ajout complexité WOW et UNIWARD

[canny.git] / complexity.tex

diff --git a/complexity.tex b/complexity.tex
index 304ab329262c10ab37a994f979ddcc8548ab1d6e..44998574f200312e1993b0f0f2402877a7475dbf 100644 (file)
--- a/complexity.tex
+++ b/complexity.tex
@@ -1,10 +1,14 @@
 
 This section aims at justifying the lightweight attribute of our approach.
 
 This section aims at justifying the lightweight attribute of our approach.

-To be more precise, we compare the complexity of our schemes to the 
- best available steganographic scheme, namely HUGO~\cite{DBLP:conf/ih/PevnyFB10}.
+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},
+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.
+In all the rest of this section, we consider a $n \times n$ square image. 

 
 

-
-In what follows, 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 
 of the difference arrays and next of the 686 features (of size 343).
 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 
 of the difference arrays and next of the 686 features (of size 343).


@@ -24,6 +28,29 @@ 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},
 $O(2.n^2(343^2 + \ln(n)))$.
 
 $O(n^2 +2.343^2 + 2\times 343^2 \times n^2 + 2.n^2 \ln(n))$, \textit{i.e},
 $O(2.n^2(343^2 + \ln(n)))$.
 

+
+
+
+Let us focus now on WOW.  
+This scheme starts to compute the residual 
+of the cover as a convolution product which is in $O(n^2\ln(n^2))$. 
+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 $O(n^2 \times n^2\ln(n^2))$.
+Moreover the suitability is computed for each wavelet level
+detail (HH, HL, LL). 
+This distortion computation step is thus in $O(6n^4\ln(n))$.
+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 $O(6n^4\ln(n) +n^2)$
+
+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 $O(6n^4\ln(n) + n^2)$.
+
+

 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.
 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.


@@ -31,8 +58,10 @@ Computing gradients is in $O(4Tn)$ since derivatives of each direction (vertical
 are in $O(2Tn)$.
 Finally, thresholding with hysteresis is in $O(n^2)$.
 The overall complexity is thus in $O((5^3+4T+1)n^2)$.
 are in $O(2Tn)$.
 Finally, thresholding with hysteresis is in $O(n^2)$.
 The overall complexity is thus in $O((5^3+4T+1)n^2)$.

-To summarize, for the embedding map construction, the complexity of Hugo is
-dramatically larger than our scheme.
+
+
+
+

 
 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 


@@ -40,12 +69,17 @@ in $O(2^h.n)$ 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. 
+

 
 
 
 
 
 
 
 
 
 

-Thanks to these complexity results, we claim that STABYLO is lightweight.