X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/canny.git/blobdiff_plain/4b63033789d12bbc41ec23cb66990c458be402b0..5178f772a789c49b261c48acc466a982d413a9a9:/ourapproach.tex?ds=sidebyside diff --git a/ourapproach.tex b/ourapproach.tex index e87c148..e411579 100644 --- 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 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. 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}). @@ -16,21 +17,20 @@ Let us first focus on the data embedding. \begin{figure*}%[t] \begin{center} - \subfloat[Data Embedding.]{ - \begin{minipage}{0.49\textwidth} + \subfloat[Data Embedding]{ + \begin{minipage}{0.4\textwidth} \begin{center} - %\includegraphics[width=5cm]{emb.pdf} - \includegraphics[scale=0.45]{emb.ps} + %\includegraphics[scale=0.45]{emb} + \includegraphics[scale=0.4]{emb} \end{center} \end{minipage} \label{fig:sch:emb} } - - \subfloat[Data Extraction.]{ +\hfill + \subfloat[Data Extraction]{ \begin{minipage}{0.49\textwidth} \begin{center} - %\includegraphics[width=5cm]{rec.pdf} - \includegraphics[scale=0.45]{rec.ps} + \includegraphics[scale=0.4]{dec} \end{center} \end{minipage} \label{fig:sch:ext} @@ -48,13 +48,17 @@ Let us first focus on the data embedding. \subsection{Security considerations}\label{sub:bbs} -Among methods of the message encryption/decryption +\JFC{To provide a self-contained article without any bias, we shor\-tly +present the selected encryption process.} +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} +we implement the asymmetric +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 +The main justification of this choice +is that 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 @@ -89,7 +93,7 @@ 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}. -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. @@ -97,30 +101,33 @@ 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 -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} + This edge detection is applied on a filtered version of the image given as input. -More precisely, only $b$ most -significant bits are concerned by this step, where -the parameter $b$ is practically set with $6$ or $7$. +More precisely, only $b$ most significant bits are concerned by this step, +where the parameter $b$ is practically set with $6$ or $7$. +Notice that 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. 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, -and the LSB of pixels if $b$ is 7. - - - +Moreover, to provide edge gradient value, +the Canny algorithm computes derivatives +in the two directions with respect to a mask of size $T$. +The higher $T$ is, the coarse the approach is. Practically, +$T$ is set with $3$, $5$, or $7$. +In our flowcharts, this step is represented by ``Edge Detection(b, T, X)''. 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 +with respect to the size +of the message $m$ to embed and the size of the cover $x$. + @@ -129,61 +136,47 @@ adapts when the size of $x$ is not sufficient for the message $m$ to embed. \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}. +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. 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 +Canny algorithm with parameters $b=7$ and $T=3$. +The message length is thus defined to be lesser 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. - In the latter, the embedding rate is defined as a percentage between the number of modified pixels and the length of the bit message. 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 +a set of edge pixels related to increasing values of $T$ and until its cardinality -is sufficient. - +is sufficient. Even in this situation, our scheme adapts +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 changed. 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. -% 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. - - - - -\subsection{Minimizing distortion with syndrome-trellis codes}\label{sub:stc} +\subsection{Minimizing distortion with Syndrome-Trellis Codes}\label{sub:stc} \input{stc} @@ -205,14 +198,26 @@ It is further referred to as \emph{STC} and is detailed in the next section. % but the whole approach can be updated to consider % the fuzzy logic edge detector. -% Next, following~\cite{Luo:2010:EAI:1824719.1824720}, our scheme automatically -% modifies Canny parameters to get a sufficiently large set of edge bits: this -% one is practically enlarged untill its size is at least twice as many larger -% than the size of embedded message. - - - -%%RAPH: paragraphe en double :-) +For a given set of parameters, +the Canny algorithm returns a numerical value and +states whether a given pixel is an edge or not. +In this article, in the Adaptive strategy +we consider that all the edge pixels that +have been selected by this algorithm have the same +distortion cost, \textit{i.e.}, $\rho_X$ is always 1 for these bits. +In the Fixed strategy, since pixels that are detected to be edge +with small values of $T$ (e.g., when $T=3$) +are more accurate than these with higher values of $T$, +we give to STC the following distortion map of the corresponding bits +$$ +\rho_X= \left\{ +\begin{array}{l} +1 \textrm{ if an edge for $T=3$,} \\ +10 \textrm{ if an edge for $T=5$,} \\ +100 \textrm{ if an edge for $T=7$.} +\end{array} +\right. +$$ @@ -222,11 +227,13 @@ 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. -More precisely, the same edge detection is applied on the $b$ first bits to +More precisely, let $b$ be the most significant bits and +$T$ be the size of the Canny mask, both be given as a key. +Thus, the same edge detection is applied on a stego content $Y$ to 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. +This inverse function takes the $\hat{H}$ matrix as a parameter. 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. @@ -245,21 +252,21 @@ Lena and the first verses are given in Fig.~\ref{fig:lena}. \begin{center} \begin{minipage}{0.49\linewidth} \begin{center} -\includegraphics[scale=0.20]{Lena.eps} +\includegraphics[scale=0.20]{lena512} \end{center} \end{minipage} \begin{minipage}{0.49\linewidth} \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 -$~$ Of my most immemorial year;\linebreak +$\qquad$ Of my most immemorial year;\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 -$~$ In the ghoul-haunted woodland of Weir. +$\qquad$ In the ghoul-haunted woodland of Weir. \end{scriptsize} \end{flushleft} \end{minipage} @@ -268,8 +275,11 @@ $~$ In the ghoul-haunted woodland of Weir. \end{figure} 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}. - +respectively 7 and 6 and $T=3$. +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} @@ -277,7 +287,7 @@ respectively 7 and 6. These edges are represented in Figure~\ref{fig:edge}. \begin{minipage}{0.49\linewidth} \begin{center} %\includegraphics[width=5cm]{emb.pdf} - \includegraphics[scale=0.20]{edge7.eps} + \includegraphics[scale=0.20]{edge7} \end{center} \end{minipage} %\label{fig:sch:emb} @@ -286,22 +296,31 @@ respectively 7 and 6. These edges are represented in Figure~\ref{fig:edge}. \begin{minipage}{0.49\linewidth} \begin{center} %\includegraphics[width=5cm]{rec.pdf} - \includegraphics[scale=0.20]{edge6.eps} + \includegraphics[scale=0.20]{edge6} \end{center} \end{minipage} %\label{fig:sch:ext} }%\hfill \end{center} - \caption{Edge detection wrt $b$} + \caption{Edge detection wrt $b$ with $T=3$} \label{fig:edge} \end{figure} -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. -Results with \emph{adaptive+STC} strategy are presented in +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 {Adaptive} and {STC} strategies are presented in Fig.~\ref{fig:lenastego}. \begin{figure}[t] @@ -310,7 +329,7 @@ Fig.~\ref{fig:lenastego}. \begin{minipage}{0.49\linewidth} \begin{center} %\includegraphics[width=5cm]{emb.pdf} - \includegraphics[scale=0.20]{lena7.eps} + \includegraphics[scale=0.20]{lena7} \end{center} \end{minipage} %\label{fig:sch:emb} @@ -319,7 +338,7 @@ Fig.~\ref{fig:lenastego}. \begin{minipage}{0.49\linewidth} \begin{center} %\includegraphics[width=5cm]{rec.pdf} - \includegraphics[scale=0.20]{lena6.eps} + \includegraphics[scale=0.20]{lena6} \end{center} \end{minipage} %\label{fig:sch:ext} @@ -341,9 +360,15 @@ V_{ij}= \left\{ 150 & \textrm{if} & \vert X_{ij} - Y_{ij} \vert = 2 \\ 225 & \textrm{if} & \vert X_{ij} - Y_{ij} \vert = 3 \end{array} -\right.. +\right. $$ This function allows to emphasize differences between contents. +Notice that since $b$ is 7 in Fig.~\ref{fig:diff7}, the embedding is binary +and this image only contains 0 and 75 values. +Similarly, if $b$ is 6 as in Fig.~\ref{fig:diff6}, the embedding is ternary +and the image contains all the values in $\{0,75,150,225\}$. + + \begin{figure}[t] \begin{center} @@ -351,19 +376,19 @@ This function allows to emphasize differences between contents. \begin{minipage}{0.49\linewidth} \begin{center} %\includegraphics[width=5cm]{emb.pdf} - \includegraphics[scale=0.20]{diff7.eps} + \includegraphics[scale=0.20]{diff7} \end{center} \end{minipage} - %\label{fig:sch:emb} + \label{fig:diff7} }%\hfill \subfloat[$b$ is 6.]{ \begin{minipage}{0.49\linewidth} \begin{center} %\includegraphics[width=5cm]{rec.pdf} - \includegraphics[scale=0.20]{diff6.eps} + \includegraphics[scale=0.20]{diff6} \end{center} \end{minipage} - %\label{fig:sch:ext} + \label{fig:diff6} }%\hfill \end{center} \caption{Differences with Lena's cover wrt $b$} @@ -372,35 +397,3 @@ This function allows to emphasize differences between contents. -\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. - - - - -