j'ai rajouté une figure avec l'erreur, la valeur du milieu est interpolée :-)

[canny.git] / ourapproach.tex

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 then presented  (Sect.~\ref{sub:extract}) while 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 STe\-ga\-no\-gra\-phy with 
Adaptive, Bbs, binarY embedding at LOw cost.
What follows are successively some details of the inner steps and the flows both inside 
 the embedding stage (Fig.~\ref{fig:sch:emb}) 
and inside 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}{0.4\textwidth}
        \begin{center}
            %\includegraphics[scale=0.45]{emb}
            \includegraphics[scale=0.4]{emb}
        \end{center}
      \end{minipage}
      \label{fig:sch:emb}
    } 
\hfill
    \subfloat[Data Extraction]{
      \begin{minipage}{0.49\textwidth}
        \begin{center}
            \includegraphics[scale=0.4]{dec}
        \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}
\JFC{To provide a self-contained article without any bias, we shortly 
pressent the retained encryption process.}
Among the methods of  message encryption/decryption 
(see~\cite{DBLP:journals/ejisec/FontaineG07} for a survey)
we implement the asymmetric 
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.
The main justification of this choice 
is that it has been 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 $x_{i-1}$ or $x_{i+L}$ 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}, and so on, 
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 far 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 fast, well known, has been studied in depth, and is implementable
on many  kinds of architectures like FPGAs, smart phones,  desktop machines, and
GPUs, we have chosen this edge detector for illustrative purpose.




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 $7$. 
Notice that 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.
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.   
Moreover, to provide edge gradient value, 
the Canny algorithm computes derivatives  
in the two directions with respect to a mask of size $T$. 
The higher $T$ is, the coarse the approach is. Practically, 
$T$ is set with $3$, $5$, or $7$.
In our flowcharts, this step is represented by ``Edge Detection(b, T, X)''.


Let $x$ be the sequence of these bits. 
The next  section presents how to adapt our scheme 
with respect to the size
of the message $m$ to embed and the size of the cover $x$.



 




\subsection{Adaptive embedding rate}\label{sub:adaptive}
Two strategies have been developed in our approach, 
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 parameters $b=7$ and $T=3$.
The message length is thus defined to be lesser than 
half of this set cardinality.
If $x$ is too short for $m$, the message is split into sufficient parts
and a new cover image should be used for the remaining part of the message. 

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 increasing values of $T$ and 
until its cardinality
is sufficient. Even in this situation, our scheme adapts
its algorithm to meet all the user's requirements. 


Once the map of possibly modified pixels is computed, 
two methods may further be applied to extract bits that 
are really changed. 
The first one randomly chooses the subset of pixels to modify by 
applying the BBS PRNG again. This method is further denoted  as a \emph{sample}.
Once this set is selected, a classical LSB replacement is applied to embed the 
stego content.
The second method considers the last significant bits of all the pixels 
inside the previous map. It next directly applies the STC 
algorithm~\cite{DBLP:journals/tifs/FillerJF11}. 
It  is further referred to as \emph{STC} and is detailed in the next section.








\subsection{Minimizing distortion with Syndrome-Trellis 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.   

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}