X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/canny.git/blobdiff_plain/b417a74f270da1ad1a8b11552a8508c45cb75085..6bc78c180bea99d493483c3cac3f559301d3af09:/ourapproach.tex diff --git a/ourapproach.tex b/ourapproach.tex index 74c1b55..36fefc3 100644 --- a/ourapproach.tex +++ b/ourapproach.tex @@ -1,33 +1,42 @@ +This section first presents the embedding scheme through its +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}) 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. -What follows are successively details of the inner steps and flows inside -both the embedding stage (Fig.~\ref{fig:sch:emb}) -and the extraction one (Fig.~\ref{fig:sch:ext}). +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}). Let us first focus on the data embedding. -\begin{figure*}[t] +\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=0.5]{emb.ps} + %\includegraphics[scale=0.45]{emb} + \includegraphics[scale=0.4]{emb} \end{center} \end{minipage} \label{fig:sch:emb} - }%\hfill - \subfloat[Data Extraction.]{ + } +\hfill + \subfloat[Data Extraction]{ \begin{minipage}{0.49\textwidth} \begin{center} - %\includegraphics[width=5cm]{rec.pdf} - \includegraphics[scale=0.5]{rec.ps} + \includegraphics[scale=0.4]{dec} \end{center} \end{minipage} \label{fig:sch:ext} }%\hfill \end{center} - \caption{The STABYLO Scheme.} + \caption{The STABYLO scheme} \label{fig:sch} \end{figure*} @@ -38,18 +47,22 @@ Let us first focus on the data embedding. -\subsection{Security Considerations} -Among methods of message encryption/decryption +\subsection{Security considerations}\label{sub:bbs} +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 -which allows to find $x_{i-1}$ and $x_{i+L}$ with a probability greater +which allows to find $x_{i-1}$ or $x_{i+L}$ with a probability greater than $1/2$. Equivalent formulations of such a property can be found. They all lead to the fact that, @@ -61,7 +74,7 @@ Starting thus with a key $k$ and the message \textit{mess} to hide, this step computes a message $m$, which is the encrypted version of \textit{mess}. -\subsection{Edge-Based Image Steganography} +\subsection{Edge-based image steganography}\label{sub:edge} The edge-based image @@ -80,57 +93,90 @@ 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. -As for as fuzzy edge methods are concerned, they are obviously based on fuzzy logic to highlight edges. +As far as fuzzy edge methods are concerned, they are obviously based on fuzzy logic to highlight edges. -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 well known and studied, fast, and implementable -on many kinds of architectures like FPGAs, smartphones, desktop machines, and +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, smart phones, desktop machines, and GPUs, we have chosen this edge detector for illustrative purpose. + + + 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. -Let $x$ be the sequence of these bits. +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 to adapt our scheme +with respect to the size +of the message $m$ to embed and the size of the cover $x$. + + + -\JFC{il faudrait comparer les complexites des algo fuzy and canny} -% 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{Adaptive embedding rate}\label{sub:adaptive} +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 parameters $b=7$ and $T=3$. +The message length is thus defined to be lesser than +half of this set cardinality. +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 increasing values of $T$ and +until its cardinality +is sufficient. Even in this situation, our scheme adapts +its algorithm to meet all the user's requirements. + + +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 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. -As argue in the introduction section, we do not adapt the parameters of the -the edge detection as in~\cite{Luo:2010:EAI:1824719.1824720} but we modify -the size of the embedding message. Practically, the lenght of $x$ -has to be at least twice as large -as the size of the embedded encrypted message. -Otherwise, a new image is used to hide the remaning part of the message. -\subsection{Minimizing Distortion with Syndrome-Treillis Codes} + +\subsection{Minimizing distortion with Syndrome-Trellis Codes}\label{sub:stc} \input{stc} @@ -152,24 +198,202 @@ Otherwise, a new image is used to hide the remaning part of the message. % 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. +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 +$$ +\rho_X= \left\{ +\begin{array}{l} +1 \textrm{ if an edge for $T=3$,} \\ +10 \textrm{ if an edge for $T=5$,} \\ +100 \textrm{ if an edge for $T=7$.} +\end{array} +\right. +$$ + + + + +\subsection{Data extraction}\label{sub:extract} +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, 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 $\hat{H}$ 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. +Finally, the Blum-Goldwasser decryption function is executed and the original +message is extracted. + + +\subsection{Running example}\label{sub:xpl} +In this example, the cover image is Lena, +which is a $512\times512$ image with 256 grayscale levels. +The message is the poem Ulalume (E. A. Poe), which is constituted by 104 lines, 667 +words, and 3,754 characters, \textit{i.e.}, 30,032 bits. +Lena and the first verses are given in Fig.~\ref{fig:lena}. + +\begin{figure} +\begin{center} +\begin{minipage}{0.49\linewidth} +\begin{center} +\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 +$\qquad$ 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 +$\qquad$ Of my most immemorial year;\linebreak +It was hard by the dim lake of Auber,\linebreak +$\qquad$ In the misty mid region of Weir—\linebreak +It was down by the dank tarn of Auber,\linebreak +$\qquad$ In the ghoul-haunted woodland of Weir. +\end{scriptsize} +\end{flushleft} +\end{minipage} +\end{center} +\caption{Cover and message examples} \label{fig:lena} +\end{figure} + +The edge detection returns 18,641 and 18,455 pixels when $b$ is +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} + \subfloat[$b$ is 7.]{ + \begin{minipage}{0.49\linewidth} + \begin{center} + %\includegraphics[width=5cm]{emb.pdf} + \includegraphics[scale=0.20]{edge7} + \end{center} + \end{minipage} + %\label{fig:sch:emb} + }%\hfill + \subfloat[$b$ is 6.]{ + \begin{minipage}{0.49\linewidth} + \begin{center} + %\includegraphics[width=5cm]{rec.pdf} + \includegraphics[scale=0.20]{edge6} + \end{center} + \end{minipage} + %\label{fig:sch:ext} + }%\hfill + \end{center} + \caption{Edge detection wrt $b$ with $T=3$} + \label{fig:edge} +\end{figure} + +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. -%%RAPH: paragraphe en double :-) +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] + \begin{center} + \subfloat[$b$ is 7.]{ + \begin{minipage}{0.49\linewidth} + \begin{center} + %\includegraphics[width=5cm]{emb.pdf} + \includegraphics[scale=0.20]{lena7} + \end{center} + \end{minipage} + %\label{fig:sch:emb} + }%\hfill + \subfloat[$b$ is 6.]{ + \begin{minipage}{0.49\linewidth} + \begin{center} + %\includegraphics[width=5cm]{rec.pdf} + \includegraphics[scale=0.20]{lena6} + \end{center} + \end{minipage} + %\label{fig:sch:ext} + }%\hfill + \end{center} + \caption{Stego images wrt $b$} + \label{fig:lenastego} +\end{figure} + + +Finally, differences between the original cover and the stego images +are presented in Fig.~\ref{fig:lenadiff}. For each pair of pixel $X_{ij}$ and $Y_{ij}$ ($X$ and $Y$ being the cover and the stego content respectively), +the pixel value $V_{ij}$ of the difference is defined with the following map +$$ +V_{ij}= \left\{ +\begin{array}{rcl} +0 & \textrm{if} & X_{ij} = Y_{ij} \\ +75 & \textrm{if} & \vert X_{ij} - Y_{ij} \vert = 1 \\ +150 & \textrm{if} & \vert X_{ij} - Y_{ij} \vert = 2 \\ +225 & \textrm{if} & \vert X_{ij} - Y_{ij} \vert = 3 +\end{array} +\right. +$$ +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} + \subfloat[$b$ is 7.]{ + \begin{minipage}{0.49\linewidth} + \begin{center} + %\includegraphics[width=5cm]{emb.pdf} + \includegraphics[scale=0.20]{diff7} + \end{center} + \end{minipage} + \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} + \end{center} + \end{minipage} + \label{fig:diff6} + }%\hfill + \end{center} + \caption{Differences with Lena's cover wrt $b$} + \label{fig:lenadiff} +\end{figure} -\subsection{Data Extraction} -The message extraction summarized in Fig.~\ref{fig:sch:ext} follows data embedding -since there exists a reverse function for all its steps. -First of all, the same edge detection is applied (on the 7 first bits) to -get the set of LSBs, -which is sufficiently large with respect to the message size given as a key. -Then the STC reverse algorithm is applied to retrieve the encrypted message. -Finally, the Blum-Goldwasser decryption function is executed and the original -message is extracted.