Partial differential equations and variational methods for geometric processing of images

Abstract

This paper arose from a minisymposium held in 2018 at the 9th International Conference on Curves and Surface in Arcachon, France, and organized by Simon Masnou and Carola-Bibiane Schonlieb. This minisymposium featured a variety of recent developments of geometric partial differential equations and variational models which are directly or indirectly related to several problems in image and data processing. The current paper gathers three contributions which are in connection with the talks of three minisymposium speakers: Blanche Buet, Jean-Marie Mirebeau, and Yves van Gennip. The first contribution (Section 1) by Yves van Gennip provides a short overview of recent activity in the field of PDEs on graphs, without aiming to be exhaustive. The main focus is on techniques related to the graph Ginzburg–Landau variational model, but some other research in the field is also mentioned at the end of the section. The second contribution (Section 2), written by Jean-Marie Mirebeau, Francois Desquilbet, Johann Dreo, and Frederic Barbaresco presents a recent numerical method devoted to computing curves that globally minimize an energy featuring both a data driven term, and a second order curvature penalizing term. Applications to image segmentation are discussed, and recent progress on radar network configuration, in which the optimal curves represent an opponent’s trajectories, is described in detail. Lastly, Section 3 is devoted to a work by Blanche Buet, Gian Paolo Leonardi, and Simon Masnou on the definition and the approximation of weak curvatures for a large class of generalized surfaces, and in particular for point clouds, based on the geometric measure theoretic notion of varifolds.