X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/canny.git/blobdiff_plain/d67ddfb670a8d832dcf94f66044f3bb2ac8add39..ebe8c2f159425cf2a2f7a696eedb593305fd19b7:/complexity.tex

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 This section aims at justifying the lightweight attribute of our approach.
-To be more precise, we compare the complexity of our schemes to the 
-state of the art steganography, 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 separately evaluate this complexity.
+In all the rest of this section, we consider a $n \times n$ square image. 
 
-
-In what follows, we consider an $n \times n$ square image. 
 First of all, HUGO starts with computing the second order SPAM Features.
-This steps is in  $O(n^2 + 2.343^2)$ 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 
 features. Pixels are thus selected according to their ability to provide
-an image whose SPAM features are close to the original one. 
-The algorithm is thus computing a distance between each computed feature, 
-andthe 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 a insertion sort which is in 
-$2.n^2 \ln(n)$.
-The overall complexity of the pixel selection is thus 
-$O(n^2 +2.343^2 + 2\times 343^2 \times n^2 + 2.n^2 \ln(n))$, \textit{i.e}
-$O(2.n^2(343^2 + \ln(n)))$.
-
-Our edge selection is based on a Canny  Filter, 
-whose complexity is in $O(2n^2.\ln(n))$ thanks to the convolution step
-which can be implemented with FFT.
-To summarize, for the embedding map construction, the complexity of Hugo is
-at least $343^2/\ln{n}$ times higher than 
-our scheme. For instance, for a squared image with 4M pixel per slide,
-this part of our algorithm is more than 14100 faster than Hugo.
-
-We are then left to express the complexity of the STC algorithm .
+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 an insertion sort, which is in 
+$2 \times n^2 \ln(n)$.
+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 $\theta((5^3+4T+1)n^2)$.
+
+
+
+
+
+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 
-matrix. Its complexity is thus negligeable compared with the embedding map
+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.
 
-Thanks to these complexity result, 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}
+
+
 
 
 
