Merge branch 'master' of ssh://info.iut-bm.univ-fcomte.fr/canny

[canny.git] / stc.tex

diff --git a/stc.tex b/stc.tex
index 0d1541a44916f9034d0691a89d59630bb5517153..5e5b66e6258df38d81475c859e714ec93b6223c1 100644 (file)
--- a/stc.tex
+++ b/stc.tex
@@ -1,5 +1,6 @@
 To make this article self-contained, this section recalls
 To make this article self-contained, this section recalls

-the basis of the Syndrome Treillis Codes  (STC).
+the basis of the Syndrome Treillis Codes  (STC). 
+\JFC{A reader who is familar with syndrome coding can skip it.}

 
 Let 
 $x=(x_1,\ldots,x_n)$ be the $n$-bits cover vector issued from an image $X$, 
 
 Let 
 $x=(x_1,\ldots,x_n)$ be the $n$-bits cover vector issued from an image $X$, 


@@ -23,9 +24,9 @@ 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).  
 
 $\rho_X(i,x,y)$ 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  $H$ be the binary Hamming matrix  
+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 \\
 \begin{array}{lllllll}
  0 & 0 & 0 & 1 & 1 & 1 & 1 \\
  0 & 1 & 1 & 0 & 0 & 1 & 1 \\


@@ -33,11 +34,11 @@ H = \left(
 \end{array}
 \right).
 $$
 \end{array}
 \right).
 $$

-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.
 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 : 
 $$
 \delta = \left( \begin{array}{l}
 as a vector : 
 $$
 \delta = \left( \begin{array}{l}


@@ -53,7 +54,7 @@ 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 
 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 

-$m = Hy$.
+$m = \dot{H}y$.

 It is then possible to embed 3 bits in 7 LSBs of pixels by modifying
 at most 1 bit.
 In the general case, communicating a message of $p$ bits in a cover of 
 It is then possible to embed 3 bits in 7 LSBs of pixels by modifying
 at most 1 bit.
 In the general case, communicating a message of $p$ bits in a cover of