abstract, conclusion

[HindawiJournalOfChaos.git] / IH / dhCI.tex

\section{Further Investigations of the dhCI Class}
\label{dhciSec}
We have recalled at the beginning of this chapter that 
chaotic iterations can be applied on the least significant
coefficients of a medium, either in spatial or in frequency domain, in order to watermark it.
The general process has been denoted by dhCI in our thesis, while
its particular instantiation with the negation function has been
later called $\mathcal{CIW}_1$ (remark that $\mathcal{CIS}_2$ and 
$\mathcal{DI}_3$ processes do not belong, stricto sensu, to
the dhCI class). Since our thesis defense,
the dhCI class has been investigated more largely, by discovering
new iteration functions and evaluating both its security and
robustness. Results of such questioning are summarized thereafter.


\subsection{Introduction}
\label{sec:intro}

The study of the dhCI class has been deepened in~\cite{bcg11b:ip} for its
theoretical aspects and in~\cite{bcg11:ij} for practical ones.  

As for the  $\mathcal{CIW}_1$ scheme, the work around the dhCI 
class focuses on non-blind  binary information hiding scheme: 
the  original host  is  required  to  extract the  binary hidden
information. This  context is indeed not
as restrictive as it could primarily appear.
Firstly, it allows to prove 
the authenticity of a document sent  through the  Internet 
(the original document is stored whereas the stego content is sent). 
Secondly, Alice and Bob can establish an hidden channel into a
streaming video 
(Alice and Bob both have the same movie, and  Alice hide information into 
the frame number $k$  iff the binary digit
number $k$ of its hidden message is  1).
Thirdly, based on a similar idea, a same
given image can be marked  several times by using various secret parameters
owned both by Alice and Bob. Thus more than one bit can be embedded into a given
image  by using  dhCI  dissimulation. Lastly,  non-blind watermarking  is
useful in  network's anonymity and intrusion  detection \cite{Houmansadr09}, and
to protect digital data sending through the Internet \cite{P1150442004}.

Before~\cite{bcg11b:ip}, stego-security~\cite{Cayre2008} and topological security 
were only proven 
on the spread spectrum watermarking~\cite{Cox97securespread,HuangFang},
and on the $\mathcal{CIW}_1$ algorithm, which is notably an instance of the
dhCI method, but which restricts itself to the negation mode (security proofs
of $\mathcal{CIS}_2$ and $\mathcal{DI}_3$ have occurred later).
We argued in~\cite{bcg11b:ip} that dhCI with other functions can 
provide algorithms as secure as the $\mathcal{CIW}_1$ one.
This work has then generalized the algorithm recalled in Section~\ref{sec:ciw1} and formalized all
its stages, independently from the iteration mode. Due to this formalization, it has then been 
possible to address the proofs of the two
security properties for a larger class of iteration modes in~\cite{bcg11b:ip}.  


Then, in The
Computer Journal~\cite{bcg11:ij}, a review of the researches on the dhCI class has been presented. 
Additionally, this article has investigated robustness aspects of the 
process: applications in frequency domains (namely DWT and DCT embedding) 
have been formalized and corresponding experiments have been
given~\cite{bcg11:ij}.  Such a study shows the applicability of the whole approach.



\subsection{Formalization of steganographic methods}
\label{sec:formalization}


The data hiding scheme presented in previous works does not constrain media to have 
a constant size. It is indeed sufficient to provide a function and a strategy 
that may be  parametrized with the size of the elements to modify. 
Parametrized strategies have already been introduced in a previous 
section, leading to the notion of \emph{strategy-adapter}. 
The \emph{mode} notion defined below achieves the same goal
but for the iteration function~\cite{bcg11b:ip}.% (until now, only the negation function
%has been regarded).  

\begin{Def}[Mode]
\label{def:mode}
A map $f$, which associates to any $n \in \mathds{N}$ an application 
$f_n : \mathds{B}^n \rightarrow \mathds{B}^n$, is called a \emph{mode}.
\end{Def}

For instance, the \emph{negation mode} is defined by the map that
assigns to every integer $n \in \mathds{N}^*$ the function 
${\neg}_n:\mathds{B}^n \to \mathds{B}^n, 
{\neg}_n(x_1, \hdots, x_n) \mapsto (\overline{x_1}, \hdots, \overline{x_n})$.


We now use the previously introduced \emph{signification function} 
to attach a weight to each term defining a digital media,
w.r.t. its position $t$, leading to the following notion of
a decomposition function~\cite{bcg11b:ip}.



\begin{Def}[Decomposition function]
Let $(u^k)^{k \in \mathds{N}}$ be a signification function, 
$\mathfrak{B}$ the set of finite binary sequences,
$\mathfrak{N}$ the set of finite integer sequences, 
$m$ and $M$ be two reals s.t. $m < M$.  
Any host $x$ may be decomposed into 
\[
(u_M,u_m,u_p,\phi_{M},\phi_{m},\phi_{p})
\in
\mathfrak{N} \times 
\mathfrak{N} \times 
\mathfrak{N} \times 
\mathfrak{B} \times 
\mathfrak{B} \times 
\mathfrak{B} 
\]
where
\begin{itemize}
\item $u_M$, $u_m$, and $u_p$ are coefficients defined in Definition  
\ref{def:msc,lsc};
\item $\phi_{M} = \left( x^{u^1_M}, x^{u^2_M}, \ldots,x^{u^{|u_M|}_M}\right)$;
 \item $\phi_{m} = \left( x^{u^1_m}, x^{u^2_m}, \ldots,x^{u^{|u_m|}_m} \right)$;
 \item $\phi_{p} =\left( x^{u^1_p}, x^{u^2_p}, \ldots,x^{u^{|u_p|}_p}\right) $.
 \end{itemize}
The function that associates the decomposed host to any digital host is 
the \emph{decomposition function}. It is 
further referred as $\textit{dec}(u,m,M)$ since it is parametrized by 
$u$, $m$ and $M$. Notice that $u$ is a shortcut for $(u^k)^{k \in \mathds{N}}$.
\end{Def} 


\begin{Def}[Recomposition]
Let 
$(u_M,u_m,u_p,\phi_{M},\phi_{m},\phi_{p}) \in 
\mathfrak{N} \times 
\mathfrak{N} \times 
\mathfrak{N} \times 
\mathfrak{B} \times 
\mathfrak{B} \times 
\mathfrak{B} 
$ s.t.
\begin{itemize}
\item the sets of elements in $u_M$, elements in $u_m$, and 
elements in $u_p$ are a partition of $\llbracket 1, n\rrbracket$;
\item $|u_M| = |\varphi_M|$, $|u_m| = |\varphi_m|$, and $|u_p| = |\varphi_p|$.  
\end{itemize}
One may associate the vector 
\[
x = 
\sum_{i=1}^{|u_M|} \varphi^i_M . e_{{u^i_M}} +  
\sum_{i=1}^{|u_m|} \varphi^i_m .e_{{u^i_m}} +  
\sum_{i=1}^{|u_p|} \varphi^i_p. e_{{u^i_p}} 
\]
\noindent where 
$(e_i)_{i \in \mathds{N}}$ is the usual basis of the $\mathds{R}-$vectorial space $\left(\mathds{R}^\mathds{N}, +, .\right)$.
The function that associates $x$ to any 
$(u_M,u_m,u_p,\phi_{M},\phi_{m},\phi_{p})$ following the above constraints 
is called the \emph{recomposition function}.
\end{Def}

The embedding consists in the replacement of the values of 
$\phi_{m}$ of $x$'s LSCs  by $y$. 
It then composes the two decomposition and
recomposition functions seen previously. More formally~\cite{bcg11b:ip}:


\begin{Def}[Embedding media]
Let $\textit{dec}(u,m,M)$ be a decomposition function,
$x$ be a host content,
$(u_M,u_m,u_p,\phi_{M},\phi_{m},\phi_{p})$ be its image by $\textit{dec}(u,m,M)$, 
and $y$ be a digital media of size $|u_m|$.
The digital media $z$ resulting on the embedding of $y$ into $x$ is 
the image of $(u_M,u_m,u_p,\phi_{M},y,\phi_{p})$
by  the recomposition function $\textit{rec}$.
\end{Def}

We have thus been able in~\cite{bcg11b:ip} to reformulate the \emph{dhCI} information hiding scheme, as follows:

\begin{Def}[Data hiding dhCI]
 \label{def:dhCI}
Let $\textit{dec}(u,m,M)$ be a decomposition function,
$f$ be a mode, 
$\mathcal{S}$ be a strategy adapter,
$x$ be an host content,\linebreak
$(u_M,u_m,u_p,\phi_{M},\phi_{m},\phi_{p})$ 
be its image by $\textit{dec}(u,m,M)$,
$q$ be a positive natural number,  
and $y$ be a digital media of size $l=|u_m|$.


The \emph{dhCI dissimulation}  maps any
$(x,y)$  to the digital media $z$ resulting on the embedding of
$\hat{y}$ into $x$, s.t.

\begin{itemize}
\item We instantiate the mode $f$ with parameter $l=|u_m|$, leading to 
  the function $f_{l}:\mathds{B}^{l} \rightarrow \mathds{B}^{l}$.
\item We instantiate the strategy adapter $\mathcal{S}$ 
with parameter $y$ (and some other ones eventually). 
This instantiation leads to the strategy $S_y \in \llbracket 1;l\rrbracket ^{\mathds{N}}$.

\item We iterate $G_{f_l}$ with initial configuration $(S_y,\phi_{m})$.
\item $\hat{y}$  is the $q$-th term.
\end{itemize}
\end{Def}


To summarize, iterations are realized on the LSCs of the
host content
(the mode gives the iterate function,  
the strategy-adapter gives its strategy), 
and the last computed configuration is re-injected into the host content, 
in place of the former LSCs.


\begin{figure}[h!]
\centering
\includegraphics[width=10cm]{IH/organigramme22.pdf}
\caption{The dhCI dissimulation scheme}
\label{fig:organigramme}
\end{figure}


We are then left to show how to formally check
whether a given digital media $z$ 
results from the dissimulation of $y$ into the digital media $x$~\cite{bcg11b:ip}. 

\begin{Def}[Marked content]
Let $\textit{dec}(u,m,M)$ be a decomposition function,
$f$ be a mode, 
$\mathcal{S}$ be a strategy adapter, 
$q$ be a positive natural number,  
and  
$y$ be a digital media, 
\linebreak $(u_M,u_m,u_p,\phi_{M},\phi_{m},\phi_{p})$ be the 
image by $\textit{dec}(u,m,M)$  of  a digital media $x$. 

Then $z$ is \emph{marked} with $y$ if
the image by $\textit{dec}(u,m,M)$ of $z$ is 
$(u_M,u_m,u_p,\phi_{M},\hat{y},\phi_{p})$ where 
$\hat{y}$ is the right member of $G_{f_l}^q(S_y,\phi_{m})$.
\end{Def}


Various decision strategies are obviously  possible to determine whether a given
image $z$ is  marked or not, depending  on the eventuality
that the considered image may have  been attacked.
For example, a  similarity percentage between $x$
and  $z$ can  be  computed, and  the result  can  be compared  to a  given
threshold. Other  possibilities are the use of  ROC curves or
the definition of a null hypothesis problem. These strategies have already
been discussed in a previous section, they can be adapted, mutatis mutandis,
to the generalized dhCI algorithm detailed above.


The next section, always extracted from~\cite{bcg11b:ip}, 
recalls some security properties and shows how the 
\emph{dhCI dissimulation} algorithm verifies them.





\subsection{Security analysis}\label{sec:security}


We have proven in~\cite{bcg11b:ip}, using the stochastic matrix theorem, that,

\begin{Th}\label{th:stego}
Let $\epsilon$ be positive,
$l$ be any size of LSCs, 
$X   \sim \mathcal{U}\left(\mathbb{B}^l\right)$,
$f_l$ be an image mode s.t. 
$\Gamma(f_l)$ is strongly connected and 
the Markov matrix associated to $f_l$ 
is doubly stochastic. 

In the instantiated \emph{dhCI dissimulation} algorithm 
with any uniformly distributed (u.d.) strategy-adapter 
which is independent from $X$,  
there exists some positive natural number $q$ s.t.
$|p(X^q)- p(X)| < \epsilon$. 
\end{Th}


\begin{Pre}   
See~\cite{bcg11b:ip}.
 \end{Pre}

Since $p(Y| K)$ is $p(X^q)$ the method is then stego-secure.
We have then focused on topological security properties, and have deduced
from the characterization recalled in Theorem~\ref{Th:Caracterisation des  IC chaotiques} that,

\begin{Prop}
Functions $f  : \mathds{B}^{n} \to
\mathds{B}^{n}$ such that the graph $\Gamma(f)$ is strongly connected lead to topologically secure 
\emph{dhCI dissimulation} algorithms.
\end{Prop}




%\subsection{Instantiation of  Steganographic Methods}
\label{instantiating}

Theorem~\ref{th:stego} relies on a u.d.   
strategy-adapter that is independent from the cover, and on an image mode 
$f_l$ whose iteration graph  $\Gamma(f_l)$ is strongly
connected and whose  Markov matrix 
is doubly stochastic. 
We have shown in~\cite{bcg11b:ip} that the CIIS strategy 
adapter~\cite{gfb10:ip} has the required 
properties and have mentioned that \cite{wcbg11:ip} has presented an iterative approach 
(which has been recalled in Section~\ref{sec:prac})
to 
generate image modes $f_l$ such that  
$\Gamma(f_l)$ is strongly connected.
Among these maps, it is obvious to check which verifies or not
the doubly stochastic constrain. 

% 
% \subsection{Conclusion}\label{sec:concl}
% 
% This work has presented a new class of information hiding
% algorithms which generalizes 
% algorithm~\cite{gfb10:ip} reduced to the negation mode.
% Its complete formalization has allowed to prove the stego-security 
% and topological security properties.
% As far as we know, this is the first time a whole class of algorithm 
% has been proven to have these two properties. 
% 
% In future work, our intention is to study the robustness of this class of 
% dhCI dissimulation schemes.
% We are to find the optimized parameters (modes, stretegy adapters, signification coefficients, iterations numbers\ldots) giving 
% the strongest robustness 
% (depending on the chosen representation domain), theoretically and practically
% by realizing comprehensive simulations.
% Finally these algorithms will be compared to other existing ones, among other
% things by regarding whether these algorithms are chaotic or not.


























%\section{Steganography: a class of secure and robust algorithms~\cite{bcg11:ij}}





\subsection{Discovering another relevant mode}
\label{sec:applications}
%\input{applications}

We can conclude from the previously summarized article
that we are left to provide:
\begin{itemize}
\item an u.d. strategy-adapter that is independent
from the cover, 
\item an image mode $f_l$ whose iteration
graph $\Gamma(f_l)$ is strongly connected and whose Markov
matrix is doubly stochastic.
\end{itemize}
We have recalled in the previous section that the $\textit{CIIS}(K,y,\alpha,l)$ strategy adapter 
has the required properties.
In all the experiments provided in~\cite{bcg11:ij}, parameters $K$ and $\alpha$ are randomly 
chosen in $] 0, 1[$ and $] 0, 0.5[$ respectively, while the 
number of iteration is set to $4\times lm$, where $lm$ is the number of LSCs 
that depends on the domain.  
 
\cite{bcg11:ij} has then used the iterative approach of Section~\ref{sec:prac} to generate image
modes $f_l$ such that $\Gamma(f_l)$ is strongly connected, which has
been proposed in~\cite{bcgr11:ip} and recalled in the first part
of this manuscript. Among these
maps, it is obvious to check which verifies or not the doubly
stochastic constrain.
We have already stated that the negation mode matches these hypotheses, so it is relevant in that context.
As a second example, we have considered in~\cite{bcg11:ij} the mode
$f_l: \mathds{B}^l \rightarrow \mathds{B}^l$ s.t. its $i$-th component is
defined by
\begin{equation}
\label{eq:fqq}
{f_l}(x)_i =
\left\{
\begin{array}{l}
\overline{x_i} \textrm{ if $i$ is odd} \\
x_i \oplus x_{i-1} \textrm{ if $i$ is even.}
\end{array}
\right.
\end{equation}

Thanks to Theorem~\ref{th:stego}, we have deduced in~\cite{bcg11:ij} that its iteration graph 
$\Gamma(f_l)$ is strongly connected, and have finally proven
that its Markov chain is doubly stochastic by induction on the length $l$.


\subsection{dhCI in frequency domains}
We recall in this section the experimental protocol applied in~\cite{bcg11:ij}.

\subsubsection{DWT embedding}
%\input{dwt}

We have firstly explained in~\cite{bcg11:ij}  how the dhCI dissimulation can be applied in
the discrete wavelets transform domain (DWT).
The Daubechies family of wavelets has been chosen: 
% a bitmap file of the famous Lena is converted into its Daubechies-1 DWT
% coefficients, which are altered by chaotic iterations. 
each DWT decomposition depends on a decomposition level and a coefficient
matrix (Figure~\ref{fig:DWTs}): $\textit{LL}$ means approximation coefficient,
when $\textit{HH},\textit{LH},\textit{HL}$ denote respectively diagonal, 
vertical, and horizontal detail coefficients. 
For example, the DWT coefficient \textit{HH}2 is the matrix equal to the 
diagonal detail coefficient of the second level of decomposition of the image.

\begin{figure}[h!]
\centerline{
\includegraphics[width=4cm]{IH/CompJ/DWTs.eps}
}
\caption{Wavelets coefficients.}
\label{fig:DWTs}
\end{figure}





The choice of the detail level is motivated by finding
a good compromise between robustness and invisibility.
Choosing low or high frequencies in DWT domain leads either to a very
fragile watermarking without robustness (especially when facing a
JPEG 2000 compression attack) or to a large degradation of the host
content. 
In order to have a robust but discrete DWT embedding, 
the second detail level 
(\textit{i.e.}, $\textit{LH}2,\textit{HL}2,\textit{HH}2$) 
that corresponds to the middle frequencies,
has been retained in~\cite{bcg11:ij}.




Let us consider the Daubechies wavelet coefficients of a third
level decomposition as represented in Figure~\ref{fig:DWTs}. 
We then have translated these float coefficients into their 32-bits values, and have
defines in~\cite{bcg11:ij} the  significance function $u$ that associates to any index $k$ in this sequence of bits the following numbers:
\begin{itemize}
\item $u^k = -1$ if $k$ is one of the three last bits of any index of
  coefficients in  $\textit{LH}2$, $\textit{HL}2$, or in $\textit{HH}2$;
\item $u^k = 0$ if $k$ is an index of a coefficient in  
  $\textit{LH}1$, $\textit{HL}1$, or in $\textit{HH}1$;
\item $u^k = 1$ otherwise.
\end{itemize}

According to the definition of significance of coefficients 
(Def.~\ref{def:msc,lsc}), if $(m,M)$ is $(-0.5,0.5)$,  LSCs are the
last three bits of coefficients in 
$\textit{HL}2$, $\textit{HH}2$, and $\textit{LH}2$.
Thus, decomposition and recomposition functions are fully defined
and dhCI dissimulation scheme can now be applied.

Figure \ref{fig:DWT} shows the result of a
dhCI dissimulation embedding into DWT domain. 
The original is the image 5007 of the BOSS contest~\cite{Boss10}.
Watermark $y$ is given in Fig.~\ref{(b) Watermark}.
From a random selection of 50 images into the database from the BOSS 
contest~\cite{Boss10}, we have applied in~\cite{bcg11:ij} the dhCI algorithm 
with mode $f_l$ 
defined in the previous section and with the negation mode. 

\begin{figure}[ht]
  \centering
  \subfigure[Original Image.]{\includegraphics[width=5cm]
    {IH/CompJ/5007.eps}\label{(a) Original 5007}}\hspace{1cm}
  \subfigure[Watermark $y$.]{\includegraphics[width=1cm]{IH/CompJ/invader.eps}\label{(b) Watermark}}\hspace{1cm}
  \subfigure[Watermarked Image.]{\includegraphics[width=5cm]{IH/CompJ/5007_bis.eps}\label{(c) Watermarked 5007}}

\caption{Data hiding in DWT domain}
\label{fig:DWT}
\end{figure}


\subsubsection{DCT embedding}
%\input{dct}

We have then explored the discrete cosinus transform (DCT) frequency domain embedding in~\cite{bcg11:ij}, by
following the protocol detailed below.

Let us denote by $x$ the original image of size $H \times L$, and by $y$
the hidden message, supposed here to be a binary image of size $H' \times L'$. %
The image $x$ is transformed from the spatial
domain to DCT domain frequency bands,
in order to embed $y$ inside it.  
To do so, the host image is firstly divided into $8 \times 8$
image blocks as given below:
$$x = \bigcup_{k=1}^{H/8} \bigcup_{k'=1}^{L/8} x(k,k').$$
Thus, for each image block,
a DCT is performed and the coefficients in the frequency bands 
are obtained as follows:
$x_{DCT}(m;n) = DCT(x(m;n))$.

To define a discrete but robust scheme, only the three following coefficients of each $8 \times 8$ block in position $(m,n)$ has 
been possibly modified in~\cite{bcg11:ij}: $x_{DCT}(m;n)_{(3,1)},$ $x_{DCT}(m;n)_{(2,2)},$ or $x_{DCT}(m;n)_{(1,3)}$. 
This choice can be reformulated as follows. 
Coefficients of each DCT matrix are re-indexed by using a southwest/northeast diagonal, such that $i_{DCT}(m,n)_1 = x_{DCT}(m;n)_{(1,1)}$, $i_{DCT}(m,n)_2 = x_{DCT}(m;n)_{(2,1)}$, $i_{DCT}(m,n)_3 = x_{DCT}(m;n)_{(1,2)}$, $i_{DCT}(m,n)_4 = x_{DCT}(m;n)_{(3,1)}$, ..., and $i_{DCT}(m,n)_{64} =$  $ x_{DCT}(m;n)_{(8,8)}$.
So the signification function can be defined in this context by:
\begin{itemize}
\item if $k$ mod $64 \in \{1,2,3\}$ and $k\leqslant H\times L$, then $u^k=1$;
\item else if $k$ mod $64 \in \{4, 5, 6\}$ and $k\leqslant H\times L$, then $u^k=-1$;
\item else $u^k = 0$.
\end{itemize}
The significance of coefficients are obtained for instance with 
$(m,M)=(-0.5,0.5)$ leading to the definitions of MSCs, LSCs, and passive coefficients.
Thus, decomposition and recomposition functions are fully defined
 and dhCI dissimulation scheme has then been applied in~\cite{bcg11:ij}.




\subsection{Image quality}
%\input{quality}
This section focuses on measuring visual quality of our steganographic method.
Traditionally, this is achieved by quantifying the similarity 
between the modified image and its reference image.
The Mean Squared Error (MSE) and the Peak Signal to Noise
Ratio (PSNR) are the most widely known tools that provide such a metric.
However, both of them do not take into account Human Visual System (HVS)
properties. 
Recent works~\cite{EAPLBC06,SheikhB06,PSECAL07,MB10} have tackled this problem 
by creating new metrics. Among them, what follows focuses on PSNR-HVS-M~\cite{PSECAL07} and BIQI~\cite{MB10}, considered as advanced visual quality metrics.  
The former efficiently combines PSNR and  visual between-coefficient contrast masking of DCT basis functions based on HVS. This metric has 
been computed in~\cite{bcg11:ij}
 by using the implementation given at~\cite{psnrhvsm11}.
The latter allows to get a blind image quality assessment measure, 
\textit{i.e.}, without any knowledge of the source distortion.
Its implementation is available at~\cite{biqi11}.


\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
Embedding & \multicolumn{2}{|c|}{DWT} 
 & \multicolumn{2}{|c|}{DCT} \\
\hline
Mode & $f_l$ & neg. & $f_l$ & neg. \\
\hline
PSNR & 42.74     & 42.76     &  52.68      &  52.41   \\
\hline
PSNR-HVS-M & 44.28  & 43.97 & 45.30 & 44.93 \\
\hline
BIQI & 35.35 & 32.78 & 41.59 & 47.47 \\
\hline
\end{tabular}
\end{center}
\caption{Quality measures of our steganography approach~\cite{bcg11:ij}}
\label{table:quality} 
\end{table}



Results of the image quality metrics obtained in~\cite{bcg11:ij}
are summarized in Table~\ref{table:quality}.
In wavelet domain, the PSNR values obtained in~\cite{bcg11:ij} are comparable to other approaches
(for instance, PSNR are 44.2 in~\cite{TCL05} and 46.5 in~\cite{DA10}), 
but  a real improvement for the discrete cosine embedding is obtained 
(PSNR is 45.17 for~\cite{CFS08}, it is always lower than 48 for~\cite{Mohanty:2008:IWB:1413862.1413865}, and always lower than 39 for~\cite{MK08}).
Among steganography approaches that evaluate PSNR-HVS-M, results of our approach 
are convincing. Firstly, optimized method developed along~\cite{Randall11} has a PSNR-HVS-M equal to 44.5 whereas our approach, with a similar PSNR-HVS-M, should be easily improved by considering optimized mode. Next, 
another approach~\cite{Muzzarelli:2010} have higher PSNR-HVS-M, certainly, but
this work does not address robustness evaluation whereas the study presented in~\cite{bcg11:ij} is complete.
Finally, as far as we know, \cite{bcg11:ij} is the first one that has evaluated the BIQI metric in a 
steganographic context. 

 

With all this material, we have then evaluated the robustness of our 
approach in~\cite{bcg11:ij}. 



\subsection{Robustness}
%\input{robustness}
Previous sections have formalized frequency domains embedding and
has focused on the negation and $f_l$ modes.
In the robustness given in this continuation, {dwt}(neg), 
{dwt}(fl), {dct}(neg), and {dct}(fl) 
respectively stand for the DWT and DCT embedding 
with the negation mode and with this instantiated mode.
 
For each experiment presented in~\cite{bcg11:ij}, a set of 50 images is randomly extracted 
from the database taken from the BOSS contest~\cite{Boss10}. 
Each cover is a $512\times 512$ grayscale digital image and the watermark $y$ 
is given in Figure~\ref{(b) Watermark}. 
Testing the robustness of the approach is achieved in~\cite{bcg11:ij} by successively applying
on watermarked images attacks like cropping, compression, and geometric 
transformations.
Differences between 
$\hat{y}$ and $\varphi_m(z)$ have then been
computed. Behind a given threshold rate, the image is said to be watermarked.  
Finally, discussion on metric quality of the approach given in~\cite{bcg11:ij} is recalled in 
Section~\ref{sub:roc}.

 

\subsubsection{Robustness against cropping}

Robustness of the approach is  evaluated by
applying different percentage of cropping: from 1\% to 81\%.
Results obtained in~\cite{bcg11:ij} are recalled in Figure~\ref{Fig:atck:dec}. 
Figure~\ref{Fig:atq:dec:img}
gives the cropped image 
where 36\% of the image is removed, while Figure~\ref{Fig:atq:dec:curves} 
presents effects of such an attack.
From this experiment, we have concluded in~\cite{bcg11:ij} that all embedding has similar 
behavior.
All the percentage differences are so far less than 50\% 
(which is the mean random error) and thus robustness is established.



\begin{figure}[ht]
  \centering
  \subfigure[Cropped image]{\includegraphics[width=0.24\textwidth]
    {IH/CompJ/5007_dec_307.eps}\label{Fig:atq:dec:img}}\hspace{2cm}
  \subfigure[Cropping effects]{
\includegraphics[width=0.5\textwidth]{IH/CompJ/atq-dec.eps}\label{Fig:atq:dec:curves}}
\caption{Cropping results}
\label{Fig:atck:dec}
\end{figure}


\subsubsection{Robustness against compression}

Robustness against compression is addressed
by studying both JPEG  and JPEG 2000 image compression.
Results  obtained in~\cite{bcg11:ij}  are respectively presented in Fig.~\ref{Fig:atq:jpg:curves}
and Fig.~\ref{Fig:atq:jp2:curves}.
Without surprise, DCT embedding which is based on DCT 
(as JPEG compression algorithm is) is  more 
adapted to JPEG compression than DWT embedding.
Furthermore, we have a similar behavior for the JPEG 2000 compression algorithm, which is based on wavelet encoding: DWT embedding naturally outperforms
DCT one in that case.  


\begin{figure}[ht]
  \centering
  \subfigure[JPEG effects]{
\includegraphics[width=0.48\textwidth]{IH/CompJ/atq-jpg.eps}\label{Fig:atq:jpg:curves}}
  \subfigure[JPEG 2000 effects]{
\includegraphics[width=0.48\textwidth]{IH/CompJ/atq-jp2.eps}\label{Fig:atq:jp2:curves}}
\caption{Compression results}
\label{Fig:atck:comp}
\end{figure}



\subsubsection{Robustness against contrast and sharpness}

Contrast and Sharpness adjustments belong to the classical set of 
filtering image attacks.
Results of such attacks are presented in 
Fig.~\ref{Fig:atq:fil} where 
Fig.~\ref{Fig:atq:cont:curve} and Fig.~\ref{Fig:atq:sh:curve} summarize 
effects of contrast and sharpness adjustment respectively~\cite{bcg11:ij}. 


\begin{figure}[ht]
  \centering
  \subfigure[Contrast effects]{
\includegraphics[width=0.48\textwidth]{IH/CompJ/atq-contrast.eps}\label{Fig:atq:cont:curve}}
  \subfigure[Sharpness effects]{
\includegraphics[width=0.48\textwidth]{IH/CompJ/atq-flou.eps}\label{Fig:atq:sh:curve}}
\caption{Filtering results}
\label{Fig:atq:fil}
\end{figure}

\subsubsection{Robustness against geometric transformations}

Among geometric transformations, we have focused in~\cite{bcg11:ij} on  
rotations, \textit{i.e.}, when two opposite rotations 
of angle $\theta$ are successively applied around the center of the image.
In these geometric transformations,  angles range from 2 to 20 
degrees.  
Results obtained in~\cite{bcg11:ij} are summed up in Figure~\ref{Fig:atq:rot}: Fig.~\ref{Fig:atq:rot:img}
gives the image of a rotation of 20 degrees whereas
Fig.~\ref{Fig:atq:rot:curve} presents effects of such an attack.
It is not a surprise that results are better for DCT embedding: this approach
is based on cosine as rotation is. 



\begin{figure}[ht]
  \centering
  \subfigure[20 degrees rotation image]{
    \includegraphics[width=0.25\textwidth]{IH/CompJ/5007_rot_10.eps}\label{Fig:atq:rot:img}}
  \subfigure[Rotation effects]{
\includegraphics[width=0.5\textwidth]{IH/CompJ/atq-rot.eps}\label{Fig:atq:rot:curve}}

\caption{Rotation attack results}
\label{Fig:atq:rot}
\end{figure}

\subsection{Evaluation of the  Embedding}
\label{sub:roc}

We are then left to set a convenient threshold that is accurate to 
determine whether an image is watermarked or not.
Starting from a set of 100 images selected among the Boss image panel,
we have computed in~\cite{bcg11:ij} the following three sets: 
the one with all the watermarked images $W$,
the one with all successively watermarked and attacked images $\textit{WA}$,
and the one with only the attacked images $A$.
Notice that the 100 attacks for each image
are selected among these detailed previously.


For each threshold $t \in \llbracket 0,55 \rrbracket$ and a given image 
$x \in \textit{WA} \cup A$, 
differences on DCT have been computed in~\cite{bcg11:ij}. 
The image has been claimed as watermarked
if these differences are less than the threshold. 
\begin{itemize}
\item In the positive case and if $x$ really belongs to 
$\textit{WA}$ it is a True Positive (TP) case.  
\item In the negative case but if $x$ belongs to 
$\textit{WA}$, it is a False Negative (FN) case.  
\item In the positive case but if $x$  belongs to 
$\textit{A}$, it is a False Positive (FP) case.  
\item Finally, in the negative case and if $x$ belongs to
$\textit{A}$, it is a True Negative (TN).  
\end{itemize}
The True (resp. False) Positive Rate  (TPR) (resp. FPR) has thus been computed 
by dividing the number of TP (resp. FP) by 100.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=9cm]{IH/CompJ/ROC.eps}
\end{center}
\caption{ROC curves for DWT or DCT embeddings}\label{fig:roc:dwt}
\end{figure}

Figure~\ref{fig:roc:dwt} recalled the obtained Receiver Operating Characteristic (ROC) 
curve. 
For the DWT, it shows that best results are obtained when the threshold 
is 45\% for the dedicated function (corresponding to the point (0.01, 0.88))
and  46\% for the negation function (corresponding to (0.04, 0.85)).
It allows to conclude that each time LSCs differences between
a watermarked image and another given image  $i'$ are less than 45\%, we can claim that 
$i'$ is an attacked version of the original watermarked content.
For the two DCT embedding, best results have been obtained when the threshold 
is 44\% (corresponding to the points (0.05, 0.18) and (0.05, 0.28)).

We have thus conclude some confidence intervals for all the evaluated 
attacks in~\cite{bcg11:ij}. The
approach is resistant to:
\begin{itemize}
\item all the cropping where percentage is less than 85;
\item compression where quality ratio is greater 
  than 82 with DWT embedding and 
  where quality ratio is greater than 67 with DCT one;
\item contrast when strengthening belongs to $[0.76,1.2]$ 
(resp. $[0.96,1.05]$)  in DWT (resp. in DCT) embedding;
\item all the rotation attacks with DCT embedding and a rotation where
angle is less than 13 degrees with DWT one.
\end{itemize}


% \subsection{Conclusion}\label{sec:concl}
% %\input{conclusion}
% This paper has proposed a new class of secure and robust information hiding 
% algorithms.
% It has been entirely formalized, thus allowing both its theoretical security 
% analysis, and the computation of numerous variants encompassing spatial and 
% frequency domain embedding.
% After having presented the general algorithm with detail, we have given
% conditions for choosing mode and strategy-adapter making the whole
% class  stego-secure or $\epsilon$-stego-secure.
% To our knowledge, this is the first time such a result has been established.
% 
% 
% Applications in frequency domains (namely DWT and DCT domains) have finally be
% formalized.
% Complete experiments have allowed us 
% first to evaluate how invisible is the steganographic method (thanks to the PSNR computation) and next to verify the robustness property against attacks.
% Furthermore, the use of ROC curves for DWT embedding have revealed very high rates
% between True positive and False positive results.
% 
% In future work, our intention is to find the best image mode with respect to  
% the combination between  DCT and DWT based steganography
% algorithm. Such a combination topic has already been addressed
% (\textit{e.g.}, in~\cite{al2007combined}), but never with objectives
% we have set.


% Additionally, we will try to discover new topological properties for the dhCI
% dissimulation schemes.
% Consequences of these chaos properties will be drawn in the context of 
% information hiding security.
% We will especially focus on the links between topological properties and classes
% of attacks, such as KOA, KMA, EOA, or CMA.
% 
% Moreover, these algorithms will be compared to other existing ones, among other
% things by testing whether these algorithms are chaotic or not.
% Finally we plan to verify the robustness of our approach 
% against statistical steganalysis methods~\cite{GFH06,ChenS08,DongT08,FridrichKHG11a}.
% 
%