quelques modifs

[canny.git] / complexity.tex

diff --git a/complexity.tex b/complexity.tex
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+++ b/complexity.tex
@@ -2,15 +2,16 @@
 This section aims at justifying the lightweight attribute of our approach.
 To be more precise, we compare the complexity of our schemes to some of 
 current state of the art of 
 This section aims at justifying the lightweight attribute of our approach.
 To be more precise, we compare the complexity of our schemes to some of 
 current state of the art of 

-steganographic scheme, namely HUGO~\cite{DBLP:conf/ih/PevnyFB10},
+steganographic schemes, namely HUGO~\cite{DBLP:conf/ih/PevnyFB10},

 WOW~\cite{conf/wifs/HolubF12}, and UNIWARD~\cite{HFD14}.
 WOW~\cite{conf/wifs/HolubF12}, and UNIWARD~\cite{HFD14}.

-Each of these scheme starts with the computation of the distortion cost 
+Each of these schemes starts with the computation of the distortion cost 

 for each pixel switch and is later followed by the STC algorithm.
 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.
-In all the rest of this section, we consider a $n \times n$ square image. 
+Since this last step is shared by all, 
+we separately evaluate this complexity.
+In all the remainder of this section, we consider a $n \times n$ square image. 

 
 First of all, HUGO starts with computing the second order SPAM Features.
 
 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 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 
 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 +19,47 @@ 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,
 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 $O(2\times 343^2 \times n^2)$.
-Ranking these results may be achieved with an insertion sort, which is in 
-$2.n^2 \ln(n)$.
+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
 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 
 
 
 
 
 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.
 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). 
 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 
 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 
 
 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 $O(5^3 n^2)$.
-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 $O((5^3+4T+1)n^2)$.
+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 $\theta((5^3+4T+1)n^2)$.

 
 
 
 
 
 


@@ -65,18 +67,25 @@ 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 
 
 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.
 
 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 $O(n^4\ln(n))$.
-Thanks to these complexity results, we claim that STABYLO is lightweight. 
-
-
-
+The Fig.~\ref{fig:compared}
+summarizes the complexity of the embedding map construction, for
+WOW/UNIWARD, HUGO, and STABYLO. 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 the algorithms
+is dramatically larger than the one of the STABYLO scheme. 
+Thanks to these complexity results, we claim that our approach 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}