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Exhaustive Search (ES) and re-synchronization through Template Matching (TM) are two of the most commonly invoked solutions against geometric attacks, however doubts exist on whether they represent an effective solution to the problem. The former, in fact, dramatically increases the false detection probability, whereas the latter must take into account the probability of synchronization errors. In order to get some insight into the real effectiveness of ES and TM against watermark de-synchronization, the following simple case study can be considered.
Let the host feature sequence f and the watermarking signal w be modelled as two i.i.d. Gaussian sequences independent of each other, and let us assume that watermarking is achieved by simply adding a scaled version of w to f. In the TM case, a second i.i.d. Gaussian signal (the template) s is also added to f, by paying attention to split the available energy between w and s. We assume that the attacker cyclically shifts the marked vector by an unknown amount resulting in an attacked sequence r. According to the ES approach the detector computes the correlation between r and all the cyclically shifted versions of w. If at least one of such correlations is above the detection threshold, then the watermark presence is revealed. On its side, a TM detector first exploits the presence of s within r in order to re-synchronize w and r, then it applies a standard correlation-based detector. It is easy to show that, in this simple scenario, ES outperforms TM. It can also be shown that as the length of f increases the performance of both methods improve, in that both the false and missed detection probabilities tend to zero.
Despite its simplicity, this simple case study opens up to many interesting investigation directions.
A first question regards the effectiveness of ES detection: apart from complexity issues, is ES the best approach to cope with geometric attacks ? According to the simple case study outlined above the answer seems to be yes, however many important points must be taken into account before giving a final answer: does anything change when a wider class of de-synchronization attacks is allowed? Other approaches to cope with geometric attacks exist, e.g. the use of self-synchronizing – periodic – watermarks, is any of them better than ES ? Does something change when security issues are brought into the picture? In the example above, spread spectrum, 1-bit watermarking is adopted, what about the informed watermarking case?
A second question concerns the possibility of ever defeating geometric attacks. The case study considered above seems to point out that as long as the size of the search space does not increase exponentially, ES (and TM) is an asymptotic-effective solution against de-synchronization. Can this result be demonstrated in a more general set-up? Is it possible to obtain better results by means of informed watermarking? It is not difficult to think about a de-synchronization attack whose dimensionality grows exponentially with the size of the feature sequence, however this may be difficult when perceptual constraints are taken into account. Does such a deadly attack exist?