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Section: Scientific Foundations

Watermarking as a problem of communication with side information

Digital watermarking aims at hiding discrete messages into multimedia content. The watermark must not spoil the regular use of the content, i.e., the watermark should be non perceptible. Hence, the embedding is usually done in a transformed domain where a human perception model is exploited to assess the non perceptibility criterion. The watermarking problem can be regarded as a problem of creating a communication channel within the content. This channel must be secure and robust to usual content manipulations like lossy compression, filtering, geometrical transformations for images and video. When designing a watermarking system, the first issue to be addressed is the choice of the transform domain, i.e., the choice of the signal components that will host the watermark data. An extraction function E(.) going from the content space Im19 $\#119966 $ to the components space, isomorphic to Im20 $\#119825 ^N$ , must then first be defined.

The embedding process actually transforms a host vector Im21 $\#119829 $ into a watermarked vector Im22 $\#119829 _\#119856 $ . The perceptual impact of the watermark embedding in this domain must be quantified and constrained to remain below a certain level. The measure of perceptual distortion is usually defined as a cost function Im23 ${d(\#119829 _\#119856 -\#119829 )}$ in Im20 $\#119825 ^N$ constrained to be lower than a given distortion bound dw . Attack noise will be added to the watermark vector. In order to evaluate the robustness of the watermarking system and design counter-attack strategies, the noise induced by the different types of attack (e.g. compression, filtering, geometrical transformations, ...) must be modelled. The distortion induced by the attack must also remain below a distortion bound Im24 ${d{(\#119829 _\#119834 -\#119829 )}\lt d_a}$ . Beyond this distortion bound, the content is considered to be non usable any more. Watermark detection and extraction techniques will then exploit the knowledge of the statistical distribution of the vectors Im21 $\#119829 $ . Given the above mathematical model, one has then to design a suitable communication scheme. Direct sequence spread spectrum techniques are often used. The chip rate sets the trade-off between robustness and capacity for a given embedding distortion. This can be seen as a labelling process S(.) mapping a discrete message Im25 ${m\#8712 \#8499 }$ onto a signal in Im20 $\#119825 ^N$ .

The decoding function S-1(.) is then applied to the received signal Im26 $\#119829 _\#119834 $ in which the watermark interferes with two sources of noise: the original host signal (Im21 $\#119829 $ ) and the attack (Im27 $\#119808 $ ). The problem is then to find the pair of functions {S(.), S-1(.)} that will allow to optimise the communication channel under the distortion constraints {dt, da} . This amounts to maximizing the probability to decode correctly the hidden message:

Im28 ${max\mtext Prob{[S^{-1}{(S{(m)}+\#119829 +\#119808 )}=m]}~\mtext under~\mtext constraints~{d_t,d_a}}$

A new paradigm stating that the original host signal Im21 $\#119829 $ shall be considered as a channel state only known at the embedding side rather than a source of noise, appeared recently. The watermark signal thus depends on the channel state: Im29 ${\#119826 =S(m,\#119829 )}$ . This new paradigm known as communication with side information, sets the theoretic foundations for the design of new communication schemes with increased capacity.


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