sec:experiments}.
+Ce chapitre a Ã©tÃ© publiÃ© dans~\cite{ccfg16:ip}.
 
 
 
-\section{Derivatives in an Image}\label{sec:gradient}
+\section{Des dÃ©rivÃ©es dans une image}\label{sec:gradient}
 
-This section first recalls links between level curves, gradient, and
-Hessian matrix (Section~\ref{sub:general}).
-It next analyses them using kernels from signal theory 
-(Section~\ref{sub:class:1} and Section~\ref{sub:class:2}).
+Cette section rappelle d'abord les liens entre lignes de niveau, gradient et 
+matrice hessienne puis analyse ensuite leur construction Ã  l'aide
+de noyaux de la thÃ©orie du signal.
 
 
-\subsection{Hessian Matrix}\label{sub:general}
-Let us consider that an image can be seen as a numerical function $P$ that 
-associates a value $P(x,y)$ to each pixel of coordinates $(x,y)$.
-The variations of this function in $(x_0,y_0)$ 
-can be evaluated thanks to its gradient
-$\nabla{P}$, which is the vector whose two components 
-are the partial derivatives in $x$ and in $y$ of $P$: 
-
-\[\nabla{P}(x_0,y_0) = \left(\frac{\partial P}{\partial x}(x_0,y_0),\frac{\partial P}{\partial y}(x_0,y_0)\right).
-\]
-
-In the context of two variables, the gradient vector
-points to the direction where the function has the highest increase.
-Pixels with close values thus follow level curve that is orthogonal 
-to the one of highest increase.
-
-The variations of the gradient vector are expressed in the 
-Hessian matrix $H$ of second-order partial derivatives of $P$.
+\subsection{Matrice hessienne}\label{sub:general}
+On considÃ¨re qu'une image peut Ãªtre assimilÃ©e Ã  une fonction de $\R^+\times \R^+$
+dans $\R^+$ telle que la valeur $P(x,y)$ est associÃ©e Ã  chaque  pixel de coordonnÃ©es $(x,y)$.
+Les variations d'une telle fonction en $(x_0,y_0)$ peuvent Ãªtre Ã©valuÃ©es 
+grÃ¢ce au gradient 
+$\nabla{P}(x_0,y_0) = \left(\frac{\partial P}{\partial x}(x_0,y_0),\frac{\partial P}{\partial y}(x_0,y_0)\right).
+$
+Le vecteur gradient pointe dans la direction oÃ¹ la fonction a le plus fort accroissement.
+Des pixels ayant des valeurs voisines sont sur des lignes de niveaux qui sont orthogonales
+Ã  ce vecteur.
 
+Les variations du vecteur gradient s'expriment usuellement Ã  l'aide de la matrice
+hessienne $H$ des dÃ©rivÃ©es partielles de second ordre de $P$.
 \[
 H = \begin{bmatrix} 
 \dfrac{\partial^2 P}{\partial x^2} & 
@@ -73,87 +69,77 @@ H = \begin{bmatrix}
 \end{bmatrix}. 
 \]
 
-
-
-
-In one pixel $(x_0,y_0)$, the larger the absolute values of this matrix are,
-the more the gradient is varying around $(x_0,y_0)$.
-We are then left to evaluate such an Hessian matrix.
-
-This task is not as easy as it appears since natural images are not defined 
-with differentiable functions from $\R^2$ to $\R$.
-Following subsections provide various approaches to compute these 
-Hessian matrices.
-
-\subsection{Classical Gradient Image Approaches}\label{sub:class:1}
-In the context of image values, the most used approaches to evaluate gradient vectors are the well-known ``Sobel'', ``Prewitt'', ``Central Difference'', and ``Intermediate Difference'' ones.
+En un pixel $(x_0,y_0)$, plus les valeurs de cette matrice sont Ã©loignÃ©es de zÃ©ro,
+plus le gradient varie en ce point. Ãvaluer ce type de matrice est ainsi primordial
+en stÃ©ganographie. Cependant cette tÃ¢che n'est pas aussi triviale qu'elle n'y 
+paraÃ®t puisque les images naturelles ne sont pas  dÃ©finies Ã  l'aide 
+de fonctions diffÃ©rentiables de $\R^+\times \R^+$
+dans $\R^+$. 
+La suite montre comment obtenir des approximations de telles matrices. 
+
+\subsection{Approches classiques pour Ã©valuer le gradient dans des images}\label{sub:class:1}
+Dans ce contexte, les approches les plus utilisÃ©es pour Ã©valuer un gradient 
+sont ``Sobel'', ``Prewitt'', ``DiffÃ©rence centrale'' et `` DiffÃ©rence intermÃ©diaire''.
+Chacune de ces  approches applique un produit de convolution $*$ entre un noyau $K$
+(rappelÃ© dans le tableau~\ref{table:kernels:usual}) et une fenÃªtre $A$ de taille 
+$3\times 3$. Le rÃ©sultat 
+ $A * K$ est une approximation du gradient horizontal
+\textit{i.e.}, $\dfrac{\partial P}{\partial x}$.
+Soit $K'$ le rÃ©sultat de la rotation d'un angle $\pi/2$ appliquÃ©e Ã  $K$.
+La composante verticale du gradient,  $\dfrac{\partial P}{\partial y}$ est obtenue 
+de maniÃ¨re similaire en Ã©valuant $A * K'$. Lorsqu'on applique ceci sur toute 
+la matrice image, on obtient  une matrice de mÃªme taille pour chacune des 
+dÃ©rivÃ©es partielles. 
+
+Les deux Ã©lÃ©ments de la premiÃ¨re colonne (respectivement de la seconde) 
+de la matrice hessienne
+sont le rÃ©sultat du calcul du gradient sur la matrice $\dfrac{\partial P}{\partial x}$
+(resp. sur la matrice  $\dfrac{\partial P}{\partial y}$).
 
 
 \begin{table}[ht]
-\caption{Kernels of usual image gradient operators\label{table:kernels:usual}
+\caption{Noyaux usuels pour Ã©valuer des gradients d'images\label{table:kernels:usual}
 }
 \centering
-\scriptsize
 \begin{tabular}{|c|c|c|}
     \hline
-    Name&   Sobel & Prewitt \\
+    Nom&   Sobel & Prewitt \\
     \hline
-    Kernel & $\textit{Ks}= \begin{bmatrix} -1 & 0 & +1 \\ -2 & 0 & +2 \\ -1 & 0 & +1 \end{bmatrix} $ &
+    Noyau & $\textit{Ks}= \begin{bmatrix} -1 & 0 & +1 \\ -2 & 0 & +2 \\ -1 & 0 & +1 \end{bmatrix} $ &
     $\textit{Kp}= \begin{bmatrix} -1 & 0 & +1 \\ -1 & 0 & +1 \\ -1 & 0 & +1 \end{bmatrix} $\\
     \hline\hline
-    Name & Central & Intermediate \\
-            & Difference &Difference \\ 
+    Nom & DiffÃ©rence  & DiffÃ©rence  \\
+            & centrale & IntermÃ©diaire \\ 
     \hline
-    Kernel & $\textit{Kc}= \begin{bmatrix} 0&0&0 \\ -\dfrac{1}{2} & 0 & +\dfrac{1}{2} \\ 0&0&0 \end{bmatrix} $ &
+    Noyau & $\textit{Kc}= \begin{bmatrix} 0&0&0 \\ -\dfrac{1}{2} & 0 & +\dfrac{1}{2} \\ 0&0&0 \end{bmatrix} $ &
     $\textit{Ki}= \begin{bmatrix} 0&0&0 \\ 0 & -1 & 1 \\ 0&0&0 \end{bmatrix} $ \\
     \hline
 \end{tabular}
 \end{table}
 
 
-Each of these approaches applies a convolution product $*$ between a  kernel $K$ (recalled in Table~\ref{table:kernels:usual}) and 
- a $3\times 3$ window of pixel values $A$. The result 
- $A * K$ is an evaluation of the horizontal gradient, 
-\textit{i.e.}, $\dfrac{\partial P}{\partial x}$
-expressed as a matrix in $\R$.
-Let $K\rl$ be the result of a $\pi/2$ rotation applied on $K$. 
-The vertical gradient $\dfrac{\partial P}{\partial y}$ is similarly obtained by computing $A * K\rl$, which is again expressed as a matrix in $\R$.
-
-The two elements of the first line 
-of the Hessian matrix are the result
-of applying the horizontal gradient calculus 
-first on  $\dfrac{\partial P}{\partial x}$ and next 
-on  $\dfrac{\partial P}{\partial y}$. 
-Let us study these Hessian matrices in the next section.
-
-
-\subsection{Hessian Matrices induced by Gradient Image Approaches}\label{sub:class:2}
-
-First of all, it is well known that 
-$\dfrac{\partial^2 P}{\partial x \partial y} $ is equal to 
-$\dfrac{\partial^2 P}{\partial y \partial x}$ if  
-the approach that computes the gradient and the
-one which evaluates the Hessian matrix are the same.
-For instance, in the Sobel approach, 
-it is easy to verify that the calculus of 
-$\dfrac{\partial^2 P}{\partial x \partial y}$ 
-and of 
-$\dfrac{\partial^2 P}{\partial y \partial x}$ 
-are both the result 
-of a convolution product with the Kernel 
-$\textit{Ks}''_{xy}$ given in Table~\ref{table:hessian:usual}.
-This one summarizes kernels
-$K_{x^2}''$ and 
+
+
+\subsection{Matrices hessiennes induites  par des approches 
+de gradient d'images}\label{sub:class:2}
+Il est connu que  
+$\dfrac{\partial^2 P}{\partial x \partial y} $ est Ã©gal Ã  
+$\dfrac{\partial^2 P}{\partial y \partial x}$ si 
+les mÃ©thodes qui calculent le gradient et le gradient du gradient (la matrice hessienne)
+sont les mÃªmes.
+Le tableau~\ref{table:hessian:usual} rÃ©sume les noyaux 
+$K_{x^2}''$ et
 $K_{xy}''$  
-that allow to respectively compute  
-$\dfrac{\partial^2 P}{\partial x^2}$ and
-$\dfrac{\partial^2 P}{\partial x \partial y}$ with a convolution product
-for each of the usual image gradient operator.
+qui permettent de calculer respectivement 
+$\dfrac{\partial^2 P}{\partial x^2}$ et 
+$\dfrac{\partial^2 P}{\partial x \partial y}$ comme un produit de convolution 
+pour chacun des opÃ©rateurs de gradient rappelÃ©s Ã  la section prÃ©cÃ©dente.
+
 
 \begin{table}[ht]
-\caption{Kernels of second order gradient operators\label{table:hessian:usual}
+\caption{Noyaux usuels pour Ã©valuer des gradients de second ordre d'images
+\label{table:hessian:usual}
 }\centering
-\tiny
 \begin{tabular}{|c|c|}
     \hline
     Sobel & Prewitt \\
@@ -204,8 +190,8 @@ for each of the usual image gradient operator.
     $ \\
     
     \hline\hline
-    Central & Intermediate \\
-    Difference &Difference \\ 
+    DiffÃ©rence & DiffÃ©rence  \\
+    centrale  &intermÃ©diaire \\ 
     \hline
     $
     \textit{Kc}_{x^2}''= 
@@ -252,45 +238,35 @@ for each of the usual image gradient operator.
 
 \end{table}
 
-The Sobel kernel $\textit{Ks}_{x^2}''$ allows to detect whether the central 
-pixel belongs to a ``vertical'' edge, even if this one is noisy, by considering its 
-vertical neighbours. The introduction of these vertical neighbours in this kernel is meaningful 
-in the context of finding edges, but not very accurate when the objective is to 
-precisely find the level curves of the image. 
-Moreover, all the pixels that are in the  second and the fourth column in this kernel
-are ignored.
-The Prewitt Kernel has similar drawbacks in this context.
-
-
-The Central Difference kernel $\textit{Kc}_{x^2}''$ is not influenced by 
-the vertical neighbours of the central pixel and is thus more accurate here.
-However, the kernel $\textit{Kc}_{xy}''$ again looses the values of the pixels that 
-are vertically and diagonally aligned with the central one.
-
-Finally, the Intermediate Difference kernel $\textit{Ki}_{x^2}''$ shifts 
-to the left the value of horizontal variations of $\dfrac{\partial P}{\partial x}$:
-the central pixel $(0,0)$ exactly receives the value
+Le noyau $\textit{Ks}_{x^2}''$ permet de dÃ©tecter si le 
+pixel central appartient Ã  un bord ``vertical'', mÃªme si celui-ci
+ contient du bruit,
+en considÃ©rant ses voisins verticaux. Ces derniers sont vraiment 
+pertinents dans un objectif de dÃ©tecter les bords. Cependant, leur lien avec 
+les lignes de niveau n'est pas direct. De plus tous les pixels qui sont dans la 
+deuxiÃ¨me et la quatriÃ¨me colonne de ce noyau sont ignorÃ©s.
+Le noyau de Prewitt a des propriÃ©tÃ©s similaires.
+Le noyau de diffÃ©rence centrale $\textit{Kc}_{x^2}''$ n'est pas influencÃ© par les 
+voisins verticaux du pixel central et peut paraÃ®tre plus adaptÃ© ici.
+Cependant, le noyau $\textit{Kc}_{xy}''$ perd aussi les valeurs des pixels 
+qui sont alignÃ©s verticalement et diagonalement avec le pixel central.
+Enfin, le noyau de diffÃ©rence intermÃ©diaire  $\textit{Ki}_{x^2}''$ dÃ©cale
+Ã  gauche la valeur des variations horizontales de $\dfrac{\partial P}{\partial x}$:
+Le pixel central  $(0,0)$ reÃ§oit  exactement la valeur 
 $\dfrac{P(0,2)-P(0,1)}{1} - \dfrac{P(0,1)-P(0,0)}{1}$,
-which is an approximation of 
-$\dfrac{\partial P}{\partial x}(0,1)$ and not of 
+qui est une approximation de 
+$\dfrac{\partial P}{\partial x}(0,1)$ et non de 
 $\dfrac{\partial P}{\partial x}(0,0)$.
-Furthermore the Intermediate Difference kernel $\textit{Ki}_{xy}''$ only deals 
-with pixels in the upper right corner, loosing all the other information.
-
-Due to these drawbacks, we are then left to produce another approach to find the level curves with strong accuracy.
-
-\section{Second Order Kernels for Accurate Level Curves}\label{sec:second}
-
-This step aims at finding 
-accurate level curve variations in an image. 
-We do not restrict the kernel to have a fixed size (\textit{e.g.}, $3\times3$ or $5 \times 5$ as in the 
-aforementioned schemes).
-This step is thus defined with kernels of size 
-$(2n+1)\times (2n+1)$, $n \in \{1,2,\dots,N\}$, where 
-$N$ is a parameter of the approach.
-
-The horizontal gradient variations are thus captured thanks to $(2n+1)\times (2n+1)$ square kernels
-\begin{small}
+De plus, le noyau de diffÃ©rence intermÃ©diaire $\textit{Ki}_{xy}''$ ne concerne 
+que les pixels du coin supÃ©rieur droit, en perdant toutes les autres informations.
+La section suivante propose une autre approche pour calculer les lignes de niveau avec une prÃ©cision accrue.
+
+\section{Des noyaux pour des lignes de niveau}\label{sec:second}
+On ne restreint pas aux noyaux de taille fixe (comme  $3\times3$ or $5 \times 5$ 
+dans les schÃ©mas prÃ©cÃ©dents). Au contraire, on considÃ¨re des noyaux de taille variable  
+$(2n+1)\times (2n+1)$, $n \in \{1,2,\dots,N\}$, oÃ¹
+$N$ est un paramÃ¨tre de l'approche.
+Les variations horizontales du gradient sont extraites grÃ¢ce au noyau de taille $(2n+1)\times (2n+1)$: 
 \[
 \arraycolsep=1.4pt
 \def\arraystretch{1.4}
@@ -308,24 +284,22 @@ The horizontal gradient variations are thus captured thanks to $(2n+1)\times (2n
     \end{array}
     \right)
 \]
-\end{small}
 
-When the convolution product is applied on a $(2n+1)\times(2n+1)$ window,
-the result is 
-$\dfrac{1}{2}\left(\dfrac{P(0,n)-P(0,0)}{n} - \dfrac{P(0,0)-P(0,-n)}{n}\right)$, which is indeed 
-the variation between the gradient around the central pixel. 
-This proves that this calculus is a correct approximation of 
+Lorsque le produit de convolution est appliquÃ© sur une fenÃªtre $(2n+1)\times(2n+1)$,
+le rÃ©sultat est 
+$\dfrac{1}{2}\left(\dfrac{P(0,n)-P(0,0)}{n} - \dfrac{P(0,0)-P(0,-n)}{n}\right)$, 
+qui reprÃ©sente en effet les variation horizontales de la partie horizontale 
+du gradient autour du pixel central. On obtient donc bien une approximation de 
 $\dfrac{\partial^2 P}{\partial x^2}$.
+Lorsque $n$ vaut 1, ce noyau est une version centrÃ©e du noyau horizontal de diffÃ©rence 
+intermÃ©diaire. $\textit{Ki}_{x^2}''$ Ã  un facteur  $1/2$ prÃ¨s).
+Lorsque $n$ vaut 2, on retrouve $\textit{Kc}_{x^2}''$.
 
-When $n$ is 1, this kernel is a centered version of the horizontal Intermediate Difference kernel $\textit{Ki}_{x^2}''$ modulo a multiplication by $1/2$. When $n$ is 2, this kernel is equal to $\textit{Kc}_{x^2}''$.
+Les variations verticales du gradient sont aussi obtenues en faisant subir 
+Ã  $\textit{Ky}_{x^2}''$ une rotation d'angle  $\pi/2$.
+Les variations diagonales sont obtenues Ã  l'aide du gradient 
+$\textit{Ky}_{xy}''$ dÃ©fini par:
 
-The vertical gradient variations are again obtained by applying 
-a $\pi/2$ rotation to each horizontal kernel $\textit{Ky}_{x^2}''$.
-
-The diagonal gradient variations are obtained thanks to the $(2n+1)\times (2n+1)$ square kernels 
-$\textit{Ky}_{xy}''$ defined by
-
-\begin{small}
 \[
 \arraycolsep=1.4pt
 \def\arraystretch{1.4}
@@ -367,79 +341,24 @@ $\textit{Ky}_{xy}''$ defined by
     \end{array}
     \right).
 \]
-\end{small}
 
-When $n$ is 1, $\textit{Ky}_{xy}''$ is equal to the kernel
-$\textit{Kc}_{xy}''$, and %.
-%
-%When $n$ is 1, 
-the average vertical variations of the horizontal variations are 
-\[
-\begin{array}{l}
-\dfrac{1}{4}
-\left[ 
-\left((P(0,1)-P(0,0))-(P(1,1)-P(1,0))\right)+ \right.\\
-\quad \left((P(-1,1)-P(-1,0))-(P(0,1)-P(0,0))\right)+\\
-\quad \left((P(0,0)-P(0,-1))-(P(1,0)-P(1,-1))\right)+\\
-\quad \left. \left((P(-1,0)-P(-1,-1))-(P(0,0)-P(0,-1))\right)
-\right]  \\
-=\dfrac{1}{4}
-\left[ P(1,-1) -P(1,1) - P(-1,-1) + P(-1,1)\right].
-\end{array}
-\]
-which is $\textit{Ky}_{xy}''$.
-
-Let us now consider any number $n$, $1 \le n \le N$.
-Let us first investigate the vertical variations related to 
-the horizontal vector $\vec{P_{0,0}P_{0,1}}$ 
-(respectively  $\vec{P_{0,-1}P_{0,0}}$)
-of length 1 that starts from (resp. that points to) $(0,0)$. 
-As with the case $n=1$, there are 2 new vectors of 
-length 1, namely 
-$\vec{P_{n,0}P_{n,1}}$ and $\vec{P_{-n,0}P_{-n,1}}$ 
-(resp. 
-$\vec{P_{n,-1}P_{n,0}}$, and $\vec{P_{-n,-1}P_{-n,0}}$)
-that are vertically aligned with $\vec{P_{0,0}P_{0,1}}$ 
-(resp. with $\vec{P_{0,-1}P_{0,0}}$).
-
-The vertical variation is now equal to $n$. Following the case where $n$ is 1 to compute the average variation, 
-the coefficients of the first and last line around the central 
-vertical line are thus from left to right:
-$\dfrac{1}{4n}$,
-$\dfrac{-1}{4n}$,
-$\dfrac{-1}{4n}$, and
-$\dfrac{1}{4n}$.
-
-Cases are similar with vectors $\vec{P_{0,0}P_{0,1}}$, \ldots  
-$\vec{P_{0,0}P_{0,n}}$ which respectively lead to coefficients 
-$-\dfrac{1}{4 \times 2n}$, \ldots, 
-$-\dfrac{1}{4 \times n.n}$, and the proof is omitted.
-Finally, let us consider the vector $\vec{P_{0,0}P_{0,1}}$
-and its vertical variations when $\delta y$ is $n-1$.
-As in the case where $n=1$, we thus obtain the coefficients 
-$\dfrac{1}{4 \times (n-1)n}$ and 
-$-\dfrac{1}{4 \times (n-1)n}$ 
- (resp. $-\dfrac{1}{4 \times (n-1)n}$ and 
-$\dfrac{1}{4 \times (n-1)n}$)
-in the second line (resp. in the 
-penultimate line) since the vector has length $n$
-and $\delta y$ is $n-1$.
-Coefficient in the other lines are similarly obtained and the proof is thus omitted.
-
-We are then left to compute an approximation of the partial second order derivatives 
-$\dfrac{\partial^2 P}{\partial x^2}$,  $\dfrac{\partial^2 P}{\partial y^2}$, and $\dfrac{\partial^2 P}{\partial x \partial y}$
-with the kernels, 
-$\textit{Ky}_{x^2}''$, $\textit{Ky}_{y^2}''$, and $\textit{Ky}_{xy}''$ respectively.
-However, the size of each of these kernels is varying from $3\times3$ to $(2N+1)\times (2N+1)$.
-Let us explain the approach on the former partial derivative.
-The other can be immediately deduced.
-
-Since the objective is to detect large variations, the second order derivative is approximated as 
-the maximum of the approximations. More formally, 
-let $n$, $1 \le n \le N$, be an integer number and 
-$\dfrac{\partial^2 P}{\partial x^2}_n$ be the result of applying the Kernel 
-$\textit{Ky}_{x^2}''$ of size $(2n+1)\times (2n+1)$. The derivative 
-$\dfrac{\partial^2 P}{\partial x^2}$ is defined by
+
+
+En effet, lorsque  $n$ vaut 1, $\textit{Ky}_{xy}''$ se retrouve en calculant la moyenne
+des variations horizontales de la composante verticale du gradient calculÃ© Ã  l'aide de 
+$\textit{Ky}_{y}'$. Pour cette valeur de $n$, on a
+$\textit{Ky}_{xy}'' =   \textit{Kc}_{xy}''$. 
+Pour chaque nombre  $n$, $1 <  n \le N$, $\textit{Ky}_{xy}''$ se retrouve de la mÃªme
+maniÃ¨re, c'est-Ã -dire en effectuant des moyennes de variations. 
+Une preuve de la construction se trouve dans l'article~\cite{ccfg16:ip}.
+
+
+
+L'objectif est de dÃ©tecter les grandes variations des dÃ©rivÃ©es premiÃ¨res. 
+Ainsi les dÃ©rivÃ©es secondes seront approximÃ©es comme les maximums des 
+matrices hessiennes  obtenues lorsque $n$ varie entre  $1$ et $ N$.
+
+La dÃ©rivÃ©e partielle $\dfrac{\partial^2 P}{\partial x^2}$ est dÃ©finie par 
 
 \begin{equation}
 \dfrac{\partial^2 P}{\partial x^2}
@@ -448,28 +367,30 @@ $\dfrac{\partial^2 P}{\partial x^2}$ is defined by
 \right \}. 
 \label{eq:d2p_dx2}
 \end{equation}
-
-The same iterative approach is applied to compute approximations of
+oÃ¹ $\dfrac{\partial^2 P}{\partial x^2}_n$ est le rÃ©sultat de l'application 
+du noyau $\textit{Ky}_{x^2}''$ de taille $(2n+1)\times (2n+1)$.
+La mÃªme approche itÃ©rative est appliquÃ©e pour construire les 
+approximations de 
 $\dfrac{\partial^2 P}{\partial y \partial x}$ 
-and of 
+et de 
 $\dfrac{\partial^2 P}{\partial y^2}$.
-Next section studies the suitability of approximating second order derivatives
-when considering an image as a polynomial.
+
+La section suivante Ã©tudie la pertinence d'interpoler une image par un polynome 
+lorsqu'on cherche Ã  obtenir ces dÃ©rivÃ©es secondes.
 
 
-\section{Polynomial Interpolation of Images for Hessian Matrix Computation}\label{sec:poly}
-Let $P(x,y)$ be the discrete value of the pixel $(x,y)$ in the image.
-Let $n$, $1 \le n \le N$, be an integer such that the objective is to find a polynomial interpolation 
-on the $(2n+1)\times(2n+1)$ window where the central pixel has index $(0,0)$.  
-There exists an unique polynomial $L : \R\times \R \to \R$ of degree $(2n+1)\times(2n+1)$ defined  
-such that $L(x,y)=P(x,y)$ for each pixel $(x,y)$ in this window.
-Such a polynomial is defined by 
+\section{Interpolation polynomiale pour le calcul de la matrice hessienne}\label{sec:poly}
+Soit $P(x,y)$ la  valeur du pixel $(x,y)$ et soit  $n$, $1 \le n \le N$, 
+tel que l'objectif est de trouver un polynome d'interpolation dans la fenÃªtre de taille  
+$(2n+1)\times(2n+1)$ dont le pixel central a pour indice $(0,0)$.  
+Il existe un unique polynÃ´me $L : \R\times \R \to \R$ 
+de degrÃ© $(2n+1)\times(2n+1)$ tel que  $L(x,y)=P(x,y)$ pour chaque pixel  
+$(x,y)$ de cette fenÃªtre et ce polynome est dÃ©fini par  
 \begin{equation}
 \begin{array}{l}
 L(x,y) =  
 \sum_{i=-n}^{n}
-\sum_{j=-n}^{n} \\
-\quad P(i,j)
+\sum_{j=-n}^{n}P(i,j)
 \left( 
 \prod_{\stackrel{-n\leq j'\leq n}{j'\neq j}}
 \frac{x-j'}{i-j'}
@@ -480,9 +401,7 @@ L(x,y) =
 \right)
 \end{array}
 \end{equation}
-
-It is not hard to prove that the first order horizontal derivative of the polynomial $L(x,y)$ 
-is
+On peut facilement prouver que la dÃ©rivÃ©e partielle  de $L$ selon $x$ est 
 \begin{equation}
 \begin{array}{l}
 \dfrac{\partial L}{\partial x} =  
@@ -492,8 +411,7 @@ P(i,j)
 \left( 
 \prod_{\stackrel{-n\leq j'\leq n}{j'\neq j}}
 \frac{y-j'}{j-j'}
-\right)\\
-\quad
+\right)
 \left( 
 \sum_{\stackrel{-n\leq i'\leq n}{i'\neq i}}
 \frac{1}{i-i'}
@@ -502,20 +420,17 @@ P(i,j)
 \right)
 \end{array}
 \end{equation}
-\noindent and thus to deduce that the
-second order ones are
-
-\begin{equation}
-\begin{array}{l}
-\dfrac{\partial^2 L}{\partial x^2} =  
+\noindent et ainsi en dÃ©duire que les dÃ©rivÃ©es partielles de second ordre 
+sont 
+\begin{eqnarray}
+\dfrac{\partial^2 L}{\partial x^2} &=&  
 \sum_{i=-n}^{n}
 \sum_{j=-n}^{n} 
 P(i,j)
 \left( 
 \prod_{\stackrel{-n\leq j'\leq n}{j'\neq j}}
 \frac{y-j'}{j-j'}
-\right)\\
-\quad
+\right) 
 \left( 
 \sum_{\stackrel{-n\leq i'\leq n}{i'\neq i}}
 \frac{1}{i-i'}
@@ -524,74 +439,33 @@ P(i,j)
 \prod_{\stackrel{-n\leq i'''\leq n}{i'''\neq i,i',i''}}
 \frac{x-i'''}{i-i'''}
 \right)
-\end{array}
-\label{eq:deriv:poly:x2}
-\end{equation}
-
-\begin{equation}
-\begin{array}{l}
-\dfrac{\partial^2 L}{\partial y \partial x} =  
+\label{eq:deriv:poly:x2} \\
+\dfrac{\partial^2 L}{\partial y \partial x} &= & 
 \sum_{i=-n}^{n}
-P(i,j) \\
-\quad
+P(i,j) 
 \left( 
 \sum_{\stackrel{-n\leq j'\leq n}{j'\neq j}}
 \frac{1}{j-j'}
 \prod_{\stackrel{-n\leq j''\leq n}{j''\neq j, j'}}
 \frac{y-j''}{j-j''}
-\right)\\
-\quad
+\right)
 \left( 
 \sum_{\stackrel{-n\leq i'\leq n}{i'\neq i}}
 \frac{1}{i-i'}
 \prod_{\stackrel{-n\leq i''\leq n}{i''\neq i, i'}}
 \frac{x-i''}{i-i''}
 \right)
-\end{array}
 \label{eq:deriv:poly:yx}
-\end{equation}
-
-These second order derivatives  are computed for each moving 
-window and are associated to the central pixel, \textit{i.e.}, to the pixel $(0,0)$ inside this one.
-
-Let us first simplify $\dfrac{\partial^2 L}{\partial x^2}$ when $(x,y)=(0,0)$
-defined in Equation~(\ref{eq:deriv:poly:x2}). If $j$ is not null, the index $j'$
-is going to be null and the product  
-$\left( 
-\prod_{\stackrel{-n\leq j'\leq n}{j'\neq j}}
-\frac{-j'}{j-j'}
-\right)$ is null too. 
-In this equation, we thus only consider $j=0$.
-It is obvious that the product indexed with $j'$ is thus equal to 1.
-This equation can thus be simplified in:
+\end{eqnarray}
+Ces dÃ©rivÃ©es secondes sont calculÃ©es pour chaque pixel central, \textit{i.e.} le pixel dont l'indice est  $(0,0)$ dans la fenÃªtre.
+En considÃ©rant cette particularisation, l'Ã©quation~(\ref{eq:deriv:poly:x2}) 
+se simplifie en 
 
 \begin{equation}
 \begin{array}{l}
 \dfrac{\partial^2 L}{\partial x^2} =  
 \sum_{i=-n}^{n}
-P(i,0)\\
-\quad
-\left( 
-\sum_{\stackrel{-n\leq i'\leq n}{i'\neq i}}
-\frac{1}{i-i'}
-\sum_{\stackrel{-n\leq i''\leq n}{i''\neq i,i'}}
-\frac{1}{i-i''}
-\prod_{\stackrel{-n\leq i'''\leq n}{i'''\neq i,i',i''}}
-\frac{i'''}{i'''-i}
-\right)
-\end{array}
-\label{eq:deriv:poly:x2:simpl}
-\end{equation}
-
-
-and then in:
-
-\begin{equation}
-\begin{array}{l}
-\dfrac{\partial^2 L}{\partial x^2} =  
-\sum_{i=-n}^{n}
-P(i,0)\\
-\quad
+P(i,0)
 \left( 
 \sum_{\stackrel{-n\leq i' < i'' \le n}{i',i''\neq i}}
 \frac{2}{(i-i')(i-i'')}
@@ -602,19 +476,9 @@ P(i,0)\\
 \label{eq:deriv:poly:x2:simpl:2}
 \end{equation}
 
-From this equation, the kernel allowing to evaluate horizontal 
-second order derivatives 
-can be computed for any $n$.
-It is further denoted as $Ko''_{x^2}$. 
-Instances of such matrix when $n=2$, $3$, and $4$
-are given in Table~\ref{table:sod:hori:poly}.
-
 \begin{table}[ht]
-    \caption{Kernels $Ko''_{x^2}$ 
-    for second order horizontal derivatives induced 
-    by polynomial interpolation}
+    \caption{Noyaux $Ko''_{x^2}$ pour  calculer des dÃ©rivÃ©es de second ordre Ã  partir d'interpolation polynomiale}
     \centering
-    \scriptsize
 \def\arraystretch{1.4}
     \begin{tabular}{|c|c|}
          \hline
@@ -635,11 +499,10 @@ are given in Table~\ref{table:sod:hori:poly}.
 
 
 \begin{table}[ht]
-\caption{Kernels for second order diagonal derivatives induced 
-    by polynomial interpolation \label{table:sod:diag:poly}
+\caption{ Noyaux pour les dÃ©rivÃ©es secondes en $x$ et $y$ lors de l'interpolation polynomiale\label{table:sod:diag:poly}
 }
 \centering
-\scriptsize
+%\scriptsize
 \def\arraystretch{1.5}
 \begin{tabular}{|c|c|}
     \hline
@@ -667,29 +530,26 @@ $\\
 \end{tabular}
 \end{table}
 
+Cette dÃ©rivÃ©e partielle peut s'Ã©crire comme un produit de convolution avec un noyau
+notÃ© $Ko''_{x^2}$. Des instances de tels noyaux, pour $n=2$, $3$ et  $4$
+sont donnÃ©es au tableau~\ref{table:sod:hori:poly}. De maniÃ¨re similaire,  
+le tableau~\ref{table:sod:diag:poly} donne deux exemples pour $n=1$ et $n=2$ 
+de noyaux $Ko''_{xy}$ permettant de calculer directement les dÃ©rivÃ©es 
+de second ordre selon $x$ et $y$ en $(0,0)$.
+On remarque que pour $n=1$, le noyau est Ã©gal Ã  $Kc''_{xy}$. 
 
-From Equation~(\ref{eq:deriv:poly:yx}), kernels allowing to evaluate diagonal 
-second order derivatives (\textit{i.e.}, 
-$\dfrac{\partial^2 L}{\partial y \partial x}$)  are computed.
-They are denoted as $Ko''_{xy}$.
-Table~\ref{table:sod:diag:poly} gives two examples of them when $n=1$ and $n=2$.
-Notice that for $n=1$, the kernel $Ko''_{xy}$ is equal to $Kc''_{xy}$. 
 
 
 
 
 
 
-\section{Distortion Cost}\label{sec:distortion}
-
-The distortion function has to associate to each pixel $(i,j)$ 
-the cost $\rho_{ij}$ of its modification
-by $\pm 1$.
-
-The objective is to map a small value to a pixel when all its second order derivatives 
-are high and a large value otherwise. 
-%\YF{The WOW  use the  wavelet relative distortion (UNIWARD) as a function to force the embedding operation to the noisy areas and avert the smooth areas.}
-In WOW and UNIWARD  the distortion function is based on the H\"older norm with
+\section{Fonction de distorsion}\label{sec:distortion}
+Une fonction de distorsion associe Ã  chaque pixel  $(i,j)$ 
+le coÃ»t $\rho_{ij}$ de modification  par $\pm 1$. L'objectif est d'associer une 
+valeur faible aux pixels dont toutes les dÃ©rivÃ©es secondes sont Ã©loignÃ©es de 0 
+et une valeur rÃ©dhibitoire sinon.
+Dans WOW comme dans  UNIWARD la fonction de distorsion est dÃ©finie par 
 \[
 \rho_{ij}^w = 
 \left(
@@ -698,119 +558,78 @@ In WOW and UNIWARD  the distortion function is based on the H\"older norm with
 \abs{\xi_{ij}^d}^{p} 
 \right)^{-\frac{1}{p}}
 \]
-where $p$ is a negative number and 
-$\xi_{ij}^h$ (resp. $\xi_{ij}^v$ and $\xi_{ij}^d$)
-represents the horizontal (resp. vertical and diagonal) suitability.
-A small suitability in one direction means an inaccurate position to embed a message.
-
-We propose here to adapt such a distortion cost as follows:
+oÃ¹ $p$ est un nombre nÃ©gatif et  
+$\xi_{ij}^h$ (resp. $\xi_{ij}^v$ et $\xi_{ij}^d$)
+reprÃ©sente la pertinence horizontale (resp. verticale et diagonale) de modification.
+Une faible pertinence dans une direction signifie que l'embarquement 
+dans ce pixel est inappropriÃ©.
+La fonction de distorsion que l'on a retenu est une particularisation ($p=-1$) 
+de cette derniÃ¨re:
 \[
 \rho_{ij} = 
 \left(
-\abs{\dfrac{\partial^2 P}{\partial x^2}(i,j)} +
-\abs{\dfrac{\partial^2 P}{\partial y^2}(i,j)} +
-\abs{\dfrac{\partial^2 P}{\partial y \partial x}(i,j)}
-\right)^{-\frac{1}{p}}
+\abs{\dfrac{\partial^2 P}{\partial x^2}(i,j)}^{-1} +
+\abs{\dfrac{\partial^2 P}{\partial y^2}(i,j)}^{-1} +
+\abs{\dfrac{\partial^2 P}{\partial y \partial x}(i,j)}^{-1}
+\right)
 \]
-It is not hard to check that such a function has large 
-value when at least one of its derivatives is null. Otherwise,
-the larger the derivatives are, the smaller the returned value
-is.
-
 
 
+\section{ExpÃ©rimentations}\label{sec:experiments}
 
-%Practically, it is computed as a double convolution product with Daubechies Wavelet Kernels. 
+Tout d'abord, l'ensemble du code est accessible en ligne\footnote{\url{https://github.com/stego-content/SOS}}.
+La Figure~\ref{fig:oneimage} reprÃ©sente les rÃ©sultats d'embarquement de donnÃ©es dans 
+l'image 38 du challenge BOSS~\cite{Boss10} en suivant les deux schÃ©mas basÃ©s 
+sur les dÃ©rivÃ©es secondes prÃ©sentÃ©s dans ce chapitre.
+Le taux d'embarquement $\alpha$ est fixÃ© Ã  0.4 bits par pixel et les noyaux sont 
+construits avec $N=4$. On remarque bien que les pixels dans les zones uniformes 
+et les pixels dans les bords bien dÃ©finis ne sont pas modifiÃ©s par l'approche tandis 
+qu'au contraire les zones peu prÃ©visibles (le monument par exemple) 
+concentrent les changements.
 
-%It is not hard to see that this approach 
-%does not consider the range of the values. For %instance, if one direction only has large suitability
-%absolute values, all the elements $\abs{\xi_{ij}}^{p}$ are so small, and 
-%suitable pixels cannot be distinguished from inaccurate ones. The distortion function has to 
-%provide a way to discriminate pixels even when the distribution has small deviation
 
-%Because of each pixel value ranges in $[0,255]$, each first order derivative belongs to
-%$[-255,255]$ and thus each second order derivative has a real value in 
-%$[-255,255]$.  
-%Due to the definition of the approximation of $\dfrac{\partial^2 P}{\partial x^2}$, 
-%this one is a positive or null real number lower than 255. 
 
-%The higher the second order derivatives are, 
-%the 
 
-%To enlarge the discrimination power, 
-%data are firstly linearly mapped to $[0,1]$. 
-%The tuple 
-%$(\dfrac{\partial^2 P}{\partial x^2} , \dfrac{\partial^2 P}{\partial y^2}, \dfrac{\partial^2 %P}{\partial x \partial y})$ belongs to $[0,1]\times[0,1]\times[0,1] = [0,1]^3$.
-
-%Basically, for each dimension the idea is to select a subset of pixels of size $k$ and next to retain those which have been selected three times, \textit{i.e.}, those with large values 
-%in any element of 
-%$(\dfrac{\partial^2 P}{\partial x^2} , \dfrac{\partial^2 P}{\partial y^2}, \dfrac{\partial^2 %P}{\partial x \partial y})$.
-%Since the dimension is $3$, the size $k$ can be defined by
-%\[
-%k = W \times H \times \alpha ^{\dfrac{1}{3}}
-%\]
-%where $W$ and $H$ respectively stand for the width and the length of the image (expressed in pixels)
-%and $\alpha$ is the payload. For instance if $\alpha$ is 0.1, 
-%about $0.46\%$ of the image pixels with higher values in $\dfrac{\partial^2 P}{\partial x^2}$ should be selected, and similarly with $\dfrac{\partial^2 P}{\partial y^2}$ and $\dfrac{\partial^2  P}{\partial x \partial y}$.
-
-
-% Dire comment on les ordonne, les autres et le sens
-
-\begin{figure*}
+\begin{figure}[ht]
     \centering
     \begin{tabular}{|c|c|c|}
     \hline
-    Scheme & Stego. content & Changes with cover \\
+    SchÃ©ma & Image. Stego. & DiffÃ©rence avec le support \\
     \hline
-    $Ky$ based approach &\includegraphics[scale=0.20]{images/38_dp}&
+    Approche Ã  base de $Ky$  &\includegraphics[scale=0.20]{images/38_dp}&
     \includegraphics[scale=0.20]{images/38_dp_diff} \\
     \hline
-    $Ko$ based approach & \includegraphics[scale=0.20]{images/38_poly} &
+    Approche Ã  base de  $Ko$  & \includegraphics[scale=0.20]{images/38_poly} &
     \includegraphics[scale=0.20]{images/38_poly_diff} \\
     \hline
     \end{tabular}
-    \caption{Embedding changes instance with payload  $\alpha = 0.4$}
+    \caption{Exemple de changements dus Ã  un embarquement avec  $\alpha = 0.4$}
     \label{fig:oneimage}
-\end{figure*}
-
-
-\section{Experiments}\label{sec:experiments}
-
-First of all, the whole steganographic approach code is available online\footnote{\url{https://github.com/stego-content/SOS}}.
-
-Figure~\ref{fig:oneimage} presents the results of embedding data in a cover image
-from the BOSS contest database~\cite{Boss10} with respect to the two second order derivative schemes presented in this work. 
-The $Ky$ based approach (resp. the $Ko$ based one)
-corresponds to the scheme detailed in Section~\ref{sec:second} 
-(resp. in Section~\ref{sec:poly}).
-The payload $\alpha$ is set to 0.4 and kernels are computed with $N=4$.    
-The central column outputs the embedding result whereas the right one displays 
-differences between the cover image and the stego one.
-It can be observed that pixels in smooth area (the sky, the external access steps) and pixels in clean edges (the columns, 
-the step borders) are not modified by the approach.
-On the contrary, an unpredictable area (a monument for example) concentrates pixel changes.
-
-
-\subsection{Choice of parameters}
-
-The two methods proposed in Section~\ref{sec:second} and  
-in Section~\ref{sec:poly} are based on kernels of size up to 
-$(2N+1)\times(2N+1)$. This section aims at finding the 
-value of the $N$ parameter that maximizes the security level.
-For each approach, we have built 1,000 stego images with 
-$N=2$, $4$, $6$, $8$, $10$, $12$, and $14$ where the covers belong 
-to the BOSS contest database. 
-This set contains 10,000 grayscale $512\times 512$ images in a RAW format. 
-The security of the approach has been evaluated thanks to the Ensemble Classifier~\cite{DBLP:journals/tifs/KodovskyFH12} based steganalyser,
-which is considered as a state of the art steganalyser tool.
-This steganalysis process embeds the rich model (SRM) features~\cite{DBLP:journals/tifs/FridrichK12}
-of size 34,671. 
-For a payload $\alpha$, either equal to $0.1$ or to $0.4$, average testing errors (expressed in percentages) have been studied and are summarized in
-Table~\ref{table:choice:parameter}.
+\end{figure}
+
+
+
+
+
+\subsection{Choix des paramÃ¨tres}
+
+Les deux mÃ©thodes prÃ©sentÃ©es ici dÃ©pendent de noyaux dont la taille va jusqu'Ã   
+$(2N+1)\times(2N+1)$. Cette section montre comment Ã©valuer $N$ pour maximiser 
+le niveau de sÃ©curitÃ©.
+Pour chaque approche ($N=2$, $4$, $6$, $8$, $10$, $12$ et $14$),
+1000 images stÃ©gos du challenge BOSS ont Ã©tÃ© selectionnÃ©es.
+La sÃ©curitÃ© de l'approche a Ã©tÃ© Ã©valuÃ©e avec le stÃ©ganalyseur 
+Ensemble Classifier~\cite{DBLP:journals/tifs/KodovskyFH12}.
+Pour un taux d'embarquement   $\alpha$ Ã©gal soit Ã   $0.1$ ou soit Ã   $0.4$, 
+l'erreur moyenne de test (exprimÃ©e en pourcentage) a Ã©tÃ© calculÃ©e. 
+Le tableau~\ref{table:choice:parameter} synthÃ©tise les rÃ©sultats.
+On observe que la taille $N=4$ (respectivement $N=12$) 
+permet d'obtenir des erreurs suffisamment Ã©levÃ©es pour l'approche basÃ©e sur $Ky$  
+(resp. pour celle basÃ©e sur  $Ko$). 
+Ces deux valeurs de paramÃ¨tres sont retenues par la suite.
 
 \begin{table}[ht]
-\caption{Average Testing Errors with respect to the the Kernel Size}
-\begin{scriptsize}
+\caption{Erreur moyenne de test en fonction de la taille du noyau}
 \centering
 \setlength{\tabcolsep}{3pt} 
 \begin{tabular}{|c|c|c|c|c|c|c|c|c|}
@@ -819,58 +638,48 @@ Table~\ref{table:choice:parameter}.
 \cline{3-9}
 \multicolumn{1}{c|}{}&  & $2$ & $4$&  $6$&  $8$& $10$& $12$ & $14$ \\
 \hline{}
-Average testing  
+Erreur moyenne    
 & \textit{0.1} & 39& 40.2&  39.7&  39.8& 40.1& $39.9$&  $39.8$  \\
 \cline{2-9}
-error for Kernel $K_y$ & \textit{0.4}& 15& 18.8& 19.1&  19.0& 18.6& 18.7 & 18.7 \\
+de test pour le noyau $K_y$ & \textit{0.4}& 15& 18.8& 19.1&  19.0& 18.6& 18.7 & 18.7 \\
 \hline
-Average testing & \textit{0.1} & 35.2 & 36.6&  36.7&  36.6& 37.1& 37.2 & 37.2 \\
+Erreur moyenne    & \textit{0.1} & 35.2 & 36.6&  36.7&  36.6& 37.1& 37.2 & 37.2 \\
 \cline{2-9}
-error for Kernel $K_o$ & \textit{0.4}  & 5.2 & 6.8& 7.5 & 7.9 & 8.1 & 8.2 & 7.6 \\
+de test pour le noyau $K_o$ & \textit{0.4}  & 5.2 & 6.8& 7.5 & 7.9 & 8.1 & 8.2 & 7.6 \\
 \hline
 \end{tabular}
 \label{table:choice:parameter}
-\end{scriptsize}
+
 \end{table}
 
-Thanks to these experiments, we observe that the size $N=4$ (respectively $N=12$) obtains sufficiently  large average testing errors for the $Ky$ based approach 
-(resp. for the $Ko$ based one). In what follows, these values are retained for these two methods.
-
-\subsection{Security Evaluation}
-As in the previous section, the BOSS contest database has been retained.
-To achieve a complete comparison with other steganographic tools,
-the whole database of 10,000 images has been used.
-Ensemble Classifier with SRM features is again used to evaluate the security of the approach.
-
-We have chosen 4 different payloads, 0.1, 0.2, 0.3, and 0.4, as in many steganographic evaluations.
-Three values are systematically given for each experiment:
-the area under the ROC curve (AUC),
-the average testing error (ATE),
-and the OOB error (OOB).
-
-All the results are summarized in Table~\ref{table:experiments:summary}.
-%and  in Figure~\ref{fig1}, Figure~\ref{fig2}, and Figure~\ref{fig3}.
-Let us analyse these experimental results. 
-The security approach is often lower than those observed with state of the art tools:
-for instance with payload $\alpha=0.1$, the most secure approach is WOW 
-with an average testing error equal to 0.43 whereas our approach reaches 0.38.
-However these results are  promising and for two reasons.
-First, our approaches give more resistance towards Ensemble Classifier (contrary to HUGO)
-for large payloads.
-Secondly, without any optimisation, our approach is not so far from state of the art steganographic tools.
-Finally, we explain the lack of security of the $Ko$ based approach with large payloads as follows:
-second order derivatives are indeed directly extracted from polynomial interpolation.
-This easy construction however induces large variations between the polynomial $L$ and 
-the pixel function $P$.
+
+\subsection{Ãvaluation de la sÃ©curitÃ©}
+Comme dans ce qui prÃ©cÃ¨de, la base du challenge BOSS a Ã©tÃ© retenue.
+Ici c'est cependant l'ensemble des 10000 images qui a Ã©tÃ© utilisÃ© pour Ã©valuer 
+la sÃ©curitÃ©.
+C'est aussi les caractÃ©ristiques SRM et Ensemble Classifier qui ont Ã©tÃ© utilisÃ©es 
+pour Ã©valuer la sÃ©curitÃ© de l'approche..
+Quatre taux d'embarquement  0.1, 0.2, 0.3 et  0.4
+ont Ã©tÃ© retenus. Pour chaque expÃ©rience, 
+l'aire sous la courbe de ROC (AUC),
+l'erreur moyenne de test (ATE), 
+l'erreur OOB (OOB) sont donnÃ©es et tous les rÃ©sultats sont synthÃ©tisÃ©s 
+dans le tableau~\ref{table:experiments:summary}.
+MÃªme si la sÃ©curitÃ© est souvent plus faible que celle observÃ©e 
+pour les outils les plus rÃ©cents, 
+les rÃ©sultats concernant $K_y$ sont encourageants car ils ne sont pas Ã©loignÃ©s de 
+ceux de l'Ã©tat de l'art sans aucune optimisation.
+Enfin la faible sÃ©curitÃ© de $K_o$ s'explique par le fait que le polynÃ´me interpole 
+exactement l'image en tous les points de la fenÃªtre, mais il ne tient pas forcÃ©ment 
+compte des variations dans celle-ci. Les dÃ©rivÃ©es secondes sont certes faciles 
+Ã  exprimer, mais elles ne reprÃ©sentent pas nÃ©cessairement fidÃ¨lement celles de l'image.
 
 \begin{table}
-\caption{Summary of experiments}\label{table:experiments:summary}
-\begin{small}
+\caption{Ãvaluation de la sÃ©curitÃ©}\label{table:experiments:summary}
 \centering
 \begin{tabular}{|l|l|l|l|l|}
 \hline
-
- & Payload & AUC & ATE   &  OOB \\ \hline
+ & Taux & AUC & ATE   &  OOB \\ \hline
 {WOW}   
             & 0.1 & 0.6501 & 0.4304 & 0.3974\\
             & 0.2 & 0.7583 & 0.3613 & 0.3169\\
@@ -893,13 +702,13 @@ the pixel function $P$.
             & 0.3 & 0.8720 & 0.2557 & 0.2212 \\
             & 0.4 & 0.9517 & 0.1472 & 0.1230 \\ 
  \hline
-{$Ky$ based approach} 
+{Approche Ã  base de $Ky$} 
             & 0.1 & 0.7378 & 0.3768 & 0.3306 \\
             & 0.2 & 0.8568 & 0.2839 & 0.2408 \\
             & 0.3 & 0.9176 & 0.2156 & 0.1710 \\
             & 0.4 & 0.9473 & 0.1638 & 0.1324\\
  \hline
-{$Ko$ based approach} 
+{Approche Ã  base de $Ko$} 
             & 0.1 & 0.6831 & 0.3696  & 0.3450 \\
             & 0.2 & 0.8524 & 0.1302  & 0.2408 \\
             & 0.3 & 0.9132 & 0.1023  & 0.1045 \\
@@ -907,7 +716,6 @@ the pixel function $P$.
                               \hline 
                               
 \end{tabular}
-\end{small}
 \end{table}
 
 
@@ -933,22 +741,15 @@ the pixel function $P$.
 
 
 \section{Conclusion}
-
-The first contribution of this paper is to propose of a distortion
-function which is based on second order derivatives. These
-partial derivatives allow to accurately compute 
-the level curves and thus to look favorably on pixels
-without clean level curves. 
-Two approaches to build these derivatives have been proposed.
-The first one is based on revisiting kernels usually embedded 
-in edge detection algorithms. 
-The second one is based on the polynomial approximation
-of the bitmap image.
-These two methods have been completely implemented.
-The first experiments have shown that the security level 
-is slightly inferior the one of the most stringent approaches. These first promising results encourage us to deeply investigate this research direction. 
-
-Future works aiming at improving the security of this proposal are planned as follows. The authors want first to focus on other approaches to provide second order derivatives with larger discrimination power.
-Then, the objective will be to deeply investigate whether the H\"older norm is optimal when the objective is to avoid null second order derivatives, and to give priority to the largest second order values.
-
-
+La principale contribution de ce chapitre est de proposer des 
+fonctions de distorsion basÃ©es sur des approximations de dÃ©rivÃ©es 
+secondes, l'idÃ©e sous-jacente Ã©tant qu'une zone oÃ¹ les lignes de niveau 
+ne sont pas clairement dÃ©finies est peu prÃ©visible.
+Deux approches d'approximation ont Ã©tÃ© prÃ©sentÃ©es.
+La premiÃ¨re  basÃ©e 
+sur un produit de convolution, exploite des noyaux dÃ©jÃ  intÃ©grÃ©s dans des
+algorithmes de dÃ©tection de bords.
+La seconde s'appuie sur une interpolation polynomiale de l'image.
+Ces deux mÃ©thodes ont Ã©tÃ© complÃ¨tement implantÃ©es et leur sÃ©curitÃ© 
+face Ã  des stÃ©ganalyseurs a Ã©tÃ© Ã©tudiÃ©e. Les rÃ©sultats encouragent 
+Ã  poursuivre dans cette direction.
\ No newline at end of file
