Joint Compression and Digital Watermarking:
\nInformation-Theoretic Study and Algorithms Development

In digital watermarking, a watermark is embedded into a covertext in such a way that the resulting watermarked signal is robust to certain distortion caused by either standard data processing in a friendly environment or malicious attacks in an unfriendly environment. The watermarked signal can then be used for different purposes ranging from copyright protection, data authentication,fingerprinting, to information hiding. In this thesis, digital watermarking will be investigated from both an information theoretic viewpoint and a numerical computation viewpoint. From the information theoretic viewpoint, we first study a new digital watermarking scenario, in which watermarks and covertexts are generated from a joint memoryless watermark and covertext source. The configuration of this scenario is different from that treated in existing digital watermarking works, where watermarks are assumed independent of covertexts. In the case of public watermarking where the covertext is not accessible to the watermark decoder, a necessary and sufficient condition is determined under which the watermark can be fully recovered with high probability at the end of watermark decoding after the watermarked signal is disturbed by a fixed memoryless attack channel. Moreover, by using similar techniques, a combined source coding and Gel'fand-Pinsker channel coding theorem is established, and an open problem proposed recently by Cox et al is solved. Interestingly, from the sufficient and necessary condition we can show that, in light of the correlation between the watermark and covertext, watermarks still can be fully recovered with high probability even if the entropy of the watermark source is strictly above the standard public watermarking capacity. We then extend the above watermarking scenario to a case of joint compression and watermarking, where the watermark and covertext are correlated, and the watermarked signal has to be further compressed. Given an additional constraint of the compression rate of the watermarked signals, a necessary and sufficient condition is determined again under which the watermark can be fully recovered with high probability at the end of public watermark decoding after the watermarked signal is disturbed by a fixed memoryless attack channel. The above two joint compression and watermarking models are further investigated under a less stringent environment where the reproduced watermark at the end of decoding is allowed to be within certain distortion of the original watermark. Sufficient conditions are determined in both cases, under which the original watermark can be reproduced with distortion less than a given distortion level after the watermarked signal is disturbed by a fixed memoryless attack channel and the covertext is not available to the watermark decoder. Watermarking capacities and joint compression and watermarking rate regions are often characterized and/or presented as optimization problems in information theoretic research. However, it does not mean that they can be calculated easily. In this thesis we first derive closed forms of watermarking capacities of private Laplacian watermarking systems with the magnitude-error distortion measure under a fixed additive Laplacian attack and a fixed arbitrary additive attack, respectively. Then, based on the idea of the Blahut-Arimoto algorithm for computing channel capacities and rate distortion functions, two iterative algorithms are proposed for calculating private watermarking capacities and compression and watermarking rate regions of joint compression and private watermarking systems with finite alphabets. Finally, iterative algorithms are developed for calculating public watermarking capacities and compression and watermarking rate regions of joint compression and public watermarking systems with finite alphabets based on the Blahut-Arimoto algorithm and the Shannon's strategy.

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