On the role of non-local Menger curvature in image processing

Curvature is a fundamental component in differential geometry. It is used extensively in signal, image and shape processing, as a feature and in segmentation flows and regularization processes. In this paper we extend the notion of curvature in two ways. First, we present the Menger curvature which goes beyond classical curves and Riemannian manifolds to general metric spaces and is rigorously defined on a variety of discrete settings. We further extend the curvature to become a non-local entity using an adaptive, non-local integration measure, allowing curvature to be computed in a robust manner. Examples on natural and textural images highlight potential applications of these new concepts.