Toward Optimal Prediction Error Expansion-Based Reversible Image Watermarking

Reversible image watermarking is a technique that allows the cover image to remain unmodified after watermark extraction. Prediction error expansion-based schemes are currently the most efficient and widely used class of reversible image watermarking techniques. In this paper, first, we prove that the bounded capacity distortion minimization problem for prediction error expansion-based reversible watermarking schemes is NP-hard, and the corresponding decision version of the problem is NP-complete. Then, we prove that the dual problem of bounded distortion capacity maximization problem for prediction error expansion-based reversible watermarking schemes is NP-hard, and the corresponding decision problem is NP-complete. Furthermore, taking advantage of the integer linear programming formulations of the optimization problems, we find the optimal performance metric values for a given image, using concepts from the optimal linear prediction theory. Our technique allows the calculation of these performance metric limit without assuming any particular prediction scheme. The experimental results for several common benchmark images are consistent with the calculated performance limits validate our approach.