[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}) 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 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}{0.49\textwidth}
        \begin{center}
          %\includegraphics[width=5cm]{emb.pdf}
          \includegraphics[scale=0.45]{emb.ps}
        \end{center}
      \end{minipage}
      \label{fig:sch:emb}
    } 

    \subfloat[Data Extraction.]{
      \begin{minipage}{0.49\textwidth}
        \begin{center}
          %\includegraphics[width=5cm]{rec.pdf}
          \includegraphics[scale=0.45]{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 the 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 $x_{i-1}$ 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}, \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 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, 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 presents 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 a high threshold.
The message length is thus defined to be less than 
half of this set cardinality.
If $x$ is then 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 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 a \emph{sample}.
Once this set is selected, a classical LSB replacement is applied to embed the 
stego content.
The second method is a direct application of the 
STC algorithm~\cite{DBLP:journals/tifs/FillerJF11}. 
It  is further referred to as \emph{STC} and is detailed in the next 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-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.   

% 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 the data embedding approach 
since there exists a reverse function for all its steps.

More precisely, the same edge detection is applied on the $b$ first bits  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 $H$ 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]{Lena.eps}
\end{center}
\end{minipage}
\begin{minipage}{0.49\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 18,641 and 18,455 pixels when $b$ is
respectively 7 and 6. These edges are represented in Figure~\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=0.20]{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=0.20]{edge6.eps}
        \end{center}
      \end{minipage}
      %\label{fig:sch:ext}
    }%\hfill
  \end{center}
  \caption{Edge detection wrt $b$}
  \label{fig:edge}
\end{figure}



Only 9,320 bits (resp. 9,227 bits) are available for embedding 
in the former configuration where $b$ is 7 (resp. where $b$ is 6).
In both cases, 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=0.20]{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=0.20]{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 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.

\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.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=0.20]{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}



\section{Complexity Analysis}\label{sub:complexity}
This section aims at justifying the leightweight attribute of our approach.
To be more precise, we compare the complexity of our schemes to the 
state of the art steganography, namely HUGO~\cite{DBLP:conf/ih/PevnyFB10}.


In what folllows, we consider an $n \times n$ square image. 
First of all, HUGO starts with computing the second order SPAM Features.
This steps is in  $O(n^2 + 2.343^2)$ due to the calculation 
of the difference arrays and next of the 686 features (of size 343).
Next for each pixel, the distortion measure is calculated by +1/-1 modifying
its value and computing again the SPAM 
features. Pixels are thus selected according to their ability to provide
an image whose SPAM features are close to the original one. 
The algorithm is thus computing a distance between each Feature, 
which is at least in $O(343)$ and an overall distance between these 
metrics which is in $O(686)$. Computing the distance is thus in 
$O(2\time 343^2)$ and this mdification is thus in $O(2\time 343^2 \time n^2)$.
Ranking these results may be achieved with a insertion sort which is in $2.n^2 \ln(n)$.
The overall complexity of the pixel selection is thus 
$O(n^2 +2.343^2 + 2\time 343^2 \time n^2 + 2.n^2 \ln(n))$, \textit{i.e}
$O(2.n^2(343^2 + \ln(n)))$.

Our edge selection is based on a Canny  Filter, 
whose complexity is in $O(2n^2.\ln(n))$ thanks to the convolution step
which can be implemented with FFT.
The complexity of Hugo is  at least $343^2/\ln{n}$ times higher than our scheme.