This research work takes place in the field of information hiding, considerably developed these last two decades. The proposed method for 
steganography considers digital images as covers.
It belongs to the well-known large category
of spatial least significant bits (LSBs) replacement schemes.
Let us recall that, in this LSB replacement category, a subset of all the LSBs of the cover image is modified 
with a secret bit stream depending on: a secret key, the cover, and the message to embed.
In this well-studied steganographic approach,
if we consider that a LSB is the last bit of each pixel value,  
pixels with an even value (resp. an odd value) 
are never decreased (resp. increased), 
thus such schemes may break the 
structural symmetry of the host images.
And these structural alterations can be detected by 
well-designed statistical investigations, leading to well-known 
steganalysis methods~\cite{DBLP:journals/tsp/DumitrescuWW03,DBLP:conf/mmsec/FridrichGD01,Dumitrescu:2005:LSB:1073170.1073176}.

Let us recall too that this drawback 
can be fixed by considering the LSB matching (LSBM) subcategory, in which
a $+1$ or $-1$ is randomly added to the cover pixel's LSB value 
only if this one does not correspond to the secret bit.
%TODO : modifier ceci
By considering well-encrypted hidden messages, the probabilities of increasing or decreasing the value of pixels  are equal. Then usual statistical approaches 
cannot be applied here to discover stego-contents in LSBM.
The most accurate detectors for this matching are universal steganalysers such as~\cite{LHS08,DBLP:conf/ih/Ker05,FK12},
which classify images according to extracted features from neighboring elements of residual noise.  


Finally, LSB matching revisited (LSBMR) has recently been introduced in~\cite{Mielikainen06}. 
It works as follows: for a given pair of pixels, the LSB
of the first pixel carries a first bit of the secret message, while the parity relationship
(odd/even combination) of the two pixel values carries
a second bit of the message. 
By doing so, the modification
rate of pixels can decrease from 0.5 to 0.375 bits/pixel
(bpp) in the case of a maximum embedding rate, meaning fewer
changes in the cover image at the same payload compared to both
LSBR and LSBM. It is also shown in~\cite{Mielikainen06} that such a new
scheme can avoid the LSB replacement style asymmetry, and
thus it should make the detection slightly more difficult than in the
LSBM approach. % based on our experiments





Additionally to (efficiently) modifying LSBs, there is also a need to select pixels whose value 
modification minimizes a distortion function.
This distortion may be computed thanks to feature vectors that
 are embedded for instance in the steganalysers
referenced above. 
The Highly Undetectable steGO (HUGO) method~\cite{DBLP:conf/ih/PevnyFB10},
WOW~\cite{conf/wifs/HolubF12}, and UNIWARD~\cite{HFD14}
are some of the  most efficient instances of such a scheme.

HUGO takes into account so-called SPAM features.
Thus a distortion measure for each
pixel is individually determined as the sum of the differences between
the features of the SPAM computed from the cover and from the stego images.
The features embedded in WOW and UNIWARD are based on Wavelet-based 
directional filter. Thus, similarly, the distortion function is 
the sum  of the differences between these wavelet coefficients  
computed from the cover and from the stego images.
Due to this distortion measures, HUGO, WOW and UNIWARD allow
to embed messages that are $7$ times longer than the former
ones with the same level of 
indetectability as LSB matching. 
However, this improvement has a larger computation cost, mainly due to
 the distortion function
computation.


There remains a large place between random selection of LSB and feature based modification of pixel values.
We argue that modifying edge pixels is an acceptable compromise. 
Edges form the outline of an object: they are the boundaries between overlapping objects or between an object
and its background. When producing the stego-image, a small modification of some pixel values in such edges should not impact the image quality, which is a requirement when
attempting to be undetectable. Indeed,
in a cover image, edges already break the  continuity of pixels' intensity map and thus already present large variations with their neighboring 
pixels. In other words, minor changes in regular areas are more dramatic than larger modifications in edge ones. 
Our first proposal is thus to embed message bits into edge shapes while preserving other smooth regions. 

Edge based steganographic schemes have already been  studied, 
the most interesting 
approaches being detailed in~\cite{Luo:2010:EAI:1824719.1824720} and 
in~\cite{DBLP:journals/eswa/ChenCL10}.
In the former, the authors present the Edge Adaptive
Image Steganography based on LSB matching revisited further denoted as  
EAISLSBMR. This approach selects sharper edge
 regions with respect 
to a given embedding rate: the larger the number of bits to be embedded is, the coarser
the edge regions are.
Then the data hiding algorithm is achie\-ved by applying LSBMR on some of the pixels of these regions. 
The authors show that their proposed method is more efficient than all the LSB, LSBM, and LSBMR approaches 
through extensive experiments.
However, it has been shown that the distinguishing error with LSB embedding is lower than 
the one with some binary embedding~\cite{DBLP:journals/tifs/FillerJF11}.
We thus propose to take advantage of this optimized embedding, provided they are not too time consuming.
In the latter, an hybrid edge detector is presented followed by an ad hoc
embedding. 
The Edge detection is computed by combining fuz\-zy logic~\cite{Tyan1993} 
and Canny~\cite{Canny:1986:CAE:11274.11275} approaches. 
The goal of this combination 
is to enlarge the set of modified bits to increase the payload of the data hiding scheme. 


One can notice that all the previously referenced 
sche\-mes~\cite{Luo:2010:EAI:1824719.1824720,DBLP:journals/eswa/ChenCL10,DBLP:conf/ih/PevnyFB10}
produce stego contents 
by only considering the payload, not the type of image signal: the higher the payload is, 
the better the approach is said to be.
For instance, studied payloads range from 0.04 to 0.4 modified bits per pixel.
Contrarily, we argue that some images should not be taken 
as a cover because of the nature of their signals.
Consider for instance a uniformly black image: a very tiny modification of its 
pixels can be easily detected.
Practically speaking, if Alice would send
a hidden message to Bob, she would never consider
such kind of image and a high embedding rate.
\JFC{This desire to be adaptive has been 
studied too in~\cite{LiFengyongZhang14}, 
but in JPEG frequency domain}. 
The approach we propose here is thus to provide a small complexity 
self adaptive algorithm
with an acceptable payload, which 
depends on the  cover signal. 
The payload is further said to  
 be acceptable if it  allows to embed a sufficiently 
long message in the cover signal. 
Practically speaking, our approach is efficient enough for
payloads close to 0.06 bit per pixel which allows to embed 
messages of length larger than 15,728 bits in an 
image of size $512\times 512$ pixels. 

% Message extraction is achieved by computing the same
% edge detection pixels set for the cover and the stego image. 
% The edge detection algorithm is thus not applied  on all the bits of the image,
% but to exclude the LSBs which are modified. 

Finally, even if the steganalysis discipline
 has known  great innovations these last years, it is currently impossible to prove rigorously
that a given hidden message cannot be recovered by an attacker.
This is why we add to our scheme a reasonable
message encryption stage, to be certain that,
even in the worst case scenario, the attacker
will not be able to obtain the original message content.
Doing so makes our steganographic protocol, to a certain extend, an asymmetric one.

To sum up, well-studied and experimented
techniques of signal processing (adaptive edges detection), 
coding theory (syndrome-trellis codes), and cryptography 
(Blum-Goldwas\-ser encryption protocol) are combined in this research work.
The objective is to  compute an efficient steganographic
sche\-me, whose principal characteristic is to take into 
consideration the cover image and to be compatible with small computation resources.  

The remainder of this document is organized as follows. 
Section~\ref{sec:ourapproach} presents the details of the proposed steganographic scheme and applies it on a running example. Among its technical description, 
its adaptive aspect is emphasized.
Section~\ref{sub:complexity} presents the overall complexity of our approach
and compares it to HUGO, WOW, and UNIWARD.
Section~\ref{sec:experiments} shows experiments on image quality, steganalysis evaluation, and compares them to the state of the art steganographic schemes.
Finally, concluding notes and future work are given in Section~\ref{sec:concl}.



