Semifragile hierarchical watermarking in a set theoretic framework

We introduce a set theoretic framework for watermarking and illustrate its effectiveness by designing a hierarchical semi-fragile watermark that is tolerant to compression and allows tamper localization. Using a quad-tree representation, a spatial resolution hierarchy is established on the image and a watermark is embedded corresponding to each node of the hierarchy. The watermarked image is determined so as to jointly satisfy the multiple constraints of watermark detectability, imperceptibility, and robustness to compression using the method of projections onto convex sets. The spatial hierarchy of watermarks provides a graceful trade-off between robustness and localization under JPEG compression: mild JPEG compression preserves watermarks at all levels of the hierarchy allowing fine localization of malicious changes while aggressive JPEG compression preserves watermarks at coarser levels of the hierarchy still assuring overall image integrity but giving up the capability for localization. Experimental results are presented to illustrate the effectiveness of the method.