A qualitative analysis and numerical simulations of a nonlinear second-order anisotropic diffusion problem with non-homogeneous Cauchy–Neumann boundary conditions

摘要

The paper is concerned with a qualitative analysis for a nonlinear second-order parabolic problem, subject to non-homogeneous Cauchy–Neumann boundary conditions, extending the types already studied. Under some certain assumptions, we prove the existence, estimate, regularity and uniqueness of a classical solution. The considered nonlinear second-order anisotropic diffusion model is then particularized for an image restoration task. The resulted PDE-based model is solved numerically by constructing a finite-difference based approximation algorithm that is consistent to the model and converges fast to its solution. An effective detail-preserving image filtering scheme that removes successfully the white additive Gaussian noise while overcoming the unintended effects is thus obtained. Our successful image restoration and method comparison results are also discussed in this paper.