+
+% 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.
+
+For a given set of parameters,
+the Canny algorithm returns a numerical value and
+states whether a given pixel is an edge or not.
+In this article, in the Adaptive strategy
+we consider that all the edge pixels that
+have been selected by this algorithm have the same
+distortion cost, \textit{i.e.}, $\rho_X$ is always 1 for these bits.
+In the Fixed strategy, since pixels that are detected to be edge
+with small values of $T$ (e.g., when $T=3$)
+are more accurate than these with higher values of $T$,
+we give to STC the following distortion map of the corresponding bits
+$$
+\rho_X= \left\{
+\begin{array}{l}
+1 \textrm{ if an edge for $T=3$,} \\
+10 \textrm{ if an edge for $T=5$,} \\
+100 \textrm{ if an edge for $T=7$.}
+\end{array}
+\right.
+$$
+
+
+
+
+\subsection{Data extraction}\label{sub:extract}
+The message extraction summarized in Fig.~\ref{fig:sch:ext}
+follows the data embedding approach
+since there exists a reverse function for all its steps.
+
+More precisely, let $b$ be the most significant bits and
+$T$ be the size of the Canny mask, both be given as a key.
+Thus, the same edge detection is applied on a stego content $Y$ to
+produce the sequence $y$ of LSBs.
+If the STC approach has been selected in embedding, the STC reverse
+algorithm is directly executed to retrieve the encrypted message.
+This inverse function takes the $\hat{H}$ matrix as a parameter.
+Otherwise, \textit{i.e.}, if the \emph{sample} strategy is retained,
+the same random bit selection than in the embedding step
+is executed with the same seed, given as a key.
+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 $512\times512$ image with 256 grayscale levels.
+The message is the poem Ulalume (E. A. Poe), which is constituted by 104 lines, 667
+words, and 3,754 characters, \textit{i.e.}, 30,032 bits.
+Lena and the first verses are given in Fig.~\ref{fig:lena}.
+
+\begin{figure}
+\begin{center}
+\begin{minipage}{0.49\linewidth}
+\begin{center}
+\includegraphics[scale=0.20]{lena512}
+\end{center}
+\end{minipage}
+\begin{minipage}{0.49\linewidth}
+\begin{flushleft}
+\begin{scriptsize}
+The skies they were ashen and sober;\linebreak
+$\qquad$ The leaves they were crisped and sere—\linebreak
+$\qquad$ The leaves they were withering and sere;\linebreak
+It was night in the lonesome October\linebreak
+$\qquad$ Of my most immemorial year;\linebreak
+It was hard by the dim lake of Auber,\linebreak
+$\qquad$ In the misty mid region of Weir—\linebreak
+It was down by the dank tarn of Auber,\linebreak
+$\qquad$ 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 18,641 and 18,455 pixels when $b$ is
+respectively 7 and 6 and $T=3$.
+These edges are represented in Figure~\ref{fig:edge}.
+When $b$ is 7, it remains one bit per pixel to build the cover vector.
+This configuration leads to a cover vector of size 18,641 if b is 7
+and 36,910 if $b$ is 6.
+
+\begin{figure}[t]
+ \begin{center}
+ \subfloat[$b$ is 7.]{
+ \begin{minipage}{0.49\linewidth}
+ \begin{center}
+ %\includegraphics[width=5cm]{emb.pdf}
+ \includegraphics[scale=0.20]{edge7}
+ \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=0.20]{edge6}
+ \end{center}
+ \end{minipage}
+ %\label{fig:sch:ext}
+ }%\hfill
+ \end{center}
+ \caption{Edge detection wrt $b$ with $T=3$}
+ \label{fig:edge}
+\end{figure}
+
+
+
+The STC algorithm is optimized when the rate between message length and
+cover vector length is lower than 1/2.
+So, only 9,320 bits are available for embedding
+in the configuration where $b$ is 7.
+
+When $b$ is 6, we could have considered 18,455 bits for the message.
+However, first experiments have shown that modifying this number of bits is too
+easily detectable.
+So, we choose to modify the same amount of bits (9,320) and keep STC optimizing
+which bits to change among the 36,910 ones.
+
+In the two cases, about the third part of the poem is hidden into the cover.
+Results with {Adaptive} and {STC} strategies 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=0.20]{lena7}
+ \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=0.20]{lena6}
+ \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 pair of pixel $X_{ij}$ and $Y_{ij}$ ($X$ and $Y$ being the cover and the stego content respectively),
+the pixel value $V_{ij}$ of the difference is defined with the following map
+$$
+V_{ij}= \left\{
+\begin{array}{rcl}
+0 & \textrm{if} & X_{ij} = Y_{ij} \\
+75 & \textrm{if} & \vert X_{ij} - Y_{ij} \vert = 1 \\
+150 & \textrm{if} & \vert X_{ij} - Y_{ij} \vert = 2 \\
+225 & \textrm{if} & \vert X_{ij} - Y_{ij} \vert = 3
+\end{array}
+\right.
+$$
+This function allows to emphasize differences between contents.
+Notice that since $b$ is 7 in Fig.~\ref{fig:diff7}, the embedding is binary
+and this image only contains 0 and 75 values.
+Similarly, if $b$ is 6 as in Fig.~\ref{fig:diff6}, the embedding is ternary
+and the image contains all the values in $\{0,75,150,225\}$.
+
+
+
+\begin{figure}[t]
+ \begin{center}
+ \subfloat[$b$ is 7.]{
+ \begin{minipage}{0.49\linewidth}
+ \begin{center}
+ %\includegraphics[width=5cm]{emb.pdf}
+ \includegraphics[scale=0.20]{diff7}
+ \end{center}
+ \end{minipage}
+ \label{fig:diff7}
+ }%\hfill
+ \subfloat[$b$ is 6.]{
+ \begin{minipage}{0.49\linewidth}
+ \begin{center}
+ %\includegraphics[width=5cm]{rec.pdf}
+ \includegraphics[scale=0.20]{diff6}
+ \end{center}
+ \end{minipage}
+ \label{fig:diff6}
+ }%\hfill
+ \end{center}
+ \caption{Differences with Lena's cover wrt $b$}
+ \label{fig:lenadiff}
+\end{figure}
+
+
+
