An algorithm for high-precision numerical computation of Zernike moments is presented. The algorithm, based on the introduced polar pixel tiling scheme, does not exhibit the geometric error and numerical integration error which are inherent in conventional methods based on Cartesian coordinates. This yields a dramatic improvement of the Zernike moments accuracy in terms of their reconstruction and invariance properties. The introduced image tiling requires an interpolation algorithm which turns out to be of the second order importance compared to the discretization error. Various comparisons are made between the accuracy of the proposed method and that of commonly used techniques. The results reveal the great advantage of our approach.
In spread spectrum based watermarking schemes, it is a challenging task to embed multiple bits of information into the host signal. M -ary modulation has been proposed as an effective approach to multibit watermarking. It has been proved that an M -ary modulation based watermarking system outperforms significantly a binary modulation based watermarking system. However, in the existing M -ary modulation based algorithms, the value of M is restricted to be less than 256, because as M increases, the computation workload for data extraction advances exponentially. In this paper, we propose an efficient M -ary modulation scheme, i.e., M -ary phase modulation, which reduces the computation in data extraction to a very low level. With this scheme, it is practical to implement an M -ary modulation based algorithm with a high value of M , e.g., M = 2 20 . This is significant for a watermarking system, because it can either greatly increase the data capacity of a watermark given the necessary watermark robustness, or considerably improve the watermark robustness given the amount of information of the watermark. The superiority of the proposed scheme is verified by simulation results.
In image watermarking, the watermark robustness to geometric transformations is still an open problem. Using invariant image features to carry the watermark is an effective approach to addressing this problem. In this paper, a multibit geometrically robust image watermarking algorithm using Zernike moments is proposed. Some Zernike moments of an image are seleted, and their magnitudes are dither-modulated to embed an array of bits. The watermarked image is obtained via reconstruction from the modified moments and those left intact. In watermark extraction, the embedded bits are estimated from the invariant magnitudes of the Zernike moments using a minimum distance decoder. Simulation results show that the hidden message can be decoded at low error rates, robust against image rotation, scaling and flipping, and as well, a variety of other distortions such as lossy compression.
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