Using ill-posed matrix in correlation-based digital watermarking

摘要

One of the straightforward ways to add a watermark to an image in the spatial domain is to add a pseudo-random noise pattern to the original image. The noise pattern can be generated based on a seed. To detect the watermark in an image, the image is correlated with the noise pattern and the correlation is compared to a preset threshold. Important considerations of the above mentioned correlation-based watermarking techniques are the probability of correct detection and the probability of false alarm. In this paper, we present a method of using "ill-posed" operator to pre-process the noise pattern. The watermark is obtained by pre-multiplying a noise pattern by the inverse of an "ill-posed" operator. An "ill-posed" operator has a large conditional number, i.e., the ratio of the largest singular value to the smallest singular value. Because of the large conditional number, the inverse of an "ill-posed" operator has a large change in the output when the input changes slightly. In watermarking, the "ill-posedness" can be exploited to improve the performance of correlation-based watermarking because of the pseudo-random patterns generated by different seeds have very low correlation with each other and this feature is amplified by the inverse of the "ill-posed" operator. The "ill-posed" operator can be obtained from a wide range of fields such as heat profusion, acoustic wave propagation, and Laplacian equation. Compared with the standard correlation-based watermark, the new watermark has smaller payload and approximately the same probability of correct detection. In addition the new watermark has much lower probability of false alarm. In the paper, we describe the "ill-posed" operator in details and use examples to demonstrate the performance of the watermark.