Mimetic differences and variational approach to Perona-Malik problem

Cuti Gutiérrez H.; Gonzales Herrera M.; Torres Ledesma C.E.; Venturo Orbegoso L. · Journal of Computational and Applied Mathematics (2026) · DOI: 10.1016/j.cam.2026.117731

Resumen

Digital images play a fundamental role in multiple fields, with image smoothing being a key task in image processing. The Perona-Malik model introduced a nonlinear anisotropic diffusion equation that enables noise removal while preserving edges, overcoming the limitations of traditional linear methods. Several authors have proposed extensions and improvements to this model, incorporating techniques such as K-means, local entropy, nonlocal total variation, and nonlinear or nonlocal formulations, achieving better preservation of edges, textures, and fine details in noisy images. In this work, we propose a new algorithm to solve the Perona-Malik diffusion problem using the mimetic finite difference method, which preserves essential properties of differential operators and facilitates the treatment of boundary conditions, avoiding common errors associated with ghost points or interpolation. Differences with previous approaches: no Gaussian smoothing is applied, and the threshold parameter in the diffusion function remains fixed. Evaluation of results through objective metrics: mean squared error and peak signal-to-noise ratio. Finally, we also address the existence of nonnegative solutions to the stationary Perona-Malik problem with homogeneous Dirichlet boundary conditions, employing variational methods and critical point theory.

Instituciones: Universidad Nacional de Trujillo; Universidad Nacional del Altiplano; Universidad Nacional Pedro Ruiz Gallo