Functional representation of deformable surfaces for geometry processing. (Representation fonctionnelle des surfaces deformables pour l'analyse et la synthese geometrique)
Creating and understanding deformations of surfaces is a recurring theme in geometry processing. As smooth surfaces can be represented in many ways from point clouds to triangle meshes, one of the challenges is being able to compare or deform consistently discrete shapes independently of their representation. A possible answer is choosing a flexible representation of deformable surfaces that can easily be transported from one structure to another.Toward this goal, the functional map framework proposes to represent maps between surfaces and, to further extents, deformation of surfaces as operators acting on functions. This approach has been recently introduced in geometry processing but has been extensively used in other fields such as differential geometry, operator theory and dynamical systems, to name just a few. The major advantage of such point of view is to deflect challenging problems, such as shape matching and deformation transfer, toward functional analysis whose discretization has been well studied in various cases. This thesis investigates further analysis and novel applications in this framework. Two aspects of the functional representation framework are discussed.First, given two surfaces, we analyze the underlying deformation. One way to do so is by finding correspondences that minimize the global distortion. To complete the analysis we identify the least and most reliable parts of the mapping by a learning procedure. Once spotted, the flaws in the map can be repaired in a smooth way using a consistent representation of tangent vector fields.The second development concerns the reverse problem: given a deformation represented as an operator how to deform a surface accordingly? In a first approach, we analyse a coordinate free encoding of the intrinsic and extrinsic structure of a surface as functional operator. In this framework a deformed shape can be recovered up to rigid motion by solving a set of convex optimization problems. Second, we consider a linearized version of the previous method enabling us to understand deformation fields as acting on the underlying metric. This allows us to solve challenging problems such as deformation transfer are solved using simple linear systems of equations.