The asymptotical and Gaussian setup is an artificial sandbox: In practice, n is large but not infinite, and signals are not stationary, white and Gaussian distributed. These assumptions are here to simplify the analysis. Nevertheless, this setup should not be too disconnected from reality and we should pay attention to specificities of some watermarking applications:
• Variance σ2
X is not fixed. Content to be watermarked have a huge diversity at least when
we consider multimedia contents such as images, audio and video clips. The theoretical model is also wrong when pretending that, from one content to another, features vectors share the same statistical model. To make a small step towards the real world, we may assume that the detector does not know σX2 because this variance is not fixed from one piece of content to another.
• The embedding power P is not fixed. For the same reason of diversity, pieces of multimedia content have different masking properties, which impact the non-perceptibility of the watermark signal. In academic papers dealing with image or video watermarking, the embedding constraint is often stated as a targeted PSNR in between the host and the watermarked content. This fixes P . In audio watermarking, the target is given by a SNR, which makes P varying. In the same way, real world applications aiming at pristine quality of watermarked content use complex human perceptual model analyzing the masking properties (both in frequency and in time/space domains). It is then less wrong to consider that P varies from one content to another. Power P is positive, certainly small compared to σ2X, and above all, unknown to the detector. Running the same human perceptual model at the detection to obtain an estimation of P is hazardous, especially under mean attacks. • The embedding constraint on expectation (7.2) raises the question of what is random.
There are a priori two sources of randomness. First, watermarking is usually operated by a secret key. This chapter does not deal with mechanism making the embedding and detection private. The secret key does not appear in the notations for the sake of simplicity.
7.4. Specificities of watermarking 85
Efp
Efn
EfpR
EfpR EfpR= E
Figure 7.4: The watermark detector operates at Efp = E. Three characteristics are depicted as we
consider three noise powers. The watermark is deemed robust against the ‘greenattack’ (ERfp> E) but not against the ‘magenta attack’ (EfpR< E). The ‘red attack’ is the limiting case (EfpR = E) whose noise power defines ¯σZ2.
The question is whether the secret key plays the role of a source of randomness. Drawing randomly a secret key at each embedding call is not possible in many applications because there is no auxiliary channel between the embedder and the detector to ‘synchronize’ the secret key. For instance, the watermark detector inside Blu-Ray disc players has a fixed secret key which has been drawn once for all by ‘Hollywood movie industry’. Second, the host signal X is another source of randomness when the embedding is side-informed: the watermark signal W = w(X) depends on X and the embedding constraint is an expectation of kWk2 over X.
• Variance σ2
Z is not fixed. There is also a wide diversity of attacks. Modelling an attack
by the addition of a white Gaussian noise is pure theory. But, the biggest misconception may be that the embedder and the detector know the noise power. In practice, the goal is to be as robust as possible: Pfn should smoothly degrades as σ2Z gets stronger.
• False positives matter more than false negatives. In many applications, a false positive means accusing someone innocent (i.e. in copyright protection) or stopping the playback of a content whereas the user has the right to do so (i.e. in copy protection). Probability Pfp
is required to be very small, which, in our theoretical setup, means Efp set to a high value.
On the other hand, watermarking is usually a dissuasive means: the probability to catch the attacker or to prevent the illegal use of a pirated content should be strictly positive; but dissuasion doesn’t need Pfn to be exponentially vanishing.
• Requirement on the false positive probability. Whereas Pfn cannot be under control
since the power of the attack is a priori unknown, requirements usually set a level for the probability of false positive detection Pfp. What matters in practice is to tune the
watermarking scheme to meet this requirement. This translates in our theoretical setup as the possibility to set values for a series of thresholds {τn} such that, asymptotically, a given
error exponent Efp is achieved.
The penultimate point outlines the fact that the right hand side of the characteristic (large Efp but small Efn) is the most interesting part. This stresses the importance of the quantity EfpR
and its dependence on (P, σ2 X, σ2Z).
The last point suggests the following evaluation criterion: The watermark designer decides to operate at Efp = E (E > 0). He/she needs n ≈ −log(Pfp)/E samples to meet the requirement
X
Z
Embedder W Detector d ∈ {0, 1}
(a) (b)
Figure 7.5: Case 1: Non side informed embedder and blind detector (switches (a) and (b) are open). Case 2: Side informed embedder and blind detector (switch (a) closed, (b) open). Case 3: Non blind detector (switch (b) closed, (a) open or closed).
on the false positive probability. The design of the watermark detector must enforce this error exponent. If EfpR > E for a given attack and embedding power, then the operator is sure that Efn = F (E) > 0. The watermark is deemed robust. If EfpR< E, then F (E) = 0 and the watermark
may not be dissuasive enough. Fig. 7.4 illustrates the differences. The game is not to maximize Efn for the given set of parameters (Efp, P, σX2, σ2Z). It makes more sense to maximize the range
of σ2
Z for which Efn> 0 at a required Efp = E.
Proposition 7.4.1 A meaningful definition of robustness in our context is the following: For a given setup (E, P, σ2X, σZ2), a watermarking scheme is deemed robust if EfpR ≥ E. The maximum noise power for which this inequality holds is denoted by ¯σZ2.
To conclude, the next chapters analyze the error exponent characteristic of several water- marking schemes. We shall pay attention to which parameters are needed on the embedding and detection sides to achieve these theoretical performances.