25/10/2018 - 15:00 Mr Jocelyn Meyron (Laboratoire Jean Kuntzmann) Salle 106 - Batiment IMAG
In this seminar, we show how optimal transport theory can be used to build a common framework to solve many optical component design problems. In a first part, we present optimal transport, its main formulations and the motivations behind it. We then describe in more details the so-called semi-discrete setting where the target measure is supported on a point cloud and explain the main numerical method we will use namely the damped Newton's algorithm. We then look at a particular case where the source measure is supported on a triangulation in R^3 and show the convergence with linear speed of the damped Newton's method. The convergence is a direct consequence of the regularity and strict motonicity of the Kantorovich functional. We also mention some applications such as optimal quantization of a probability density over a surface or remeshing. In a second part, we describe the relation between optimal transport and optical component design. In particular, we show how we can recast such problems into a non-linear system of equations that is a discretization of the so-called Monge-Ampère equation. This formulation allows us to develop a generic, parameter-free and efficient algorithm. We finish by showing numerous simulated and fabricated examples.