Variational Approaches to Digital Geometry Processing

Abstract

This habilitation thesis presents a series of contributions in the field of digital geometry processing. These contributions offer concepts and algorithms for surface reconstruction, surface approximation, quadrangle surface tiling and isotropic tetrahedron mesh generation. The narrative aims at highlighting the common feature shared among our contributions: we adopt a variational methodology throughout this document, in the sense that we tackle each digital geometric problem by casting it as an energy minimization so that low levels of these energies correspond to good solutions of the problem. The main motivation behind such formulations is a significantly increased quality and robustness, sometimes at the price of heavier computations than greedy algorithms. The data structures and concepts involved in our work lie between computational geometry, geometric computing, and numerical computing. A general summary also provides a vision of the many remaining challenges in the field.Cette thèse d'habilitation présente une synthèse de contributions dans le domaine du traitement numérique de la géométrie sous la forme de concepts et d'algorithmes pour la reconstruction de surfaces, l'approximation de surfaces, le remaillage quadrangle de surfaces et la génération de maillages