Numerical Schemes of Shock Filter Models for Image Enhancement and Restoration

Considerable interest has recently been given to signal processing models based on partial differential equations. Successively improved models based on hyperbolic partial differential equation types are proposed in the literature. These models yield interesting results; however, it would be of great interest to generalize them in order to increase their efficiency. In this paper, we propose a generalized shock filter model for one-dimensional signal restoration. After justifying the existence and uniqueness of the solutions in an adequate vector space, we propose an effective numerical scheme to discretize the proposed model, and derive a two-dimensional numerical scheme directly from the one-dimensional model following a space-split strategy. We then prove a stability result for both schemes. We conclude our study by providing high-quality experimental results for one- and two-dimensional signal enhancement and restoration, and showing the influence the shock speed control has on processing time.