quelques modifs

[canny.git] / stc.tex

diff --git a/stc.tex b/stc.tex
index db985853fa8dd98b21b8c676d242e8f4d65e4f63..03733f5a3c27996ce0d85875e1a6c99992d353f8 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). 
+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 $X$, 


@@ -19,13 +20,13 @@ $m$ for a given binary matrix $H$.
 
 Let us explain this embedding on a small illustrative example where
 $m$ and $x$ are respectively  a 3 bits column
 
 Let us explain this embedding on a small illustrative example where
 $m$ and $x$ are respectively  a 3 bits column

-vector and a 7 bits column vector and where 
+vector and a 7 bits column vector, and where 

 $\rho_X(i,x,y)$ is equal to 1 for any $i$, $x$, $y$
 $\rho_X(i,x,y)$ is equal to 1 for any $i$, $x$, $y$

-(\textit{i.e.}, $\rho_X(i,x,y) = 0$ if $x = y$ and $0$ otherwise).  
+(\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}


@@ -53,19 +54,19 @@ 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 
 $n=2^p-1$ pixels needs $1-1/2^p$ average changes.
 
 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 
 $n=2^p-1$ pixels needs $1-1/2^p$ average changes.
 

-This Hamming embeding is really efficient to very small payload and is 
-not well suited when the size of the message is larger, as in real situation.
+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  
 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  

-presented by Filler \emph{et al.} in~\cite{FillerJF11}
+presented by Filler \emph{et al.} \JFC{in~\cite{FillerJF11,liu2014syndrome}}

 is a practical solution to this complexity. Thanks to this contribution,
 the solving algorithm has a linear complexity with respect to $n$.
 
 is a practical solution to this complexity. Thanks to this contribution,
 the solving algorithm has a linear complexity with respect to $n$.