High-order anisotropic diffusion operators in spaces of variable exponents and application to image inpainting and restoration problems

Abstract

We consider a class of nonstandard high-order PDEs models, based on the p(⋅)-Laplace operator with variable exponents, for image denoising and image inpainting problems. These models are obtained from the minimization of a family of anisotropic energies that range between the so called TV−H−1 and the biharmonic models. The PDEs system consists of quasi-linear equations that we solve with a fixed point iterative method and we prove the convergence of the iterative process. To solve the problems, we consider an algorithm which includes a practical and efficient adaptive strategy for the choice of the exponent 1<p≤2, which allows us to fit to the multi-scale nature of the images. We present several numerical examples to test our approach and to make some comparisons with existing methods.