avant complexite

[canny.git] / ourapproach.tex

diff --git a/ourapproach.tex b/ourapproach.tex
index e87c1486bc31d3942abf9acd70e48c53077c7733..636b8bae766709b1d2a4384b1dd72ac6e9f0f69e 100644 (file)
--- a/ourapproach.tex
+++ b/ourapproach.tex
@@ -3,12 +3,13 @@ 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 then 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}) while 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.
+STABYLO, which stands for STe\-ga\-no\-gra\-phy 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}).
 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}).


@@ -16,7 +17,7 @@ Let us first focus on the data embedding.
 
 \begin{figure*}%[t]
   \begin{center}
 
 \begin{figure*}%[t]
   \begin{center}

-    \subfloat[Data Embedding.]{
+    \subfloat[Data Embedding]{

       \begin{minipage}{0.49\textwidth}
         \begin{center}
           %\includegraphics[width=5cm]{emb.pdf}
       \begin{minipage}{0.49\textwidth}
         \begin{center}
           %\includegraphics[width=5cm]{emb.pdf}


@@ -26,7 +27,7 @@ Let us first focus on the data embedding.
       \label{fig:sch:emb}
     } 
 
       \label{fig:sch:emb}
     } 
 

-    \subfloat[Data Extraction.]{
+    \subfloat[Data Extraction]{

       \begin{minipage}{0.49\textwidth}
         \begin{center}
           %\includegraphics[width=5cm]{rec.pdf}
       \begin{minipage}{0.49\textwidth}
         \begin{center}
           %\includegraphics[width=5cm]{rec.pdf}


@@ -48,13 +49,13 @@ Let us first focus on the data embedding.
 
 
 \subsection{Security considerations}\label{sub:bbs}
 
 
 \subsection{Security considerations}\label{sub:bbs}

-Among methods of the message encryption/decryption 
+Among the methods of  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.
 (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 
+It has been 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 


@@ -89,7 +90,7 @@ 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}.
 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, 
+In first order methods like Sobel, Canny~\cite{Canny:1986:CAE:11274.11275}, and so on, 

 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.


@@ -97,7 +98,7 @@ As far as fuzzy edge methods are concerned, they are obviously based on fuzzy lo
 
 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
 
 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
+on many  kinds of architectures like FPGAs, smart phones,  desktop machines, and

 GPUs, we have chosen this edge detector for illustrative purpose.
 
 %\JFC{il faudrait comparer les complexites des algo fuzy and canny}
 GPUs, we have chosen this edge detector for illustrative purpose.
 
 %\JFC{il faudrait comparer les complexites des algo fuzy and canny}


@@ -112,15 +113,15 @@ 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, 
 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.
+and the LSBs 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 presents how our scheme 
-adapts  when the size of $x$  is not sufficient for the message $m$ to embed.
+The next  section presents how to adapt our scheme 
+  when the size of $x$  is not sufficient for the message $m$ to embed.

 
 
  
 
 
  


@@ -129,7 +130,7 @@ 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, 
+Two strategies have been developed in our approach, 

 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.
 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.


@@ -137,7 +138,7 @@ 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.
 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
+If $x$ is 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. 
 
  


@@ -147,39 +148,26 @@ 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
 Practically, the Canny algorithm generates  
 a set of edge pixels related to a threshold that is decreasing 
 until its cardinality

-is sufficient. 
+is sufficient. Even in this situation, our scheme is adapting 
+its algorithm to meet all the user's requirements. 

 
 
 
 

-
-Two methods may further be applied to select bits that 
-will be modified. 
+Once the map of possibly modified pixels is computed, 
+two methods may further be applied to extract bits that 
+are really 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 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}. 
+The second method considers the last significant bits of all the pixels 
+inside the previous map. It next directly applies the STC 
+algorithm~\cite{DBLP:journals/tifs/FillerJF11}. 

 It  is further referred to as \emph{STC} and is detailed in the next section.
 
 
 
 
 
 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.   
-
-
-
-

 
 
 
 
 
 


@@ -252,14 +240,14 @@ Lena and the first verses are given in Fig.~\ref{fig:lena}.
 \begin{flushleft}
 \begin{scriptsize}
 The skies they were ashen and sober;\linebreak
 \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
+$\qquad$ The leaves they were crisped and sere—\linebreak
+$\qquad$ The leaves they were withering and sere;\linebreak

 It was night in the lonesome October\linebreak
 It was night in the lonesome October\linebreak

-$~$ Of my most immemorial year;\linebreak
+$\qquad$ Of my most immemorial year;\linebreak

 It was hard by the dim lake of Auber,\linebreak
 It was hard by the dim lake of Auber,\linebreak

-$~$ In the misty mid region of Weir—\linebreak
+$\qquad$ In the misty mid region of Weir—\linebreak

 It was down by the dank tarn of Auber,\linebreak
 It was down by the dank tarn of Auber,\linebreak

-$~$ In the ghoul-haunted woodland of Weir.
+$\qquad$ In the ghoul-haunted woodland of Weir.

 \end{scriptsize}
 \end{flushleft}
 \end{minipage}
 \end{scriptsize}
 \end{flushleft}
 \end{minipage}


@@ -269,7 +257,9 @@ $~$ In the ghoul-haunted woodland of Weir.
 
 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}.
 
 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}.

-
+When $b$ is 7, it remains one bit per pixel to build the cover vector.
+This configuration leads to a cover vector of size  18,641 if b is 7 
+and 36,910 if $b$ is 6.  

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


@@ -298,9 +288,18 @@ respectively 7 and 6. These edges are represented in Figure~\ref{fig:edge}.
 
 
 
 
 
 

-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.
+The STC algorithm is optimized when the rate between message length and 
+cover vector length is lower than 1/2. 
+So, only 9,320 bits  are available for embedding 
+in the  configuration where $b$ is 7.
+
+When $b$ is 6, we could have considered 18,455 bits for the message.
+However, first experiments have shown that modifying this number of bits is too 
+easily detectable. 
+So, we choose to modify the same amount of bits (9,320) and keep STC optimizing
+which bits to change among  the 36,910 ones.
+
+In the two 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}.
 


@@ -371,36 +370,3 @@ This function allows to emphasize differences between contents.
 \end{figure}
 
 
 \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. 
-
-
-
-
-