From b04e55fbbcdcbca04098f1bf05ec4473e47b5c51 Mon Sep 17 00:00:00 2001
From: couchot <jf.couchot@gmail.com>
Date: Wed, 29 Oct 2014 16:23:02 +0100
Subject: [PATCH 1/1] =?utf8?q?retouche=20complexit=C3=A9?=
MIME-Version: 1.0
Content-Type: text/plain; charset=utf8
Content-Transfer-Encoding: 8bit

---
 complexity.eps                      | 4097 +++++++++++++++++++++++++++
 complexity.tex                      |   68 +-
 experiments.tex                     |   19 +-
 ourapproach.tex                     |    4 +-
 stc/exp/python/complexityDisplay.py |   37 +
 5 files changed, 4184 insertions(+), 41 deletions(-)
 create mode 100644 complexity.eps
 create mode 100644 stc/exp/python/complexityDisplay.py

diff --git a/complexity.eps b/complexity.eps
new file mode 100644
index 0000000..8eb76d2
--- /dev/null
+++ b/complexity.eps
@@ -0,0 +1,4097 @@
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diff --git a/complexity.tex b/complexity.tex
index 4499857..d19f6e2 100644
--- a/complexity.tex
+++ b/complexity.tex
@@ -6,11 +6,12 @@ steganographic scheme, namely HUGO~\cite{DBLP:conf/ih/PevnyFB10},
 WOW~\cite{conf/wifs/HolubF12}, and UNIWARD~\cite{HFD14}.
 Each of these scheme starts with the computation of the distortion cost 
 for each pixel switch and is later followed by the STC algorithm.
-Since this last step is shared by all, we do not add its complexity.
+Since this last step is shared by all, 
+we separately eavluate this complexity.
 In all the rest of this section, we consider a $n \times n$ square image. 
 
 First of all, HUGO starts with computing the second order SPAM Features.
-This steps is in  $O(n^2 + 2\times 343^2)$ due to the calculation 
+This steps is in  $\theta(n^2 + 2\times 343^2)$ due to the calculation 
 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 
@@ -18,46 +19,46 @@ 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 $O(343)$, and an overall distance between these 
-metrics, which is in $O(686)$. Computing the distance is thus in 
-$O(2\times 343^2)$ and this modification
-is thus in $O(2\times 343^2 \times n^2)$.
+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 an insertion sort, which is in 
-$2.n^2 \ln(n)$.
+$2 \times n^2 \ln(n)$.
 The overall complexity of the pixel selection is finally
-$O(n^2 +2.343^2 + 2\times 343^2 \times n^2 + 2.n^2 \ln(n))$, \textit{i.e},
-$O(2.n^2(343^2 + \ln(n)))$.
+$\theta(n^2 +2 \times 343^2 + 2\times 343^2 \times n^2 + 2 \times n^2 \ln(n))$, \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 $O(n^2\ln(n^2))$. 
+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 $O(n^2 \times n^2\ln(n^2))$.
+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 $O(6n^4\ln(n))$.
+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 $O(n^2)$.
-To summarize, the complixity is in $O(6n^4\ln(n) +n^2)$
+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   $O(n^2 \times n^2\ln(n^2))$ and a sum $D$ of 
-these $W$ which is in $O(n^2)$. 
-This distortion computation step is thus in $O(6n^4\ln(n) + n^2)$.
+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 $O(5^3 n^2)$.
+$n \times n$ 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 $O(4Tn)$ since derivatives of each direction (vertical or horizontal) 
-are in $O(2Tn)$.
-Finally, thresholding with hysteresis is in $O(n^2)$.
-The overall complexity is thus in $O((5^3+4T+1)n^2)$.
+Computing gradients is in $\theta(4Tn)$ since derivatives of each direction (vertical or horizontal) 
+are in $\theta(2Tn)$.
+Finally, thresholding with hysteresis is in $\theta(n^2)$.
+The overall complexity is thus in $\theta((5^3+4T+1)n^2)$.
 
 
 
@@ -65,18 +66,25 @@ The overall complexity is thus in $O((5^3+4T+1)n^2)$.
 
 We are then left to express the complexity of the STC algorithm.
 According to~\cite{DBLP:journals/tifs/FillerJF11}, it  is 
-in $O(2^h.n)$ where $h$ is the size of the duplicated 
+in $\theta(2^h.n)$ where $h$ is the size of the duplicated 
 matrix. Its complexity is thus negligible compared with the embedding map
 construction.
 
-To summarize, for the embedding map construction, the complexity of Hugo, WOW 
-and UNIWARD are dramatically larger than the one of our scheme: 
-STABYLO is in $O(n^2)$ 
-whereas HUGO is in $O(n^2\ln(n)$, and WOW and UNIWARD are in $O(n^4\ln(n))$.
+The Fig.~\ref{fig:compared}
+summarizes the complexity of the embedding map construction, for Hugo, Wow 
+and Uniward. It deals with square images
+of size $n \times n$ when $n$ ranges from 
+512 to 4096. The $y$-coordinate is expressed in a logarithm scale.
+It shows that the complexity of all algorithms
+is dramatically larger than the one of the STABYLO scheme. 
 Thanks to these complexity results, we claim that STABYLO is lightweight. 
-
-
-
+\begin{figure}
+\begin{center}
+\includegraphics[scale=0.4]{complexity}
+\end{center}
+\caption{Complexity evaluation of Wow/Uniward, Hugo, and Stabylo}
+\label{fig:compared} 
+\end{figure}
 
 
 
diff --git a/experiments.tex b/experiments.tex
index a92c72f..6660072 100644
--- a/experiments.tex
+++ b/experiments.tex
@@ -85,24 +85,25 @@ If $b$ is 6, these values are respectively equal to
 
 \begin{table*}
 \begin{center}
-\begin{tabular}{|c|c|c||c|c|c|c|c|c|}
+\begin{small}
+\begin{tabular}{|c|c|c||c|c|c|c|c|c|c|c|c|c|}
 \hline
-Schemes & \multicolumn{4}{|c|}{STABYLO} & \multicolumn{2}{|c|}{HUGO}& \multicolumn{2}{|c|}{EAISLSBMR} \\
+Schemes & \multicolumn{4}{|c|}{STABYLO} & \multicolumn{2}{|c|}{HUGO}& \multicolumn{2}{|c|}{EAISLSBMR} & \multicolumn{2}{|c|}{WOW} & \multicolumn{2}{|c|}{UNIWARD}\\
 \hline
-Embedding &   Fixed & \multicolumn{3}{|c|}{Adaptive (about 6.35\%)} &  \multicolumn{2}{|c|}{Fixed}& \multicolumn{2}{|c|}{Fixed} \\
+Embedding &   Fixed & \multicolumn{3}{|c|}{Adaptive (about 6.35\%)} &  Fixed &Adaptive & Fixed &Adaptive & Fixed &Adaptive & Fixed &Adaptive \\
 \hline
-Rate &   10\% &  + sample &  +STC(7) & +STC(6) &  10\%&6.35\%& 10\%&6.35\%\\ 
+Rate &   10\% &  + sample &  +STC(7) & +STC(6) &  10\%&6.35\%& 10\%&6.35\%& 10\%&6.35\%& 10\%&6.35\%\\ 
 \hline
-PSNR & 61.86 & 63.48 &  66.55 (\textbf{-0.8\%}) &  63.7  & 64.65 & {67.08} & 60.8 & 62.9\\ 
+PSNR & 61.86 & 63.48 &  66.55 (\textbf{-0.8\%}) &  63.7  & 64.65 & {67.08} & 60.8 & 62.9&65.9 & 68.3 & 65.8 & 69.2\\ 
 \hline
 PSNR-HVS-M & 72.9 & 75.39 & 78.6 (\textbf{-0.8\%}) & 75.5  & 76.67 & {79.23} & 71.8  & 74.3\\ 
 %\hline
 %BIQI & 28.3 & 28.28 & 28.4 & 28.28 & 28.28 & 28.2 & 28.2\\ 
 \hline
-wPSNR & 77.47 & 80.59 & 86.43(\textbf{-1.6\%})& 86.28  & 83.03 & {87.8} & 76.7 & 80.6\\ 
+wPSNR & 77.47 & 80.59 & 86.43(\textbf{-1.6\%})& 86.28  & 83.03 & {88.6} & 76.7 & 83& 83.8 & 90.4 & 85.2 & 91.9\\ 
 \hline
 \end{tabular}
-
+\end{small}
 \begin{footnotesize}
 \vspace{2em}
 Variances given in bold font express the quality differences between 
@@ -177,7 +178,7 @@ considered as state of the art steganalysers in the spatial domain~\cite{FK12}.
 
 \begin{table*}
 \begin{center}
-%\begin{small}
+\begin{small}
 \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}
 \hline
 Schemes & \multicolumn{4}{|c|}{STABYLO} & \multicolumn{2}{|c|}{HUGO}& \multicolumn{2}{|c|}{EAISLSBMR} &  \multicolumn{2}{|c|}{WOW} &  \multicolumn{2}{|c|}{UNIWARD}\\
@@ -192,7 +193,7 @@ Ensemble Classifier & 0.35 & 0.44 & 0.47 & 0.47     & 0.48 &  0.49  &  0.43  & 0
 
 \hline
 \end{tabular}
-%\end{small}
+\end{small}
 \end{center}
 \caption{Steganalysing STABYLO\label{table:steganalyse}} 
 \end{table*}
diff --git a/ourapproach.tex b/ourapproach.tex
index 636b8ba..ec73f7d 100644
--- a/ourapproach.tex
+++ b/ourapproach.tex
@@ -21,7 +21,7 @@ Let us first focus on the data embedding.
       \begin{minipage}{0.49\textwidth}
         \begin{center}
           %\includegraphics[width=5cm]{emb.pdf}
-          \includegraphics[scale=0.45]{emb.ps}
+          \includegraphics[scale=0.45]{emb}
         \end{center}
       \end{minipage}
       \label{fig:sch:emb}
@@ -31,7 +31,7 @@ Let us first focus on the data embedding.
       \begin{minipage}{0.49\textwidth}
         \begin{center}
           %\includegraphics[width=5cm]{rec.pdf}
-          \includegraphics[scale=0.45]{rec.ps}
+          \includegraphics[scale=0.45]{rec}
         \end{center}
       \end{minipage}
       \label{fig:sch:ext}
diff --git a/stc/exp/python/complexityDisplay.py b/stc/exp/python/complexityDisplay.py
new file mode 100644
index 0000000..3be2500
--- /dev/null
+++ b/stc/exp/python/complexityDisplay.py
@@ -0,0 +1,37 @@
+import math as m 
+import numpy as np
+import matplotlib.pyplot as plt
+
+
+tb = 512
+rhugo,rwow,runiw,rcanny=[],[],[],[]
+T=3
+
+for k in range(1,7):
+    n = tb*k
+    rhugo +=[2*(n**2)*(343**2+m.log(n))]
+    rwow  +=[6*(n**4)*m.log(n)+n**2]
+    rcanny +=[(n**2)*(5**3+4*T+1)]
+
+
+print rhugo
+print rwow
+print rcanny
+
+
+
+n=np.linspace(512,4096,10)
+phugo, = plt.plot(n,2*pow(n,2)*(343**2+np.log(n)),marker='o',label="L1")  
+pwow,  =  plt.plot(n,6*pow(n,4)*np.log(n)+pow(n,2),marker='v',label="L2")
+pcanny, = plt.plot(n,pow(n,2)*(5**3+4*T+1),marker='x',label="L3")
+plt.yscale('log')
+plt.ylabel('nb. of primary operations')
+plt.xlabel("nb of lines and of rows in each image")
+#plt.legend(handles=[phugo], loc=1)
+#plt.legend([phugo, pwow], ["Hugo", "Wow/Uniward"])
+l1 = plt.legend([pwow,phugo,pcanny], ["Wow/Uniward","Hugo","Stabylo"], loc=0)
+plt.gca().add_artist(l1)
+#plt.gca().add_artist(l2)
+#plt.gca().add_artist(l3)
+plt.savefig("complexity.eps")
+#plt.show()
-- 
2.39.5