complexity

[canny.git] / ourapproach.tex

diff --git a/ourapproach.tex b/ourapproach.tex
index b007dd00eb14f4b016664f6bce01f0aee64783e8..1b2ee1a619182b8899607356d17db4c7e2d74370 100644 (file)
--- a/ourapproach.tex
+++ b/ourapproach.tex
@@ -3,18 +3,19 @@ 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 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 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
 
 
 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}).
+STABYLO, which stands for STeganography 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. 
 
 Let us first focus on the data embedding. 
 

-\begin{figure*}[t]
+\begin{figure*}%[t]

   \begin{center}
     \subfloat[Data Embedding.]{
       \begin{minipage}{0.49\textwidth}
   \begin{center}
     \subfloat[Data Embedding.]{
       \begin{minipage}{0.49\textwidth}


@@ -24,7 +25,8 @@ Let us first focus on the data embedding.
         \end{center}
       \end{minipage}
       \label{fig:sch:emb}
         \end{center}
       \end{minipage}
       \label{fig:sch:emb}

-    }%\hfill
+    } 
+

     \subfloat[Data Extraction.]{
       \begin{minipage}{0.49\textwidth}
         \begin{center}
     \subfloat[Data Extraction.]{
       \begin{minipage}{0.49\textwidth}
         \begin{center}


@@ -35,7 +37,7 @@ Let us first focus on the data embedding.
       \label{fig:sch:ext}
     }%\hfill
   \end{center}
       \label{fig:sch:ext}
     }%\hfill
   \end{center}

-  \caption{The STABYLO Scheme.}
+  \caption{The STABYLO scheme}

   \label{fig:sch}
 \end{figure*}
 
   \label{fig:sch}
 \end{figure*}
 


@@ -46,8 +48,8 @@ Let us first focus on the data embedding.
 
 
 
 
 
 

-\subsection{Security Considerations}\label{sub:bbs}
-Among methods of message encryption/decryption 
+\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} 
 (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} 


@@ -57,7 +59,7 @@ 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 
 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
+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,
 than $1/2$.
 Equivalent formulations of such a property can
 be found. They all lead to the fact that,


@@ -69,7 +71,7 @@ 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}.
 
 
 this step computes a message $m$, which is the encrypted version  of \textit{mess}.
 
 

-\subsection{Edge-Based Image Steganography}\label{sub:edge}
+\subsection{Edge-based image steganography}\label{sub:edge}

 
 
 The edge-based image
 
 
 The edge-based image


@@ -92,14 +94,14 @@ In first order methods like Sobel, Canny~\cite{Canny:1986:CAE:11274.11275}, \ldo
 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.
 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.
+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 well known and studied, fast, and implementable
+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.
 
 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}
+%\JFC{il faudrait comparer les complexites des algo fuzy and canny}

 
 
 This edge detection is applied on a filtered version of the image given 
 
 
 This edge detection is applied on a filtered version of the image given 


@@ -118,7 +120,7 @@ and the LSB of pixels if $b$ is 7.
 
 
 Let $x$ be the sequence of these bits. 
 
 
 Let $x$ be the sequence of these bits. 

-The next  section section presents how our scheme 
+The next  section presents how our scheme 

 adapts  when the size of $x$  is not sufficient for the message $m$ to embed.
 
 
 adapts  when the size of $x$  is not sufficient for the message $m$ to embed.
 
 


@@ -127,16 +129,16 @@ adapts  when the size of $x$  is not sufficient for the message $m$ to embed.
 
 
 
 
 
 

-\subsection{Adaptive Embedding Rate}\label{sub:adaptive}
+\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 
 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.
+Canny algorithm with a high threshold.

 The message length is thus defined to be less than 
 half of this set cardinality.
 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 split into sufficient parts
+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. 
 
  
 and a new cover image should be used for the remaining part of the message. 
 
  


@@ -153,7 +155,7 @@ 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 
 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}.
+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 
 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 


@@ -182,7 +184,7 @@ 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}
+\subsection{Minimizing distortion with syndrome-trellis codes}\label{sub:stc}

 \input{stc}
 
 
 \input{stc}
 
 


@@ -216,7 +218,7 @@ It  is further referred to as \emph{STC} and is detailed in the next section.
 
 
 
 
 
 

-\subsection{Data Extraction}\label{sub:extract}
+\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.
 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.


@@ -226,26 +228,28 @@ 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.
 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,
+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.
 
 
 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 
+\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
 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 3754 characters, \textit{i.e.}  30032 bits.
-Lena and the the first verses are given in Fig.~\ref{fig:lena}.
+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{figure}
 \begin{center}

-\begin{minipage}{0.4\linewidth}
-\includegraphics[width=3cm]{Lena.eps}
+\begin{minipage}{0.49\linewidth}
+\begin{center}
+\includegraphics[scale=0.20]{Lena.eps}
+\end{center}

 \end{minipage}
 \end{minipage}

-\begin{minipage}{0.59\linewidth}
+\begin{minipage}{0.49\linewidth}

 \begin{flushleft}
 \begin{scriptsize}
 The skies they were ashen and sober;\linebreak
 \begin{flushleft}
 \begin{scriptsize}
 The skies they were ashen and sober;\linebreak


@@ -264,8 +268,8 @@ $~$ In the ghoul-haunted woodland of Weir.
 \caption{Cover and message examples} \label{fig:lena}
 \end{figure}
 
 \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}
+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{figure}[t]


@@ -274,7 +278,7 @@ respectively 7 and 6. These edges are represented in Fig.~\ref{fig:edge}
       \begin{minipage}{0.49\linewidth}
         \begin{center}
           %\includegraphics[width=5cm]{emb.pdf}
       \begin{minipage}{0.49\linewidth}
         \begin{center}
           %\includegraphics[width=5cm]{emb.pdf}

-          \includegraphics[scale=0.15]{edge7.eps}
+          \includegraphics[scale=0.20]{edge7.eps}

         \end{center}
       \end{minipage}
       %\label{fig:sch:emb}
         \end{center}
       \end{minipage}
       %\label{fig:sch:emb}


@@ -283,21 +287,21 @@ respectively 7 and 6. These edges are represented in Fig.~\ref{fig:edge}
       \begin{minipage}{0.49\linewidth}
         \begin{center}
           %\includegraphics[width=5cm]{rec.pdf}
       \begin{minipage}{0.49\linewidth}
         \begin{center}
           %\includegraphics[width=5cm]{rec.pdf}

-          \includegraphics[scale=0.15]{edge6.eps}
+          \includegraphics[scale=0.20]{edge6.eps}

         \end{center}
       \end{minipage}
       %\label{fig:sch:ext}
     }%\hfill
   \end{center}
         \end{center}
       \end{minipage}
       %\label{fig:sch:ext}
     }%\hfill
   \end{center}

-  \caption{Edge Detection wrt $b$.}
+  \caption{Edge detection wrt $b$}

   \label{fig:edge}
 \end{figure}
 
 
 
   \label{fig:edge}
 \end{figure}
 
 
 

-Only 9320 bits (resp. 9227 bits) are available for embedding 
+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 the former configuration where $b$ is 7 (resp. where $b$ is 6).

-In the both case, about the third part of the poem is hidden into the cover.
+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}.
 
 Results with \emph{adaptive+STC} strategy are presented in 
 Fig.~\ref{fig:lenastego}.
 


@@ -307,7 +311,7 @@ Fig.~\ref{fig:lenastego}.
       \begin{minipage}{0.49\linewidth}
         \begin{center}
           %\includegraphics[width=5cm]{emb.pdf}
       \begin{minipage}{0.49\linewidth}
         \begin{center}
           %\includegraphics[width=5cm]{emb.pdf}

-          \includegraphics[scale=0.15]{lena7.eps}
+          \includegraphics[scale=0.20]{lena7.eps}

         \end{center}
       \end{minipage}
       %\label{fig:sch:emb}
         \end{center}
       \end{minipage}
       %\label{fig:sch:emb}


@@ -316,31 +320,31 @@ Fig.~\ref{fig:lenastego}.
       \begin{minipage}{0.49\linewidth}
         \begin{center}
           %\includegraphics[width=5cm]{rec.pdf}
       \begin{minipage}{0.49\linewidth}
         \begin{center}
           %\includegraphics[width=5cm]{rec.pdf}

-          \includegraphics[scale=0.15]{lena6.eps}
+          \includegraphics[scale=0.20]{lena6.eps}

         \end{center}
       \end{minipage}
       %\label{fig:sch:ext}
     }%\hfill
   \end{center}
         \end{center}
       \end{minipage}
       %\label{fig:sch:ext}
     }%\hfill
   \end{center}

-  \caption{Stego Images wrt $b$.}
+  \caption{Stego images wrt $b$}

   \label{fig:lenastego}
 \end{figure}
 
 
 Finally, differences between the original cover and the stego images  
   \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 pixel $X_{ij}$ and  $Y_{ij}$ ($X$ and $Y$ being the cover and the stego content respectively), 
+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
 $$
 V_{ij}= \left\{
 \begin{array}{rcl}
 0 & \textrm{if} &  X_{ij} = Y_{ij} \\
 the pixel value $V_{ij}$ of the difference is defined with the following map
 $$
 V_{ij}= \left\{
 \begin{array}{rcl}
 0 & \textrm{if} &  X_{ij} = Y_{ij} \\

-75 & \textrm{if} &  \abs{ X_{ij} - Y_{ij}} = 1 \\
-150 & \textrm{if} &  \abs{ X_{ij} - Y_{ij}} = 2 \\
-225 & \textrm{if} &  \abs{ X_{ij} - Y_{ij}} = 3 
+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..
 $$
 \end{array}
 \right..
 $$

-This function allows to emphasize differences between content.
+This function allows to emphasize differences between contents.

 
 \begin{figure}[t]
   \begin{center}
 
 \begin{figure}[t]
   \begin{center}


@@ -348,7 +352,7 @@ This function allows to emphasize differences between content.
       \begin{minipage}{0.49\linewidth}
         \begin{center}
           %\includegraphics[width=5cm]{emb.pdf}
       \begin{minipage}{0.49\linewidth}
         \begin{center}
           %\includegraphics[width=5cm]{emb.pdf}

-          \includegraphics[scale=0.15]{diff7.eps}
+          \includegraphics[scale=0.20]{diff7.eps}

         \end{center}
       \end{minipage}
       %\label{fig:sch:emb}
         \end{center}
       \end{minipage}
       %\label{fig:sch:emb}


@@ -357,12 +361,15 @@ This function allows to emphasize differences between content.
       \begin{minipage}{0.49\linewidth}
         \begin{center}
           %\includegraphics[width=5cm]{rec.pdf}
       \begin{minipage}{0.49\linewidth}
         \begin{center}
           %\includegraphics[width=5cm]{rec.pdf}

-          \includegraphics[scale=0.15]{diff6.eps}
+          \includegraphics[scale=0.20]{diff6.eps}

         \end{center}
       \end{minipage}
       %\label{fig:sch:ext}
     }%\hfill
   \end{center}
         \end{center}
       \end{minipage}
       %\label{fig:sch:ext}
     }%\hfill
   \end{center}

-  \caption{Differences  with Lena's Cover  wrt $b$.}
+  \caption{Differences  with Lena's cover  wrt $b$}

   \label{fig:lenadiff}
 \end{figure}
   \label{fig:lenadiff}
 \end{figure}

+
+
+