Optimal transport for secure spread-spectrum watermarking of still images

This paper studies the impact of secure watermark embedding in digital images by proposing a practical implementation of secure spread-spectrum watermarking using distortion optimization. Because strong security properties (key-security and subspace-security) can be achieved using natural watermarking (NW) since this particular embedding lets the distribution of the host and watermarked signals unchanged, we use elements of transportation theory to minimize the global distortion. Next, we apply this new modulation, called transportation NW (TNW), to design a secure watermarking scheme for grayscale images. The TNW uses a multiresolution image decomposition combined with a multiplicative embedding which is taken into account at the distribution level. We show that the distortion solely relies on the variance of the wavelet subbands used during the embedding. In order to maximize a target robustness after JPEG compression, we select different combinations of subbands offering the lowest Bit Error Rates for a target PSNR ranging from 35 to 55 dB and we propose an algorithm to select them. The use of transportation theory also provides an average PSNR gain of 3.6 dB on PSNR with respect to the previous embedding for a set of 2000 images.