Exploring the Cross-Correlation as a Means for Detecting Digital Watermarks and Its Reformulation Into the Fractional Calculus Framework

In this paper, we explored the application of the fractional calculus to digital watermarking. In order to reach this goal, we chose a set of watermarking systems, and then, we altered the watermark detection step of each system. The novelty of our proposal is that we deviated from the trend of proposing new embedding domains computed by using fractional calculus principles. Instead, we proposed to reformulate the detection equation set. Then, this paper contributes to a new equation set for detecting watermarks based on fractional calculus principles. In addition, we provided the proper order for the derivative; this equation set replaces the previous one, given as a result that the detection rate increased up to 88% for the best case compared to the original equation set. The main modification was to replace the system's equations to another set that we derived by using the Riemann-Liouville fractional operator. To verify that if this approach is effective, we performed tests on grayscale images of size 512 × 512; the test covered several attacks including noise addition, JPEG compression, and scaling. The results showed that the modified systems detected more watermarks than their counterparts. The proposed scheme presented better results for 15 out of 21 different tests and the other 6 ended in a draw. It is important to remark that the behavior holds for other embedding domains, for example, the discrete cosine transform coefficients and the singular values decomposition values, which suggests the regularity of the approach.