X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/canny.git/blobdiff_plain/f9feb4cb5a39609ffdf39b086e8d796592a71da5..7aeb1ab6cc41dd51a6dbe4c844c8983809205186:/stc.tex

diff --git a/stc.tex b/stc.tex
index e98589f..8da1573 100644
--- a/stc.tex
+++ b/stc.tex
@@ -1,13 +1,14 @@
+To make this article self-contained, this section recalls
+the basis of the Syndrome Treillis Codes  (STC).
+
 Let 
-$x=(x_1,\ldots,x_n)$ be the $n$-bits cover vector of the image $X$, 
+$x=(x_1,\ldots,x_n)$ be the $n$-bits cover vector issued from an image $X$, 
 $m$ be the message to embed, and 
 $y=(y_1,\ldots,y_n)$ be the $n$-bits stego vector.
 The usual additive embedding impact of replacing $x$ by $y$ in $X$
 is given by a distortion function 
 $D_X(x,y)= \Sigma_{i=1}^n \rho_X(i,x,y)$, where the function $\rho_X$
 expresses the cost of replacing $x_i$ by $y_i$ in $X$.
-Let us consider that $x$ is fixed: 
-this is for instance the LSBs of the image edge bits. 
 The objective is thus to find $y$ that minimizes $D_X(x,y)$.
 
 Hamming embedding proposes a solution to this problem. 
@@ -17,12 +18,14 @@ Furthermore this code provides a vector $y$ s.t. $Hy$ is equal to
 $m$ for a given binary matrix $H$. 
 
 Let us explain this embedding on a small illustrative example where
-$\rho_X(i,x,y)$ is identically equal to 1,
-whereas $m$ and $x$ are respectively  a 3 bits column
-vector and a 7 bits column vector. 
-Let then $H$ be the binary Hamming matrix  
+$m$ and $x$ are respectively  a 3 bits column
+vector and a 7 bits column vector, and where 
+$\rho_X(i,x,y)$ is equal to 1 for any $i$, $x$, $y$
+(\textit{i.e.}, $\rho_X(i,x,y) = 0$ if $x = y$ and $1$ otherwise).  
+
+Let  $\dot{H}$ be the binary Hamming matrix  
 $$
-H = \left(
+\dot{H} = \left(
 \begin{array}{lllllll}
  0 & 0 & 0 & 1 & 1 & 1 & 1 \\
  0 & 1 & 1 & 0 & 0 & 1 & 1 \\
@@ -30,11 +33,11 @@ H = \left(
 \end{array}
 \right).
 $$
-The objective is to modify $x$ to get $y$ s.t. $m = Hy$.
+The objective is to modify $x$ to get $y$ s.t. $m = \dot{H}y$.
 In this algebra, the sum and the product respectively correspond to
 the exclusive \emph{or} and to the \emph{and} Boolean operators.
-If $Hx$ is already equal to $m$, nothing has to be changed and $x$ can be sent.
-Otherwise we consider the difference $\delta = d(m,Hx)$, which is expressed 
+If $\dot{H}x$ is already equal to $m$, nothing has to be changed and $x$ can be sent.
+Otherwise we consider the difference $\delta = d(m,\dot{H}x)$, which is expressed 
 as a vector : 
 $$
 \delta = \left( \begin{array}{l}
@@ -50,33 +53,34 @@ We denote by $\overline{x}^j$ the vector  obtained by
 switching the $j-$th component of $x$, 
 that is, $\overline{x}^j = (x_1 , \ldots, \overline{x_j},\ldots, x_n )$.
 It is not hard to see that if $y$ is $\overline{x}^j$, then 
-$m = Hy$.
-It is then possible to embed 3 bits in only 7 LSBs of pixels by modifying
+$m = \dot{H}y$.
+It is then possible to embed 3 bits in 7 LSBs of pixels by modifying
 at most 1 bit.
-In the general case, communicating $n$ message bits in 
-$2^n-1$ pixels needs $1-1/2^n$ average changes.
-
+In the general case, communicating a message of $p$ bits in a cover of 
+$n=2^p-1$ pixels needs $1-1/2^p$ average changes.
 
-
-Unfortunately, for any given $H$, finding $y$ that solves $Hy=m$ and  
+This Hamming embedding is really efficient to very small payload and is 
+not well suited when the size of the message is larger, as in real situations.
+The matrix $H$ should be changed to deal with higher payload.
+Moreover, for any given $H$, finding $y$ that solves $Hy=m$ and  
 that minimizes $D_X(x,y)$, has an exponential complexity with respect to $n$. 
-The Syndrome-Trellis Codes  (STC) 
-presented by Filler \emph{et al.} in~\cite{DBLP:conf/mediaforensics/FillerJF10} 
+The Syndrome-Trellis Codes  
+presented by Filler \emph{et al.} in~\cite{FillerJF11}
 is a practical solution to this complexity. Thanks to this contribution,
 the solving algorithm has a linear complexity with respect to $n$.
 
-First of all, Filler et al. compute the matrix $H$
-by placing a small sub-matrix $\hat{H}$ of size $h × w$ next
-to each other and shifted down by one row. 
+First of all, Filler \emph{et al.} compute the matrix $H$
+by placing a small sub-matrix $\hat{H}$ next
+to each other and by shifting down by one row. 
 Thanks to this special form of $H$, one can represent
-every solution of  $m=Hy$ as a path through a trellis.
+any solution of  $m=Hy$ as a path through a trellis.
 
-Next, the  process of finding $y$ consists of two stages: a forward and a backward part.
+Next, the  process of finding $y$ consists in two stages: a forward and a backward part.
 \begin{enumerate}
-\item Forward construction of the trellis that depends on $\hat{H}$, on $x$, on $m$, and on $\rho$.
+\item Forward construction of the trellis that depends on $\hat{H}$, on $x$, on $m$, and on $\rho$. This step is linear in $n$.
 \item Backward determination of $y$ that minimizes $D$, starting with 
-the complete path having the minimal weight.
+the complete path having the minimal weight. This corresponds to traversing 
+a graph and has a complexity which is linear in $n$. 
 \end{enumerate} 
 
 
-