[canny.git] / complexity.tex

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},
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. 

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).
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 $O(343)$, 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 
$2.n^2 \ln(n)$.
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},
$O(2.n^2(343^2 + \ln(n)))$.




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.
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 $O((5^3+4T+1)n^2)$.
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 
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.






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