High-order anisotropic diffusion operators in spaces of variable exponents and application to image inpainting and restoration problems (Q1729266)

The authors consider a class of nonstandard high-order PDEs (partial differential equations), based on the $p(\cdot)$-Laplace operator with variable exponents, for image denoising and image inpainting problems. These equations are obtained from the minimization of a family of anisotropic energies that range between the so-called $T V - H^{- 1}$ and the biharmonic models. The PDE system consists of quasi-linear equations in which fixed point iterative methods are usually used to solve and to prove the convergence of the iterative process. To solve the problems, an algorithm which includes a practical and efficient adaptive strategy for the choice of the exponent $1 < p \leq 2$ is considered. Several numerical examples are presented to test the following approach and to make some comparisons with existing methods.