+
+% Edge Based Image Steganography schemes
+% already studied~\cite{Luo:2010:EAI:1824719.1824720,DBLP:journals/eswa/ChenCL10,DBLP:conf/ih/PevnyFB10} differ
+% how they select edge pixels, and
+% how they modify these ones.
+
+% First of all, let us discuss about compexity of edge detetction methods.
+% Let then $M$ and $N$ be the dimension of the original image.
+% According to~\cite{Hu:2007:HPE:1282866.1282944},
+% even if the fuzzy logic based edge detection methods~\cite{Tyan1993}
+% have promising results, its complexity is in $C_3 \times O(M \times N)$
+% whereas the complexity on the Canny method~\cite{Canny:1986:CAE:11274.11275}
+% is in $C_1 \times O(M \times N)$ where $C_1 < C_3$.
+% \JFC{Verifier ceci...}
+% In experiments detailled in this article, the Canny method has been retained
+% but the whole approach can be updated to consider
+% the fuzzy logic edge detector.
+
+% Next, following~\cite{Luo:2010:EAI:1824719.1824720}, our scheme automatically
+% modifies Canny parameters to get a sufficiently large set of edge bits: this
+% one is practically enlarged untill its size is at least twice as many larger
+% than the size of embedded message.
+
+
+
+%%RAPH: paragraphe en double :-)
+
+
+
+
+\subsection{Data Extraction}\label{sub:extract}
+The message extraction summarized in Fig.~\ref{fig:sch:ext} follows data embedding
+since there exists a reverse function for all its steps.
+First of all, the same edge detection is applied (on the 7 first bits) to
+get the set of LSBs,
+which is sufficiently large with respect to the message size given as a key.
+Then the STC reverse algorithm is applied to retrieve the encrypted message.
+Finally, the Blum-Goldwasser decryption function is executed and the original
+message is extracted.
+
+
+\subsection{Running Example}\label{sub:xpl}
+In this example, the cover image is Lena
+which is a 512*512 image with 256 grayscale levels.
+The message is the poem Ulalume (E. A. Poe), which is constituted by 104 lines, 667
+words, and 3754 characters, \textit{i.e.} 30032 bits.
+Lena and the the first verses are given in Fig.~\ref{fig:lena}.
+
+\begin{figure}
+\begin{center}
+\begin{minipage}{0.4\linewidth}
+\includegraphics[width=3cm]{Lena.eps}
+\end{minipage}
+\begin{minipage}{0.59\linewidth}
+\begin{flushleft}
+\begin{scriptsize}
+The skies they were ashen and sober;\linebreak
+$~$ The leaves they were crisped and sere—\linebreak
+$~$ The leaves they were withering and sere;\linebreak
+It was night in the lonesome October\linebreak
+$~$ Of my most immemorial year;\linebreak
+It was hard by the dim lake of Auber,\linebreak
+$~$ In the misty mid region of Weir—\linebreak
+It was down by the dank tarn of Auber,\linebreak
+$~$ In the ghoul-haunted woodland of Weir.
+\end{scriptsize}
+\end{flushleft}
+\end{minipage}
+\end{center}
+\caption{Cover and message examples} \label{fig:lena}
+\end{figure}
+
+The edge detection returns 18641 and 18455 pixels when $b$ is
+respectively 7 and 6. These edges are represented in Fig.~\ref{fig:edge}
+
+
+\begin{figure}[t]
+ \begin{center}
+ \subfloat[$b$ is 7.]{
+ \begin{minipage}{0.49\linewidth}
+ \begin{center}
+ %\includegraphics[width=5cm]{emb.pdf}
+ \includegraphics[scale=0.15]{edge7.eps}
+ \end{center}
+ \end{minipage}
+ %\label{fig:sch:emb}
+ }%\hfill
+ \subfloat[$b$ is 6.]{
+ \begin{minipage}{0.49\linewidth}
+ \begin{center}
+ %\includegraphics[width=5cm]{rec.pdf}
+ \includegraphics[scale=0.15]{edge6.eps}
+ \end{center}
+ \end{minipage}
+ %\label{fig:sch:ext}
+ }%\hfill
+ \end{center}
+ \caption{Edge Detection wrt $b$.}
+ \label{fig:edge}
+\end{figure}
+
+
+
+In the former configuration, only 9320 bits are available
+for embeding whereas in the latter we have 9227.
+In the both case, about the third part of the poem is hidden into the cover.
+Results with \emph{adaptive+STC} strategy are presented in
+Fig.~\ref{fig:lenastego}.
+
+\begin{figure}[t]
+ \begin{center}
+ \subfloat[$b$ is 7.]{
+ \begin{minipage}{0.49\linewidth}
+ \begin{center}
+ %\includegraphics[width=5cm]{emb.pdf}
+ \includegraphics[scale=0.15]{lena7.eps}
+ \end{center}
+ \end{minipage}
+ %\label{fig:sch:emb}
+ }%\hfill
+ \subfloat[$b$ is 6.]{
+ \begin{minipage}{0.49\linewidth}
+ \begin{center}
+ %\includegraphics[width=5cm]{rec.pdf}
+ \includegraphics[scale=0.15]{lena6.eps}
+ \end{center}
+ \end{minipage}
+ %\label{fig:sch:ext}
+ }%\hfill
+ \end{center}
+ \caption{Stego Images wrt $b$.}
+ \label{fig:lenastego}
+\end{figure}
+
+
+Finally, differences between the original cover and the stego images
+are presented in Fig.~\ref{fig:lenadiff}. For each pixel pair of picel $X_{ij}$
+$Y_{ij}$, $X$ and $Y$ being the cover and the stego content respectively,
+The pixel value $V_{ij}$ of the difference is defined with the following map
+$$
+V_{ij}= \left\{
+\begin{array}{rcl}
+0 & \textrm{if} & X_{ij} = Y_{ij} \\
+75 & \textrm{if} & \abs{ (X_{ij} - Y_{ij})} = 1 \\
+75 & \textrm{if} & \abs{ (X_{ij} - Y_{ij})} = 2 \\
+225 & \textrm{if} & \abs{ (X_{ij} - Y_{ij})} = 1
+\end{array}
+\right.
+$$.
+This function allows to emphase differences between content.
+
+\begin{figure}[t]
+ \begin{center}
+ \subfloat[$b$ is 7.]{
+ \begin{minipage}{0.49\linewidth}
+ \begin{center}
+ %\includegraphics[width=5cm]{emb.pdf}
+ \includegraphics[scale=0.15]{diff7.eps}
+ \end{center}
+ \end{minipage}
+ %\label{fig:sch:emb}
+ }%\hfill
+ \subfloat[$b$ is 6.]{
+ \begin{minipage}{0.49\linewidth}
+ \begin{center}
+ %\includegraphics[width=5cm]{rec.pdf}
+ \includegraphics[scale=0.15]{diff6.eps}
+ \end{center}
+ \end{minipage}
+ %\label{fig:sch:ext}
+ }%\hfill
+ \end{center}
+ \caption{Differences with Lena's Cover wrt $b$.}
+ \label{fig:lenadiff}
+\end{figure}
