Regularization, P-Laplace operator, Image processing

Regularization is a principle, concerning a wide range of science domains. Several methods, using this technique, have been proposed. However, there are some limitations to the functionals used in regularization. To remove them, the idea is to replace standard local operators by their nonlocal versions. Our approach is focused on discrete p-Laplace regularization on weighted graphs. As the name implies, this process incorporates weighted p-Laplace operator, which has become increasingly demanded in image processing. In images, pixels have a specific organization expressed by their spatial connectivity. Therefore, a typical structure used to represent images is a graph, where each pixel is identified with one vertex and semantically related pixels are connected by edges. The problem is to find a relevant topology of a graph. Moreover, this topology should be in a correspondence with the purpose of image processing. We give various examples and illustrate usefulness of the proposed approach on problems of noise filtering and segmentation, that are connected with the regularization.