A Simulated Annealing Algorithm for General Threshold Visual Cryptography Schemes

Reducing the pixel expansion and improving the display quality of recovered images are still major issues in visual cryptography schemes (VCSs), particularly for large k and n. Moreover, the development of a systematic and practical approach for threshold VCSs is a challenge. In this paper, a pixel-expansion-free threshold VCSs approach based on an optimization technique is proposed in order to encrypt binary secret images. In addition to contrast, we consider blackness as a performance metric in the evaluation of the display quality of recovered images. We first formulate the problem as a mathematical optimization model in order to maximize the contrast of recovered images that are subject to density-balance and blackness constraints. We then develop a simulated-annealing-based algorithm to solve this problem. Furthermore, we try to promote the contrast by slightly relaxing the density-balance constraint. The experimental results show that the proposed optimization-based approach significantly outperforms previous methods in terms of both the pixel expansion factor and the display quality of recovered images.