IH/iihmsp13/robustnessci1.tex

   1
   2
   3
   4 \subsection{Definition of robustness}\label{section:robustness-definition}
   5
   6 \textbf{TODO: Ajouter la definition de robustesse de Kalker}
   7
   8 \textbf{TODO: Ajouter la definition de sécurité de Kalker --> AU BON ENDROIT}
   9
  10 \subsection{Visual scales for robustness
  11 threshold}\label{section:robustness-scale}
  12
  13 \textbf{TODO definition}
  14
  15 Given the robustness definition, it can be said that the notion of robustness is
  16 a subjective notion depending of the context and of the watermarking application
  17 types.\newline
  18
  19 To affirm that a watermarking process is robust or not in such or such other
  20 conditions, it is necessary to define a robustness threshold. If it is
  21 considered an image as watermak, such a threshold can be subjectively defined
  22 looking for a visual difference rate scale in term of bytes differences. It has
  23 been established 3 scales for 3 types of images:
  24 \begin{enumerate}
  25   \item Greyscale images,
  26   \item Colour images,
  27   \item And white and black images.
  28 \end{enumerate}
  29
  30
  31  \textbf{Firstly, it can be remark that in the field of watermarking with an image
  32  as watermark, one watermark can be visually considered as identical to an other,
  33  if the second is the same, or correspond to the negative version of the first.}
  34
  35 \subsubsection{\textbf{Scale in greyscale domain}}
  36
  37 The scale in the greyscale domain is presented in
  38 Figure~\ref{fig:visual-scale-greyscale}~\vpageref{fig:visual-scale-greyscale}.
  39
  40 Looking the visual scale, it can be deduced that the in the greyscale domain, a
  41 watermarking process is robust for a difference rate between 0\% and 30\%, and
  42 between 70\% and 100\% for the negative image case.
  43
  44 \textbf{So in this field, the robustness threshold can be fixed to 30\%.}
  45
  46 \begin{figure*}[htb]
  47 \begin{center}
  48 \begin{minipage}[b]{.32\linewidth}
  49   \centering
  50     \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/0_lena}}
  51     \centerline{(0) Difference rate of 0\%}
  52 \end{minipage}
  53 \hfill
  54 \begin{minipage}[b]{0.32\linewidth}
  55   \centering
  56     \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/5_lena}}
  57     \centerline{(1) Difference rate of 5\%}
  58 \end{minipage}
  59 \hfill
  60 \begin{minipage}[b]{0.32\linewidth}
  61   \centering
  62     \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/10_lena}}
  63     \centerline{(2) Difference rate of 10\%}
  64 \end{minipage}
  65 \begin{minipage}[b]{.32\linewidth}
  66   \centering
  67     \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/15_lena}}
  68     \centerline{(3) Difference rate of 15\%}
  69 \end{minipage}
  70 \hfill
  71 \begin{minipage}[b]{0.32\linewidth}
  72   \centering
  73     \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/20_lena}}
  74     \centerline{(4) Difference rate of 20\%}
  75 \end{minipage}
  76 \hfill
  77 \begin{minipage}[b]{0.32\linewidth}
  78   \centering
  79     \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/25_lena}}
  80     \centerline{(5) Difference rate of 25\%}
  81 \end{minipage}
  82 \begin{minipage}[b]{.32\linewidth}
  83   \centering
  84     \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/30_lena}}
  85     \centerline{(6) Difference rate of 30\%}
  86 \end{minipage}
  87 \hfill
  88 \begin{minipage}[b]{0.32\linewidth}
  89   \centering
  90     \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/35_lena}}
  91     \centerline{(7) Difference rate of 35\%}
  92 \end{minipage}
  93 \hfill
  94 \begin{minipage}[b]{0.32\linewidth}
  95   \centering
  96     \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/40_lena}}
  97     \centerline{(8) Difference rate of 40\%}
  98 \end{minipage}
  99 \begin{minipage}[b]{.32\linewidth}
 100   \centering
 101     \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/45_lena}}
 102     \centerline{(9) Difference rate of 45\%}
 103 \end{minipage}
 104 \hfill
 105 \begin{minipage}[b]{0.32\linewidth}
 106   \centering
 107     \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/50_lena}}
 108     \centerline{(10) Difference rate of 50\%}
 109 \end{minipage}
 110 \hfill
 111 \begin{minipage}[b]{0.32\linewidth}
 112   \centering
 113     \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/55_lena}}
 114     \centerline{(11) Difference rate of 55\%}
 115 \end{minipage}
 116 \begin{minipage}[b]{.32\linewidth}
 117   \centering
 118     \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/60_lena}}
 119     \centerline{(12) Difference rate of 60\%}
 120 \end{minipage}
 121 \hfill
 122 \begin{minipage}[b]{0.32\linewidth}
 123   \centering
 124     \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/65_lena}}
 125     \centerline{(13) Difference rate of 65\%}
 126 \end{minipage}
 127 \hfill
 128 \begin{minipage}[b]{0.32\linewidth}
 129   \centering
 130     \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/70_lena}}
 131     \centerline{(14) Difference rate of 70\%}
 132 \end{minipage}
 133 \begin{minipage}[b]{.32\linewidth}
 134   \centering
 135     \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/75_lena}}
 136     \centerline{(15) Difference rate of 75\%}
 137 \end{minipage}
 138 \hfill
 139 \begin{minipage}[b]{0.32\linewidth}
 140   \centering
 141     \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/80_lena}}
 142     \centerline{(16) Difference rate of 80\%}
 143 \end{minipage}
 144 \hfill
 145 \begin{minipage}[b]{0.32\linewidth}
 146   \centering
 147     \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/85_lena}}
 148     \centerline{(17) Difference rate of 85\%}
 149 \end{minipage}
 150 \begin{minipage}[b]{.32\linewidth}
 151   \centering
 152     \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/90_lena}}
 153     \centerline{(18) Difference rate of 90\%}
 154 \end{minipage}
 155 \hfill
 156 \begin{minipage}[b]{0.32\linewidth}
 157   \centering
 158     \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/95_lena}}
 159     \centerline{(19) Difference rate of 95\%}
 160 \end{minipage}
 161 \hfill
 162 \begin{minipage}[b]{0.32\linewidth}
 163   \centering
 164     \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/100_lena}}
 165     \centerline{(20) Difference rate of 100\%}
 166 \end{minipage}
 167 \caption{Visual scale in greyscale domain for robustness threshold evaluation.}
 168 \label{fig:visual-scale-greyscale}
 169 \end{center}
 170 \end{figure*}
 171
 172
 173 \subsubsection{\textbf{Scale in colour domain}}
 174
 175 The scale in color domain is presented in
 176 Figure~\ref{fig:visual-scale-colour}~\vpageref{fig:visual-scale-colour}.
 177
 178 Looking the visual scale, it can be deduced that the in the color domain, a
 179 watermarking process is robust for a difference rate between 0\% and 40\%, and
 180 between 60\% and 100\% for the negative image case.
 181
 182 \textbf{So in this field, the robustness threshold can be fixed to 40\%.}
 183
 184 \begin{figure*}[htb]
 185 \begin{center}
 186 \begin{minipage}[b]{.32\linewidth}
 187   \centering
 188     \centerline{\includegraphics[width=2.9cm]{img/color-scale/0_singe}}
 189     \centerline{(0) Difference rate of 0\%}
 190 \end{minipage}
 191 \hfill
 192 \begin{minipage}[b]{0.32\linewidth}
 193   \centering
 194     \centerline{\includegraphics[width=2.9cm]{img/color-scale/5_singe}}
 195     \centerline{(1) Difference rate of 5\%}
 196 \end{minipage}
 197 \hfill
 198 \begin{minipage}[b]{0.32\linewidth}
 199   \centering
 200     \centerline{\includegraphics[width=2.9cm]{img/color-scale/10_singe}}
 201     \centerline{(2) Difference rate of 10\%}
 202 \end{minipage}
 203 \begin{minipage}[b]{.32\linewidth}
 204   \centering
 205     \centerline{\includegraphics[width=2.9cm]{img/color-scale/15_singe}}
 206     \centerline{(3) Difference rate of 15\%}
 207 \end{minipage}
 208 \hfill
 209 \begin{minipage}[b]{0.32\linewidth}
 210   \centering
 211     \centerline{\includegraphics[width=2.9cm]{img/color-scale/20_singe}}
 212     \centerline{(4) Difference rate of 20\%}
 213 \end{minipage}
 214 \hfill
 215 \begin{minipage}[b]{0.32\linewidth}
 216   \centering
 217     \centerline{\includegraphics[width=2.9cm]{img/color-scale/25_singe}}
 218     \centerline{(5) Difference rate of 25\%}
 219 \end{minipage}
 220 \begin{minipage}[b]{.32\linewidth}
 221   \centering
 222     \centerline{\includegraphics[width=2.9cm]{img/color-scale/30_singe}}
 223     \centerline{(6) Difference rate of 30\%}
 224 \end{minipage}
 225 \hfill
 226 \begin{minipage}[b]{0.32\linewidth}
 227   \centering
 228     \centerline{\includegraphics[width=2.9cm]{img/color-scale/35_singe}}
 229     \centerline{(7) Difference rate of 35\%}
 230 \end{minipage}
 231 \hfill
 232 \begin{minipage}[b]{0.32\linewidth}
 233   \centering
 234     \centerline{\includegraphics[width=2.9cm]{img/color-scale/40_singe}}
 235     \centerline{(8) Difference rate of 40\%}
 236 \end{minipage}
 237 \begin{minipage}[b]{.32\linewidth}
 238   \centering
 239     \centerline{\includegraphics[width=2.9cm]{img/color-scale/45_singe}}
 240     \centerline{(9) Difference rate of 45\%}
 241 \end{minipage}
 242 \hfill
 243 \begin{minipage}[b]{0.32\linewidth}
 244   \centering
 245     \centerline{\includegraphics[width=2.9cm]{img/color-scale/50_singe}}
 246     \centerline{(10) Difference rate of 50\%}
 247 \end{minipage}
 248 \hfill
 249 \begin{minipage}[b]{0.32\linewidth}
 250   \centering
 251     \centerline{\includegraphics[width=2.9cm]{img/color-scale/55_singe}}
 252     \centerline{(11) Difference rate of 55\%}
 253 \end{minipage}
 254 \begin{minipage}[b]{.32\linewidth}
 255   \centering
 256     \centerline{\includegraphics[width=2.9cm]{img/color-scale/60_singe}}
 257     \centerline{(12) Difference rate of 60\%}
 258 \end{minipage}
 259 \hfill
 260 \begin{minipage}[b]{0.32\linewidth}
 261   \centering
 262     \centerline{\includegraphics[width=2.9cm]{img/color-scale/65_singe}}
 263     \centerline{(13) Difference rate of 65\%}
 264 \end{minipage}
 265 \hfill
 266 \begin{minipage}[b]{0.32\linewidth}
 267   \centering
 268     \centerline{\includegraphics[width=2.9cm]{img/color-scale/70_singe}}
 269     \centerline{(14) Difference rate of 70\%}
 270 \end{minipage}
 271 \begin{minipage}[b]{.32\linewidth}
 272   \centering
 273     \centerline{\includegraphics[width=2.9cm]{img/color-scale/75_singe}}
 274     \centerline{(15) Difference rate of 75\%}
 275 \end{minipage}
 276 \hfill
 277 \begin{minipage}[b]{0.32\linewidth}
 278   \centering
 279     \centerline{\includegraphics[width=2.9cm]{img/color-scale/80_singe}}
 280     \centerline{(16) Difference rate of 80\%}
 281 \end{minipage}
 282 \hfill
 283 \begin{minipage}[b]{0.32\linewidth}
 284   \centering
 285     \centerline{\includegraphics[width=2.9cm]{img/color-scale/85_singe}}
 286     \centerline{(17) Difference rate of 85\%}
 287 \end{minipage}
 288 \begin{minipage}[b]{.32\linewidth}
 289   \centering
 290     \centerline{\includegraphics[width=2.9cm]{img/color-scale/90_singe}}
 291     \centerline{(18) Difference rate of 90\%}
 292 \end{minipage}
 293 \hfill
 294 \begin{minipage}[b]{0.32\linewidth}
 295   \centering
 296     \centerline{\includegraphics[width=2.9cm]{img/color-scale/95_singe}}
 297     \centerline{(19) Difference rate of 95\%}
 298 \end{minipage}
 299 \hfill
 300 \begin{minipage}[b]{0.32\linewidth}
 301   \centering
 302     \centerline{\includegraphics[width=2.9cm]{img/color-scale/100_singe}}
 303     \centerline{(20) Difference rate of 100\%}
 304 \end{minipage}
 305 \caption{Visual scale in colour domain for robustness threshold evaluation.}
 306 \label{fig:visual-scale-colour}
 307 \end{center}
 308 \end{figure*}
 309
 310 \subsubsection{\textbf{Scale in white and black domain}}
 311
 312 The scale in white and black domain is presented in
 313 Figure~\ref{fig:visual-scale-white-black}~\vpageref{fig:visual-scale-white-black}.
 314
 315 Looking the visual scale, it can be deduced that the in the  white and black domain, a
 316 watermarking process is robust for a difference rate between 0\% and 20\%, and
 317 between 80\% and 100\% for the negative image case.
 318
 319 \textbf{So in this field, the robustness threshold can be fixed to 20\%.}
 320
 321 \begin{figure*}[htb]
 322 \begin{center}
 323 \begin{minipage}[b]{.32\linewidth}
 324   \centering
 325     \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/0_invader}}
 326     \centerline{(0) Difference rate of 0\%}
 327 \end{minipage}
 328 \hfill
 329 \begin{minipage}[b]{0.32\linewidth}
 330   \centering
 331     \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/5_invader}}
 332     \centerline{(1) Difference rate of 5\%}
 333 \end{minipage}
 334 \hfill
 335 \begin{minipage}[b]{0.32\linewidth}
 336   \centering
 337     \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/10_invader}}
 338     \centerline{(2) Difference rate of 10\%}
 339 \end{minipage}
 340 \begin{minipage}[b]{.32\linewidth}
 341   \centering
 342     \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/15_invader}}
 343     \centerline{(3) Difference rate of 15\%}
 344 \end{minipage}
 345 \hfill
 346 \begin{minipage}[b]{0.32\linewidth}
 347   \centering
 348     \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/20_invader}}
 349     \centerline{(4) Difference rate of 20\%}
 350 \end{minipage}
 351 \hfill
 352 \begin{minipage}[b]{0.32\linewidth}
 353   \centering
 354     \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/25_invader}}
 355     \centerline{(5) Difference rate of 25\%}
 356 \end{minipage}
 357 \begin{minipage}[b]{.32\linewidth}
 358   \centering
 359     \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/30_invader}}
 360     \centerline{(6) Difference rate of 30\%}
 361 \end{minipage}
 362 \hfill
 363 \begin{minipage}[b]{0.32\linewidth}
 364   \centering
 365     \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/35_invader}}
 366     \centerline{(7) Difference rate of 35\%}
 367 \end{minipage}
 368 \hfill
 369 \begin{minipage}[b]{0.32\linewidth}
 370   \centering
 371     \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/40_invader}}
 372     \centerline{(8) Difference rate of 40\%}
 373 \end{minipage}
 374 \begin{minipage}[b]{.32\linewidth}
 375   \centering
 376     \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/45_invader}}
 377     \centerline{(9) Difference rate of 45\%}
 378 \end{minipage}
 379 \hfill
 380 \begin{minipage}[b]{0.32\linewidth}
 381   \centering
 382     \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/50_invader}}
 383     \centerline{(10) Difference rate of 50\%}
 384 \end{minipage}
 385 \hfill
 386 \begin{minipage}[b]{0.32\linewidth}
 387   \centering
 388     \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/55_invader}}
 389     \centerline{(11) Difference rate of 55\%}
 390 \end{minipage}
 391 \begin{minipage}[b]{.32\linewidth}
 392   \centering
 393     \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/60_invader}}
 394     \centerline{(12) Difference rate of 60\%}
 395 \end{minipage}
 396 \hfill
 397 \begin{minipage}[b]{0.32\linewidth}
 398   \centering
 399     \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/65_invader}}
 400     \centerline{(13) Difference rate of 65\%}
 401 \end{minipage}
 402 \hfill
 403 \begin{minipage}[b]{0.32\linewidth}
 404   \centering
 405     \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/70_invader}}
 406     \centerline{(14) Difference rate of 70\%}
 407 \end{minipage}
 408 \begin{minipage}[b]{.32\linewidth}
 409   \centering
 410     \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/75_invader}}
 411     \centerline{(15) Difference rate of 75\%}
 412 \end{minipage}
 413 \hfill
 414 \begin{minipage}[b]{0.32\linewidth}
 415   \centering
 416     \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/80_invader}}
 417     \centerline{(16) Difference rate of 80\%}
 418 \end{minipage}
 419 \hfill
 420 \begin{minipage}[b]{0.32\linewidth}
 421   \centering
 422     \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/85_invader}}
 423     \centerline{(17) Difference rate of 85\%}
 424 \end{minipage}
 425 \begin{minipage}[b]{.32\linewidth}
 426   \centering
 427     \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/90_invader}}
 428     \centerline{(18) Difference rate of 90\%}
 429 \end{minipage}
 430 \hfill
 431 \begin{minipage}[b]{0.32\linewidth}
 432   \centering
 433     \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/95_invader}}
 434     \centerline{(19) Difference rate of 95\%}
 435 \end{minipage}
 436 \hfill
 437 \begin{minipage}[b]{0.32\linewidth}
 438   \centering
 439     \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/100_invader}}
 440     \centerline{(20) Difference rate of 100\%}
 441 \end{minipage}
 442 \caption{Visual scale in white and black domain for robustness threshold
 443 evaluation.}
 444 \label{fig:visual-scale-white-black}
 445 \end{center}
 446 \end{figure*}
 447
 448 \subsubsection{\textbf{Conclusion concerning the robustness threshold}}
 449
 450 \textbf{From the results of the previous study, it can be concluded that the
 451 robustness threshold depends of the nature of the watermark.}\newline
 452
 453 \textbf{So for a picture as watermark, the highest robustness threshold is
 454 obtain with color images.}
 455
 456 \subsection{Presentation of the
 457 architecture}\label{section:architecture-presentation}
 458
 459 Computations have been
 460 performed on the supercomputer facilities of the Mésocentre de calcul de Franche-Comt\'{e}.
 461
 462 In order to take benefits of parallelism, we have used Jace~\cite{bhm09:ip}, a
 463 grid-enabled programming and execution environment allowing a simple and efficient implementation
 464 of parallel and distributed applications. Roughly speaking, Jace builds a virtual parallel machine by
 465 connecting a set of heterogeneous and distant computers.
 466 It schedules tasks, executes them, and returns results to the user. It
 467 also proposes a simple programming interface for the implementation of applications
 468 using a message passing model.
 469
 470 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.
 471
 472 For these experiments we have used HPC resources
 473 from Mesocentre of Franche-Comt\'{e}. This platform
 474 is currently composed of 720 cores with 11 TeraFLOPS of power.
 475
 476 The different tests have been realized on several images from the Wikimedia
 477 Commons repository~\cite{wiki:wikimedia-commons}.
 478
 479
 480 \textbf{FAIRE AUSSI DES TESTS AVEC LA BASE DE BOSS}
 481 The conditions of all tests are described for each one in the caption of the graph result.
 482
 483 %je ne sais pas comment expliquer vos configs !
 484 %In the following experiments, we have chosen different configuration files,
 485 %about 1000 images from wikimedia.....
 486
 487
 488
 489 \subsection{Tests realized}\label{sec:tests-realized-ci-1}
 490
 491 In this section, it is studied the robustness of the watermarking process
 492 $CI_1$. To do this, it has been executed two test batteries in order to evaluate
 493 the evolution of the bytes difference rate in function of the geometrical
 494 attacks parameters, between the watermark inserted and the watermark extracted
 495 after the attack.
 496
 497 In the first battery of tests we have used the same embedding key (a constant
 498 strategy,the same strategy has been used to watermark all the
 499 images in this test) to embed the watermark. In the second battery of tests,
 500 in order to see the influence of the embedding key on the robustness, we have embedded all
 501 the watermark with strategies generated randomly(a
 502 different strategy has been used to watermark each image).
 503
 504 To obtain the following results, we have realized a great number of attacks on
 505 600 images, and we have traced for each attack, two graphics (average and
 506 standard deviation), in order to determine the acceptability
 507 threshold for our watermarking process $CI_1$.
 508
 509 \subsection{Tests results}\label{sec:tests-results-ci-1}
 510
 511 The exhaustive list of the geometrical attacks tested and the
 512 corresponding graphs results is following:
 513
 514 \begin{enumerate}
 515   \item \textbf{Robustness facing a resizing attack.}\\
 516   See Figure~\ref{fig:resizing-attack-constant-strategy}~\vpageref{fig:resizing-attack-constant-strategy}.
 517   \item \textbf{Robustness facing a JPEG attack.}\\See
 518   Figure~\ref{fig:jpeg-attack-constant-strategy}~\vpageref{fig:jpeg-attack-constant-strategy}.
 519   \item \textbf{Robustness facing a Gaussian blur
 520   attack.}\\See
 521   Figure~\ref{fig:gaussian-blur-attack-constant-strategy}~\vpageref{fig:gaussian-blur-attack-constant-strategy}.
 522   \item \textbf{Robustness facing a rotation
 523   attack.}\\See
 524   Figure~\ref{fig:rotation-attack-constant-strategy}~\vpageref{fig:rotation-attack-constant-strategy}.
 525   \item \textbf{Robustness facing a  blur
 526   attack.}\\See
 527   Figure~\ref{fig:blur-attack-constant-strategy}~\vpageref{fig:blur-attack-constant-strategy}.
 528   \item \textbf{Robustness facing a contrast
 529   attack.}\\See
 530   Figure~\ref{fig:contrast-attack-constant-strategy}~\vpageref{fig:contrast-attack-constant-strategy}.
 531   \item \textbf{Robustness facing a cropping
 532   attack.}\\See
 533   Figure~\ref{fig:cropping-attack-constant-strategy}~\vpageref{fig:cropping-attack-constant-strategy}.
 534 \end{enumerate}
 535
 536
 537 % \begin{figure*}
 538 % \begin{center}
 539 % %\includegraphics[width=10.5cm]{./img/prisoner-problem.png}
 540 % \includegraphics[width=15cm]{./img/prisoner-problem}
 541 % \end{center}
 542 %
 543 % \caption{Simmons' prisoner problem~\cite{Simmons83}}
 544 % \end{figure*}
 545
 546
 547 \begin{figure*}[htb]
 548 \begin{minipage}[b]{.45\linewidth}
 549   \centering
 550     \centerline{\includegraphics[width=9cm]{graphs/graph_1_attack=redimensionnement-average}}
 551     \centerline{(a) Average}
 552 \end{minipage}
 553 \hfill
 554 \begin{minipage}[b]{0.45\linewidth}
 555   \centering
 556     \centerline{\includegraphics[width=9cm]{graphs/graph_1_attack=redimensionnement-sd}}
 557     \centerline{(b) Standard deviation}
 558 \end{minipage}
 559 \caption{Robustness of $CI_1$ facing a resizing attack.(600 images and constant strategy)}
 560 \label{fig:resizing-attack-constant-strategy}
 561 \end{figure*}
 562
 563 \begin{figure*}[htb]
 564 \begin{minipage}[b]{.45\linewidth}
 565   \centering
 566     \centerline{\includegraphics[width=9cm]{graphs/graph_2_attack=jpeg-average}}
 567     \centerline{(a) Average}
 568 \end{minipage}
 569 \hfill
 570 \begin{minipage}[b]{0.45\linewidth}
 571   \centering
 572     \centerline{\includegraphics[width=9cm]{graphs/graph_2_attack=jpeg-sd}}
 573     \centerline{(b) Standard deviation}
 574 \end{minipage}
 575 \caption{Robustness of $CI_1$ facing a JPEG compression attack.(600 images and
 576 constant strategy)}
 577 \label{fig:jpeg-attack-constant-strategy}
 578 \end{figure*}
 579
 580
 581 \begin{figure*}[htb]
 582 \begin{minipage}[b]{.45\linewidth}
 583   \centering
 584     \centerline{\includegraphics[width=9cm]{graphs/graph_3_attack=gaussien-average}}
 585     \centerline{(a) Average}
 586 \end{minipage}
 587 \hfill
 588 \begin{minipage}[b]{0.45\linewidth}
 589   \centering
 590     \centerline{\includegraphics[width=9cm]{graphs/graph_3_attack=gaussien-sd}}
 591     \centerline{(b) Standard deviation}
 592 \end{minipage}
 593 \caption{Robustness of $CI_1$ facing a Gaussian blur attack.(600 images and
 594 constant strategy)}
 595 \label{fig:gaussian-blur-attack-constant-strategy}
 596 \end{figure*}
 597
 598 \begin{figure*}[htb]
 599 \begin{minipage}[b]{.45\linewidth}
 600   \centering
 601     \centerline{\includegraphics[width=9cm]{graphs/graph_4_attack=rotation-average}}
 602     \centerline{(a) Average}
 603 \end{minipage}
 604 \hfill
 605 \begin{minipage}[b]{0.45\linewidth}
 606   \centering
 607     \centerline{\includegraphics[width=9cm]{graphs/graph_4_attack=rotation-sd}}
 608     \centerline{(b) Standard deviation}
 609 \end{minipage}
 610 \caption{Robustness of $CI_1$ facing a rotation attack.(600 images and constant
 611 strategy)}
 612 \label{fig:rotation-attack-constant-strategy}
 613 \end{figure*}
 614
 615
 616 \begin{figure*}[htb]
 617 \begin{minipage}[b]{.45\linewidth}
 618   \centering
 619     \centerline{\includegraphics[width=9cm]{graphs/graph_5_attack=flou-average}}
 620     \centerline{(a) Average}
 621 \end{minipage}
 622 \hfill
 623 \begin{minipage}[b]{0.45\linewidth}
 624   \centering
 625     \centerline{\includegraphics[width=9cm]{graphs/graph_5_attack=flou-sd}}
 626     \centerline{(b) Standard deviation}
 627 \end{minipage}
 628 \caption{Robustness of $CI_1$ facing a blur attack.(600 images and constant
 629 strategy)}
 630 \label{fig:blur-attack-constant-strategy}
 631 \end{figure*}
 632
 633
 634 \begin{figure*}[htb]
 635 \begin{minipage}[b]{.45\linewidth}
 636   \centering
 637     \centerline{\includegraphics[width=9cm]{graphs/graph_6_attack=contraste-average}}
 638     \centerline{(a) Average}
 639 \end{minipage}
 640 \hfill
 641 \begin{minipage}[b]{0.45\linewidth}
 642   \centering
 643     \centerline{\includegraphics[width=9cm]{graphs/graph_6_attack=contraste-sd}}
 644     \centerline{(b) Standard deviation}
 645 \end{minipage}
 646 \caption{Robustness of $CI_1$ facing a contrast attack.(600 images and constant
 647 strategy)}
 648 \label{fig:contrast-attack-constant-strategy}
 649 \end{figure*}
 650
 651
 652 \begin{figure*}[htb]
 653 \begin{minipage}[b]{.45\linewidth}
 654   \centering
 655     \centerline{\includegraphics[width=9cm]{graphs/graph_7_attack=decoupage-average}}
 656     \centerline{(a) Average}
 657 \end{minipage}
 658 \hfill
 659 \begin{minipage}[b]{0.45\linewidth}
 660   \centering
 661     \centerline{\includegraphics[width=9cm]{graphs/graph_7_attack=decoupage-sd}}
 662     \centerline{(b) Standard deviation}
 663 \end{minipage}
 664 \caption{Robustness of $CI_1$ facing a cropping attack.(600 images and constant
 665 strategy)}
 666 \label{fig:cropping-attack-constant-strategy}
 667 \end{figure*}
 668
 669 \subsection{Robustness evaluation}\label{sec:tests-results-ci-1}
 670
 671 From the results presented in the previous section, the robustness of the
 672 process $CI_1$ can be evaluated.
 673
 674
 675 As a first conclusion, it can be note that in the $CI_1$ watermarking process, the strategy has no
 676 influence on the robustness. Indeed, analyzing all the curves, a strong
 677 similarity between the curves obtained with a constant strategy and the curves obtained with a strategy generated randomly can
 678 be highlight.
 679
 680 As a second conclusion, it can be note that the $CI_1$ watermarking process is
 681 not robust in spatial domain facing attacks of resizing, of JPEG compression
 682 with a compression rate lesser than 97\%, of Gaussian blur with a blur rate
 683 greater than 42\%, of classic blur, of contrast, and of cropping with a number
 684 of pixels greater than 180.
 685
 686 Inversely, $CI_1$ is robust in spatial domain facing geometrical attacks of JPEG
 687 compression with a compression rate greater than 97\%, of Gaussian blur with a blur rate
 688 lesser than 42\%, of rotation with angles between 0° and 30°, and of cropping
 689 with a number of pixels lesser than 180.
 690
 691
 692 \begin{enumerate}
 693   \item \textbf{Robustness facing a resizing attack.}\\
 694   See Figure~\ref{fig:resizing-attack-constant-strategy}~\vpageref{fig:resizing-attack-constant-strategy}.
 695   \item \textbf{Robustness facing a JPEG compression
 696   attack.}\\See
 697   Figure~\ref{fig:jpeg-attack-constant-strategy}~\vpageref{fig:jpeg-attack-constant-strategy}.
 698   \item \textbf{Robustness facing a Gaussian blur
 699   attack.}\\See
 700   Figure~\ref{fig:gaussian-blur-attack-constant-strategy}~\vpageref{fig:gaussian-blur-attack-constant-strategy}.
 701   \item \textbf{Robustness facing a rotation
 702   attack.}\\See
 703   Figure~\ref{fig:rotation-attack-constant-strategy}~\vpageref{fig:rotation-attack-constant-strategy}.
 704   \item \textbf{Robustness facing a  blur
 705   attack.}\\See
 706   Figure~\ref{fig:blur-attack-constant-strategy}~\vpageref{fig:blur-attack-constant-strategy}.
 707   \item \textbf{Robustness facing a contrast
 708   attack.}\\See
 709   Figure~\ref{fig:contrast-attack-constant-strategy}~\vpageref{fig:contrast-attack-constant-strategy}.
 710   \item \textbf{Robustness facing a cropping
 711   attack.}\\See
 712   Figure~\ref{fig:cropping-attack-constant-strategy}~\vpageref{fig:cropping-attack-constant-strategy}.
 713 \end{enumerate}
 714
 715 ensuite il faut reprendre chaque courbe et énoncer une phrase du genre :
 716 " on voit qu'on est loin d'une génération aléatoire des bits qui serait environ de 50%"
 717 et donc on peut affirmer avec certitude que telle ou telle image est tatouée ou non.
 718
 719
 720
 721
 722 To conclude, at this point, only two information hiding schemes are both stego-secure and chaos-secure \cite{gfb10:ip}.
 723 The first one is based on a spread spectrum technique called Natural Watermarking.
 724 It is stego-secure when its parameter $\eta$ is equal to $1$ \cite{Cayre2008}.
 725 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}.
 726 The second scheme both chaos-secure and stego-secure is based on chaotic iterations \cite{guyeux10ter}.
 727 However, its first versions called $CI_1$ allows to embed securely only one bit per embedding parameters.
 728 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.
 729