From b00f4bba43bbc502f81d237292f4757d9fd548cc Mon Sep 17 00:00:00 2001 From: couchot Date: Thu, 24 Sep 2015 14:40:44 +0200 Subject: [PATCH 1/1] ajout de fichiers --- annexePreuvesComplexiteStego.tex | 56 + images/complexity.eps | 4097 ++++++++++++++++++++++++++++++ images/dec.eps | 662 +++++ images/emb.eps | 898 +++++++ 4 files changed, 5713 insertions(+) create mode 100644 annexePreuvesComplexiteStego.tex create mode 100644 images/complexity.eps create mode 100644 images/dec.eps create mode 100644 images/emb.eps diff --git a/annexePreuvesComplexiteStego.tex b/annexePreuvesComplexiteStego.tex new file mode 100644 index 0000000..b8f9704 --- /dev/null +++ b/annexePreuvesComplexiteStego.tex @@ -0,0 +1,56 @@ +\begin{theorem}\label{th:cplxt:hugo} +Le schéma HUGO a une complexité de l'ordre de +$\theta(2 \times n^2(343^2 + \ln(n)))$ +\end{theorem} + + +First of all, HUGO starts with computing the second order SPAM Features. +This steps is in $\theta(n^2 + 2\times 343^2)$ due to the computation +of the difference arrays and next of the 686 features (of size 343). +Next for each pixel, the distortion measure is calculated by +1/-1 modifying +its value and computing again the SPAM +features. Pixels are thus selected according to their ability to provide +an image whose SPAM features are close to the original ones. +The algorithm thus computes a distance between each feature +and the original ones, +which is at least in $\theta(343)$, and an overall distance between these +metrics, which is in $\theta(686)$. Computing the distance is thus in +$\theta(2\times 343^2)$ and this modification +is thus in $\theta(2\times 343^2 \times n^2)$. +Ranking these results may be achieved with a quick sort, which is in +$\theta(2 \times n^2 \ln(n))$ for data of size $n^2$. +The overall complexity of the pixel selection is finally +$\theta(n^2 +2 \times 343^2 + 2\times 343^2 \times n^2 + 2 \times n^2 \ln(n))$, \textit{i.e}, +$\theta(2 \times n^2(343^2 + \ln(n)))$. + + + + +Let us focus now on WOW. +This scheme starts to compute the residual +of the cover as a convolution product which is in $\theta(n^2\ln(n^2))$. +The embedding suitability $\eta_{ij}$ is then computed for each pixel +$1 \le i,j \le n$ thanks to a convolution product again. +We thus have a complexity of $\theta(n^2 \times n^2\ln(n^2))$. +Moreover the suitability is computed for each wavelet level +detail (HH, HL, LL). +This distortion computation step is thus in $\theta(6n^4\ln(n))$. +Finally a norm of these three values is computed for each pixel +which adds to this complexity the complexity of $\theta(n^2)$. +To summarize, the complixity is in $\theta(6n^4\ln(n) +n^2)$ + +What follows details the complexity of the distortion evaluation of the +UNIWARD scheme. This one is based to a convolution product $W$ of two elements +of size $n$ and is again in $\theta(n^2 \times n^2\ln(n^2))$, + and a sum $D$ of +these $W$ which is in $\theta(n^2)$. +This distortion computation step is thus in $\theta(6n^4\ln(n) + n^2)$. + + +Our edge selection is based on a Canny filter. When applied on a +$n \times n$ square image, the noise reduction step is in $\theta(5^3 n^2)$. +Next, let $T$ be the size of the Canny mask. +Computing gradients is in $\theta(4Tn^2)$ since derivatives of each direction (vertical or horizontal) +are in $\theta(2Tn^2)$. +Finally, thresholding with hysteresis is in $\theta(n^2)$. +The overall complexity is thus in $\theta((5^3+4T+1)n^2)$. diff --git a/images/complexity.eps b/images/complexity.eps new file mode 100644 index 0000000..8eb76d2 --- /dev/null +++ b/images/complexity.eps @@ -0,0 +1,4097 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Title: complexity.eps +%%Creator: matplotlib version 1.3.1, http://matplotlib.org/ +%%CreationDate: Wed Oct 29 15:58:05 2014 +%%Orientation: portrait +%%BoundingBox: 18 180 594 612 +%%EndComments +%%BeginProlog +/mpldict 8 dict def +mpldict begin +/m { moveto } bind def +/l { lineto } bind def +/r { rlineto } bind def +/c { curveto } bind def +/cl { closepath } bind def +/box { +m +1 index 0 r +0 exch r +neg 0 r +cl +} bind def +/clipbox { +box +clip +newpath +} bind def +%!PS-Adobe-3.0 Resource-Font +%%Title: Arial +%%Copyright: Typeface © The Monotype Corporation plc. 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