Recovering Sign Bits of DCT Coefficients in Digital Images as an Optimization Problem

Recovering unknown, missing, damaged, distorted or lost information in DCT coefficients is a common task in multiple applications of digital image processing, including image compression, selective image encryption, and image communications. This paper investigates recovery of a special type of information in DCT coefficients of digital images: sign bits. This problem can be modelled as a mixed integer linear programming (MILP) problem, which is NP-hard in general. To efficiently solve the problem, we propose two approximation methods: 1) a relaxation-based method that convert the MILP problem to a linear programming (LP) problem; 2) a divide-and-conquer method which splits the target image into sufficiently small regions, each of which can be more efficiently solved as an MILP problem, and then conducts a global optimization phase as a smaller MILP problem or an LP problem to maximize smoothness across different regions. To the best of our knowledge, we are the first who considered how to use global optimization to recover sign bits of DCT coefficients. We considered how the proposed methods can be applied to JPEG-encoded images and conducted extensive experiments to validate the performances of our proposed methods. The experimental results showed that the proposed methods worked well, especially when the number of unknown sign bits per DCT block is not too large. Compared with other existing methods, which are all based on simple error-concealment strategies, our proposed methods outperformed them with a substantial margin, both according to objective quality metrics (PSNR and SSIM) and also our subjective evaluation. Our work has a number of profound implications, e.g., more sign bits can be discarded to develop more efficient image compression methods, and image encryption methods based on sign bit encryption can be less secure than we previously understood.