This section first presents the embedding scheme through its 
four main steps: the data encryption (Sect.~\ref{sub:bbs}),
the cover pixel selection (Sect.~\ref{sub:edge}),
the adaptive payload considerations (Sect.~\ref{sub:adaptive}),
and how the distortion has been minimized (Sect.~\ref{sub:stc}).
The message extraction is finally presented  (Sect.\ref{sub:extract}) and a running example ends this section (Sect.~\ref{sub:xpl}). 


The flowcharts given in Fig.~\ref{fig:sch}
summarize our steganography scheme denoted by
STABYLO, which stands for STeganography with cAnny, Bbs, binarY embedding at LOw cost.
What follows are successively details of the inner steps and flows inside 
both the embedding stage (Fig.~\ref{fig:sch:emb}) 
and the extraction one (Fig.~\ref{fig:sch:ext}).
Let us first focus on the data embedding. 

\begin{figure*}[t]
  \begin{center}
    \subfloat[Data Embedding.]{
      \begin{minipage}{0.49\textwidth}
        \begin{center}
          %\includegraphics[width=5cm]{emb.pdf}
          \includegraphics[scale=0.5]{emb.ps}
        \end{center}
      \end{minipage}
      \label{fig:sch:emb}
    }%\hfill
    \subfloat[Data Extraction.]{
      \begin{minipage}{0.49\textwidth}
        \begin{center}
          %\includegraphics[width=5cm]{rec.pdf}
          \includegraphics[scale=0.5]{rec.ps}
        \end{center}
      \end{minipage}
      \label{fig:sch:ext}
    }%\hfill
  \end{center}
  \caption{The STABYLO Scheme.}
  \label{fig:sch}
\end{figure*}








\subsection{Security Considerations}\label{sub:bbs}
Among methods of message encryption/decryption 
(see~\cite{DBLP:journals/ejisec/FontaineG07} for a survey)
we implement the Blum-Goldwasser cryptosystem~\cite{Blum:1985:EPP:19478.19501}
that is based on the Blum Blum Shub~\cite{DBLP:conf/crypto/ShubBB82} 
pseudorandom number generator (PRNG) and the 
XOR binary function.
It has been indeed proven~\cite{DBLP:conf/crypto/ShubBB82} that this PRNG 
has the property of cryptographical security, \textit{i.e.}, 
for any sequence of $L$ output bits $x_i$, $x_{i+1}$, \ldots, $x_{i+L-1}$,
there is no algorithm, whose time complexity is polynomial  in $L$, and 
which allows to find $x_{i-1}$ and $x_{i+L}$ with a probability greater
than $1/2$.
Equivalent formulations of such a property can
be found. They all lead to the fact that,
even if the encrypted message is extracted, 
it is impossible to retrieve the original one in 
polynomial time.   

Starting thus with a key $k$ and the message \textit{mess} to hide, 
this step computes a message $m$, which is the encrypted version  of \textit{mess}.


\subsection{Edge-Based Image Steganography}\label{sub:edge}


The edge-based image
steganography schemes 
already presented \cite{Luo:2010:EAI:1824719.1824720,DBLP:journals/eswa/ChenCL10} differ 
in how carefully they select edge pixels, and  
how they modify them.

%Image Quality: Edge Image Steganography
%\JFC{Raphael, les fuzzy edge detection sont souvent utilisés. 
%  il faudrait comparer les approches en terme de nombre de bits retournés,
%  en terme de complexité. } \cite{KF11}
%\RC{Ben, à voir car on peut choisir le nombre de pixel avec Canny. Supposons que les fuzzy edge soient retourne un peu plus de points, on sera probablement plus détectable...  Finalement on devrait surement vendre notre truc en : on a choisi cet algo car il est performant en vitesse/qualité. Mais on peut aussi en utilisé d'autres :-)}

Many techniques have been proposed in the literature to  detect 
edges in  images (whose noise has been initially reduced). 
They can be separated in two categories: first and second order detection
methods on the one hand, and fuzzy detectors on the other  hand~\cite{KF11}.
In first order methods like Sobel, Canny~\cite{Canny:1986:CAE:11274.11275}, \ldots, 
a first-order derivative (gradient magnitude, etc.) is computed 
to search for local maxima, whereas in second order ones, zero crossings in a second-order derivative, like the Laplacian computed from the image,
are searched in order to find edges.
As for as fuzzy edge methods are concerned, they are obviously based on fuzzy logic to highlight edges.

Canny filters, on their parts, are an old family of algorithms still remaining a state-of-the-art edge detector. They can be well approximated by first-order derivatives of Gaussians.
As the Canny algorithm is well known and studied, fast, and implementable
on many  kinds of architectures like FPGAs, smartphones,  desktop machines, and
GPUs, we have chosen this edge detector for illustrative purpose.

\JFC{il faudrait comparer les complexites des algo fuzy and canny}


This edge detection is applied on a filtered version of the image given 
as input.
More precisely, only $b$ most 
significant bits are concerned by this step, where 
the parameter $b$ is practically set with $6$ or $7$. 
If set with the same value $b$, the edge detection returns thus the same 
set of pixels for both the cover and the stego image.   
In our flowcharts, this is represented by ``edgeDetection(b bits)''.
Then only the 2 LSBs of pixels in the set of edges are returned if $b$ is 6, 
and the LSB of pixels if $b$ is 7.





Let $x$ be the sequence of these bits. 
The next  section section presentsd how our scheme 
adapts  when the size of $x$  is not sufficient for the message $m$ to embed.


 




\subsection{Adaptive Embedding Rate}\label{sub:adaptive}
Two strategies have been developed in our scheme, 
depending on the embedding rate that is either \emph{adaptive} or \emph{fixed}.
In the former the embedding rate depends on the number of edge pixels.
The higher it is, the larger the message length that can be inserted is.
Practically, a set of edge pixels is computed according to the 
Canny algorithm with an high threshold.
The message length is thus defined to be less than 
half of this set cardinality.
If $x$ is then to short for $m$, the message is splitted into sufficient parts. 
In the latter, the embedding rate is defined as a percentage between the 
number of modified pixels and the length of the bit message.
This is the classical approach adopted in steganography.
Practically, the Canny algorithm generates  
a set of edge pixels related to a threshold that is decreasing 
until its cardinality
is sufficient. 



Two methods may further be applied to select bits that 
will be modified. 
The first one randomly chooses the subset of pixels to modify by 
applying the BBS PRNG again. This method is further denoted  as to \emph{sample}.
The second one is a direct application of the 
STC algorithm~\cite{DBLP:journals/tifs/FillerJF11}. 
It  is further referred to as \emph{adaptive+STC} and is detailled in the nex section.





% 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.   







\subsection{Minimizing Distortion with Syndrome-Treillis Codes}\label{sub:stc}
\input{stc}



% 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}
