Mixed Finite Element Approximation for Bivariate Perona-Malik Model Arising in 2D and 3D Image Denoising

In this paper, we present a new model of nonlinear diffusion arising in image processing field based on the Perona-Malik equation which we call the bivariate Perona-Malik model. The aim of this model is to remove image noise while preserving edges, boundaries, and textures. To solve this model we use a new algorithm based on mixed finite element method. In this context we prove mathematically the existence and uniqueness of the solution of the proposed model in a well chosen space. At last, we present the experimental results in 2D and 3D images filtering, which demonstrate the efficiency and effectiveness of our algorithm and finally, we compare it with other well known methods such as the finite difference method presented by Perona et al. (IEEE Trans Pattern Anal Mach Intell 12:629-639, 1990), the finite element method and the finite volume method studied by Handloviova et al. (J Vis Commun Image Represent 13:217-237, 2002).