Circularly orthogonal moments for geometrically robust image watermarking

Circularly orthogonal moments, such as Zernike moments (ZMs) and pseudo-Zernike moments (PZMs), have attracted attention due to their invariance properties. However, we find that for digital images, the invariance properties of some ZMs/PZMs are not perfectly valid. This is significant for applications of ZMs/PZMs. By distinguishing between the ‘good’ and ‘bad’ ZMs/PZMs in terms of their invariance properties, we design image watermarks with ‘good’ ZMs/PZMs to achieve watermark's robustness to geometric distortions, which has been considered a crucial and difficult issue in the research of digital watermarking. Simulation results show that the embedded information can be decoded at low error rates, robust against image rotation, scaling, flipping, as well as a variety of other common manipulations such as lossy compression, additive noise and lowpass filtering.