


\subsection{Definition of robustness}\label{section:robustness-definition}

\textbf{TODO: Ajouter la definition de robustesse de Kalker}

\textbf{TODO: Ajouter la definition de sécurité de Kalker --> AU BON ENDROIT}

\subsection{Visual scales for robustness
threshold}\label{section:robustness-scale}

\textbf{TODO definition}

Given the robustness definition, it can be said that the notion of robustness is
a subjective notion depending of the context and of the watermarking application
types.\newline

To affirm that a watermarking process is robust or not in such or such other
conditions, it is necessary to define a robustness threshold. If it is
considered an image as watermak, such a threshold can be subjectively defined
looking for a visual difference rate scale in term of bytes differences. It has
been established 3 scales for 3 types of images:
\begin{enumerate}
  \item Greyscale images,
  \item Colour images,
  \item And white and black images.
\end{enumerate}


 \textbf{Firstly, it can be remark that in the field of watermarking with an image
 as watermark, one watermark can be visually considered as identical to an other,
 if the second is the same, or correspond to the negative version of the first.}
 
\subsubsection{\textbf{Scale in greyscale domain}}

The scale in the greyscale domain is presented in
Figure~\ref{fig:visual-scale-greyscale}~\vpageref{fig:visual-scale-greyscale}.

Looking the visual scale, it can be deduced that the in the greyscale domain, a
watermarking process is robust for a difference rate between 0\% and 30\%, and
between 70\% and 100\% for the negative image case.

\textbf{So in this field, the robustness threshold can be fixed to 30\%.}

\begin{figure*}[htb]
\begin{center}
\begin{minipage}[b]{.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/0_lena}}
    \centerline{(0) Difference rate of 0\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/5_lena}}
    \centerline{(1) Difference rate of 5\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/10_lena}}
    \centerline{(2) Difference rate of 10\%}
\end{minipage}
\begin{minipage}[b]{.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/15_lena}}
    \centerline{(3) Difference rate of 15\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/20_lena}}
    \centerline{(4) Difference rate of 20\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/25_lena}}
    \centerline{(5) Difference rate of 25\%}
\end{minipage}
\begin{minipage}[b]{.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/30_lena}}
    \centerline{(6) Difference rate of 30\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/35_lena}}
    \centerline{(7) Difference rate of 35\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/40_lena}}
    \centerline{(8) Difference rate of 40\%}
\end{minipage}
\begin{minipage}[b]{.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/45_lena}}
    \centerline{(9) Difference rate of 45\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/50_lena}}
    \centerline{(10) Difference rate of 50\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/55_lena}}
    \centerline{(11) Difference rate of 55\%}
\end{minipage}
\begin{minipage}[b]{.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/60_lena}}
    \centerline{(12) Difference rate of 60\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/65_lena}}
    \centerline{(13) Difference rate of 65\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/70_lena}}
    \centerline{(14) Difference rate of 70\%}
\end{minipage}
\begin{minipage}[b]{.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/75_lena}}
    \centerline{(15) Difference rate of 75\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/80_lena}}
    \centerline{(16) Difference rate of 80\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/85_lena}}
    \centerline{(17) Difference rate of 85\%}
\end{minipage}
\begin{minipage}[b]{.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/90_lena}}
    \centerline{(18) Difference rate of 90\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/95_lena}}
    \centerline{(19) Difference rate of 95\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/100_lena}}
    \centerline{(20) Difference rate of 100\%}
\end{minipage}
\caption{Visual scale in greyscale domain for robustness threshold evaluation.}
\label{fig:visual-scale-greyscale}
\end{center}
\end{figure*}


\subsubsection{\textbf{Scale in colour domain}}

The scale in color domain is presented in
Figure~\ref{fig:visual-scale-colour}~\vpageref{fig:visual-scale-colour}.

Looking the visual scale, it can be deduced that the in the color domain, a
watermarking process is robust for a difference rate between 0\% and 40\%, and
between 60\% and 100\% for the negative image case.

\textbf{So in this field, the robustness threshold can be fixed to 40\%.}

\begin{figure*}[htb]
\begin{center}
\begin{minipage}[b]{.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/color-scale/0_singe}}
    \centerline{(0) Difference rate of 0\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/color-scale/5_singe}}
    \centerline{(1) Difference rate of 5\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/color-scale/10_singe}}
    \centerline{(2) Difference rate of 10\%}
\end{minipage}
\begin{minipage}[b]{.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/color-scale/15_singe}}
    \centerline{(3) Difference rate of 15\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/color-scale/20_singe}}
    \centerline{(4) Difference rate of 20\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/color-scale/25_singe}}
    \centerline{(5) Difference rate of 25\%}
\end{minipage}
\begin{minipage}[b]{.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/color-scale/30_singe}}
    \centerline{(6) Difference rate of 30\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/color-scale/35_singe}}
    \centerline{(7) Difference rate of 35\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/color-scale/40_singe}}
    \centerline{(8) Difference rate of 40\%}
\end{minipage}
\begin{minipage}[b]{.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/color-scale/45_singe}}
    \centerline{(9) Difference rate of 45\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/color-scale/50_singe}}
    \centerline{(10) Difference rate of 50\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/color-scale/55_singe}}
    \centerline{(11) Difference rate of 55\%}
\end{minipage}
\begin{minipage}[b]{.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/color-scale/60_singe}}
    \centerline{(12) Difference rate of 60\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/color-scale/65_singe}}
    \centerline{(13) Difference rate of 65\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/color-scale/70_singe}}
    \centerline{(14) Difference rate of 70\%}
\end{minipage}
\begin{minipage}[b]{.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/color-scale/75_singe}}
    \centerline{(15) Difference rate of 75\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/color-scale/80_singe}}
    \centerline{(16) Difference rate of 80\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/color-scale/85_singe}}
    \centerline{(17) Difference rate of 85\%}
\end{minipage}
\begin{minipage}[b]{.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/color-scale/90_singe}}
    \centerline{(18) Difference rate of 90\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/color-scale/95_singe}}
    \centerline{(19) Difference rate of 95\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/color-scale/100_singe}}
    \centerline{(20) Difference rate of 100\%}
\end{minipage}
\caption{Visual scale in colour domain for robustness threshold evaluation.}
\label{fig:visual-scale-colour}
\end{center}
\end{figure*}

\subsubsection{\textbf{Scale in white and black domain}}

The scale in white and black domain is presented in
Figure~\ref{fig:visual-scale-white-black}~\vpageref{fig:visual-scale-white-black}.

Looking the visual scale, it can be deduced that the in the  white and black domain, a
watermarking process is robust for a difference rate between 0\% and 20\%, and
between 80\% and 100\% for the negative image case.

\textbf{So in this field, the robustness threshold can be fixed to 20\%.}

\begin{figure*}[htb]
\begin{center}
\begin{minipage}[b]{.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/0_invader}}
    \centerline{(0) Difference rate of 0\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/5_invader}}
    \centerline{(1) Difference rate of 5\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/10_invader}}
    \centerline{(2) Difference rate of 10\%}
\end{minipage}
\begin{minipage}[b]{.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/15_invader}}
    \centerline{(3) Difference rate of 15\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/20_invader}}
    \centerline{(4) Difference rate of 20\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/25_invader}}
    \centerline{(5) Difference rate of 25\%}
\end{minipage}
\begin{minipage}[b]{.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/30_invader}}
    \centerline{(6) Difference rate of 30\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/35_invader}}
    \centerline{(7) Difference rate of 35\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/40_invader}}
    \centerline{(8) Difference rate of 40\%}
\end{minipage}
\begin{minipage}[b]{.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/45_invader}}
    \centerline{(9) Difference rate of 45\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/50_invader}}
    \centerline{(10) Difference rate of 50\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/55_invader}}
    \centerline{(11) Difference rate of 55\%}
\end{minipage}
\begin{minipage}[b]{.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/60_invader}}
    \centerline{(12) Difference rate of 60\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/65_invader}}
    \centerline{(13) Difference rate of 65\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/70_invader}}
    \centerline{(14) Difference rate of 70\%}
\end{minipage}
\begin{minipage}[b]{.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/75_invader}}
    \centerline{(15) Difference rate of 75\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/80_invader}}
    \centerline{(16) Difference rate of 80\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/85_invader}}
    \centerline{(17) Difference rate of 85\%}
\end{minipage}
\begin{minipage}[b]{.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/90_invader}}
    \centerline{(18) Difference rate of 90\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/95_invader}}
    \centerline{(19) Difference rate of 95\%}
\end{minipage}
\hfill
\begin{minipage}[b]{0.32\linewidth}
  \centering
    \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/100_invader}}
    \centerline{(20) Difference rate of 100\%}
\end{minipage}
\caption{Visual scale in white and black domain for robustness threshold
evaluation.}
\label{fig:visual-scale-white-black}
\end{center}
\end{figure*}

\subsubsection{\textbf{Conclusion concerning the robustness threshold}}

\textbf{From the results of the previous study, it can be concluded that the
robustness threshold depends of the nature of the watermark.}\newline

\textbf{So for a picture as watermark, the highest robustness threshold is
obtain with color images.}

\subsection{Presentation of the
architecture}\label{section:architecture-presentation}

Computations have been
performed on the supercomputer facilities of the Mésocentre de calcul de Franche-Comt\'{e}.

In order to take benefits of parallelism, we have used Jace~\cite{bhm09:ip}, a
grid-enabled programming and execution environment allowing a simple and efficient implementation 
of parallel and distributed applications. Roughly speaking, Jace builds a virtual parallel machine by 
connecting a set of heterogeneous and distant computers.
It schedules tasks, executes them, and returns results to the user. It
also proposes a simple programming interface for the implementation of applications
using a message passing model.

In our case, tasks are independent Python programs with different parameters. Jace takes care to execute them in parallel ensuring optimal load balancing and fault tolerance.

For these experiments we have used HPC resources 
from Mesocentre of Franche-Comt\'{e}. This platform 
is currently composed of 720 cores with 11 TeraFLOPS of power.

The different tests have been realized on several images from the Wikimedia
Commons repository~\cite{wiki:wikimedia-commons}.


\textbf{FAIRE AUSSI DES TESTS AVEC LA BASE DE BOSS}
The conditions of all tests are described for each one in the caption of the graph result.

%je ne sais pas comment expliquer vos configs !
%In the following experiments, we have chosen different configuration files,
%about 1000 images from wikimedia.....



\subsection{Tests realized}\label{sec:tests-realized-ci-1}

In this section, it is studied the robustness of the watermarking process
$CI_1$. To do this, it has been executed two test batteries in order to evaluate
the evolution of the bytes difference rate in function of the geometrical
attacks parameters, between the watermark inserted and the watermark extracted
after the attack.

In the first battery of tests we have used the same embedding key (a constant
strategy,the same strategy has been used to watermark all the
images in this test) to embed the watermark. In the second battery of tests,
in order to see the influence of the embedding key on the robustness, we have embedded all
the watermark with strategies generated randomly(a
different strategy has been used to watermark each image). 

To obtain the following results, we have realized a great number of attacks on
600 images, and we have traced for each attack, two graphics (average and
standard deviation), in order to determine the acceptability
threshold for our watermarking process $CI_1$.

\subsection{Tests results}\label{sec:tests-results-ci-1}

The exhaustive list of the geometrical attacks tested and the
corresponding graphs results is following:

\begin{enumerate}
  \item \textbf{Robustness facing a resizing attack.}\\
  See Figure~\ref{fig:resizing-attack-constant-strategy}~\vpageref{fig:resizing-attack-constant-strategy}.
  \item \textbf{Robustness facing a JPEG attack.}\\See
  Figure~\ref{fig:jpeg-attack-constant-strategy}~\vpageref{fig:jpeg-attack-constant-strategy}.
  \item \textbf{Robustness facing a Gaussian blur
  attack.}\\See
  Figure~\ref{fig:gaussian-blur-attack-constant-strategy}~\vpageref{fig:gaussian-blur-attack-constant-strategy}.
  \item \textbf{Robustness facing a rotation
  attack.}\\See
  Figure~\ref{fig:rotation-attack-constant-strategy}~\vpageref{fig:rotation-attack-constant-strategy}.
  \item \textbf{Robustness facing a  blur
  attack.}\\See
  Figure~\ref{fig:blur-attack-constant-strategy}~\vpageref{fig:blur-attack-constant-strategy}.
  \item \textbf{Robustness facing a contrast
  attack.}\\See
  Figure~\ref{fig:contrast-attack-constant-strategy}~\vpageref{fig:contrast-attack-constant-strategy}.
  \item \textbf{Robustness facing a cropping
  attack.}\\See
  Figure~\ref{fig:cropping-attack-constant-strategy}~\vpageref{fig:cropping-attack-constant-strategy}.
\end{enumerate}


% \begin{figure*}
% \begin{center}
% %\includegraphics[width=10.5cm]{./img/prisoner-problem.png}
% \includegraphics[width=15cm]{./img/prisoner-problem}
% \end{center}
% 
% \caption{Simmons' prisoner problem~\cite{Simmons83}}
% \end{figure*}


\begin{figure*}[htb]
\begin{minipage}[b]{.45\linewidth}
  \centering
    \centerline{\includegraphics[width=9cm]{graphs/graph_1_attack=redimensionnement-average}}
    \centerline{(a) Average}
\end{minipage}
\hfill
\begin{minipage}[b]{0.45\linewidth}
  \centering
    \centerline{\includegraphics[width=9cm]{graphs/graph_1_attack=redimensionnement-sd}}
    \centerline{(b) Standard deviation}
\end{minipage}
\caption{Robustness of $CI_1$ facing a resizing attack.(600 images and constant strategy)}
\label{fig:resizing-attack-constant-strategy}
\end{figure*}

\begin{figure*}[htb]
\begin{minipage}[b]{.45\linewidth}
  \centering
    \centerline{\includegraphics[width=9cm]{graphs/graph_2_attack=jpeg-average}}
    \centerline{(a) Average}
\end{minipage}
\hfill
\begin{minipage}[b]{0.45\linewidth}
  \centering
    \centerline{\includegraphics[width=9cm]{graphs/graph_2_attack=jpeg-sd}}
    \centerline{(b) Standard deviation}
\end{minipage}
\caption{Robustness of $CI_1$ facing a JPEG compression attack.(600 images and
constant strategy)}
\label{fig:jpeg-attack-constant-strategy}
\end{figure*}


\begin{figure*}[htb]
\begin{minipage}[b]{.45\linewidth}
  \centering
    \centerline{\includegraphics[width=9cm]{graphs/graph_3_attack=gaussien-average}}
    \centerline{(a) Average}
\end{minipage}
\hfill
\begin{minipage}[b]{0.45\linewidth}
  \centering
    \centerline{\includegraphics[width=9cm]{graphs/graph_3_attack=gaussien-sd}}
    \centerline{(b) Standard deviation}
\end{minipage}
\caption{Robustness of $CI_1$ facing a Gaussian blur attack.(600 images and
constant strategy)}
\label{fig:gaussian-blur-attack-constant-strategy}
\end{figure*}

\begin{figure*}[htb]
\begin{minipage}[b]{.45\linewidth}
  \centering
    \centerline{\includegraphics[width=9cm]{graphs/graph_4_attack=rotation-average}}
    \centerline{(a) Average}
\end{minipage}
\hfill
\begin{minipage}[b]{0.45\linewidth}
  \centering
    \centerline{\includegraphics[width=9cm]{graphs/graph_4_attack=rotation-sd}}
    \centerline{(b) Standard deviation}
\end{minipage}
\caption{Robustness of $CI_1$ facing a rotation attack.(600 images and constant
strategy)}
\label{fig:rotation-attack-constant-strategy}
\end{figure*}


\begin{figure*}[htb]
\begin{minipage}[b]{.45\linewidth}
  \centering
    \centerline{\includegraphics[width=9cm]{graphs/graph_5_attack=flou-average}}
    \centerline{(a) Average}
\end{minipage}
\hfill
\begin{minipage}[b]{0.45\linewidth}
  \centering
    \centerline{\includegraphics[width=9cm]{graphs/graph_5_attack=flou-sd}}
    \centerline{(b) Standard deviation}
\end{minipage}
\caption{Robustness of $CI_1$ facing a blur attack.(600 images and constant
strategy)}
\label{fig:blur-attack-constant-strategy}
\end{figure*}


\begin{figure*}[htb]
\begin{minipage}[b]{.45\linewidth}
  \centering
    \centerline{\includegraphics[width=9cm]{graphs/graph_6_attack=contraste-average}}
    \centerline{(a) Average}
\end{minipage}
\hfill
\begin{minipage}[b]{0.45\linewidth}
  \centering
    \centerline{\includegraphics[width=9cm]{graphs/graph_6_attack=contraste-sd}}
    \centerline{(b) Standard deviation}
\end{minipage}
\caption{Robustness of $CI_1$ facing a contrast attack.(600 images and constant
strategy)}
\label{fig:contrast-attack-constant-strategy}
\end{figure*}


\begin{figure*}[htb]
\begin{minipage}[b]{.45\linewidth}
  \centering
    \centerline{\includegraphics[width=9cm]{graphs/graph_7_attack=decoupage-average}}
    \centerline{(a) Average}
\end{minipage}
\hfill
\begin{minipage}[b]{0.45\linewidth}
  \centering
    \centerline{\includegraphics[width=9cm]{graphs/graph_7_attack=decoupage-sd}}
    \centerline{(b) Standard deviation}
\end{minipage}
\caption{Robustness of $CI_1$ facing a cropping attack.(600 images and constant
strategy)}
\label{fig:cropping-attack-constant-strategy}
\end{figure*}

\subsection{Robustness evaluation}\label{sec:tests-results-ci-1}

From the results presented in the previous section, the robustness of the
process $CI_1$ can be evaluated.


As a first conclusion, it can be note that in the $CI_1$ watermarking process, the strategy has no
influence on the robustness. Indeed, analyzing all the curves, a strong
similarity between the curves obtained with a constant strategy and the curves obtained with a strategy generated randomly can
be highlight.

As a second conclusion, it can be note that the $CI_1$ watermarking process is
not robust in spatial domain facing attacks of resizing, of JPEG compression
with a compression rate lesser than 97\%, of Gaussian blur with a blur rate
greater than 42\%, of classic blur, of contrast, and of cropping with a number
of pixels greater than 180.

Inversely, $CI_1$ is robust in spatial domain facing geometrical attacks of JPEG
compression with a compression rate greater than 97\%, of Gaussian blur with a blur rate
lesser than 42\%, of rotation with angles between 0° and 30°, and of cropping
with a number of pixels lesser than 180.


\begin{enumerate}
  \item \textbf{Robustness facing a resizing attack.}\\
  See Figure~\ref{fig:resizing-attack-constant-strategy}~\vpageref{fig:resizing-attack-constant-strategy}.
  \item \textbf{Robustness facing a JPEG compression
  attack.}\\See
  Figure~\ref{fig:jpeg-attack-constant-strategy}~\vpageref{fig:jpeg-attack-constant-strategy}.
  \item \textbf{Robustness facing a Gaussian blur
  attack.}\\See
  Figure~\ref{fig:gaussian-blur-attack-constant-strategy}~\vpageref{fig:gaussian-blur-attack-constant-strategy}.
  \item \textbf{Robustness facing a rotation
  attack.}\\See
  Figure~\ref{fig:rotation-attack-constant-strategy}~\vpageref{fig:rotation-attack-constant-strategy}.
  \item \textbf{Robustness facing a  blur
  attack.}\\See
  Figure~\ref{fig:blur-attack-constant-strategy}~\vpageref{fig:blur-attack-constant-strategy}.
  \item \textbf{Robustness facing a contrast
  attack.}\\See
  Figure~\ref{fig:contrast-attack-constant-strategy}~\vpageref{fig:contrast-attack-constant-strategy}.
  \item \textbf{Robustness facing a cropping
  attack.}\\See
  Figure~\ref{fig:cropping-attack-constant-strategy}~\vpageref{fig:cropping-attack-constant-strategy}.
\end{enumerate}

ensuite il faut reprendre chaque courbe et énoncer une phrase du genre :
" on voit qu'on est loin d'une génération aléatoire des bits qui serait environ de 50%"
et donc on peut affirmer avec certitude que telle ou telle image est tatouée ou non.




To conclude, at this point, only two information hiding schemes are both stego-secure and chaos-secure \cite{gfb10:ip}.
The first one is based on a spread spectrum technique called Natural Watermarking.
It is stego-secure when its parameter $\eta$ is equal to $1$ \cite{Cayre2008}. 
Unfortunately, this scheme is neither robust, nor able to face an attacker in KOA and KMA setups, due to its lack of a topological property called expansivity \cite{gfb10:ip}.
The second scheme both chaos-secure and stego-secure is based on chaotic iterations \cite{guyeux10ter}.
However, its first versions called $CI_1$ allows to embed securely only one bit per embedding parameters. 
The objective of the next sections is to improve the $CI_1$ scheme presented in \cite{guyeux10ter}, in such a way that more than one bit can be embedded.

