École d’été 2

Calcul des variations, transport optimal, théorie géométrique de la mesure: théorie et applications

11 au 15 juillet 2016

À Lyon
ICJ, Université Claude Bernard Lyon 1, Villeurbanne, France

Cette école est organisée dans le cadre de la période thématique du même nom structurée en 2 écoles d’été et une conférence internationale.

Trois conférences sont au programme :

Variational Methods for Computer Vision

Variational methods are among the most classical and established methods to solve a multitude of problems arising in computer vision and image processing. Over the last years, they have evolved substantially, giving rise to some of the most powerful methods for optic flow estimation, image segmentation and 3D reconstruction, both in terms of accuracy and in terms of computational speed. In this tutorial, I will introduce the basic concepts of variational methods. I will then focus on problems of geometric optimization including image segmentation and 3D reconstruction. I will show how the regularization terms can be adapted to incorporate statistically learned knowledge about our world. Subsequently, I will discuss techniques of convex relaxation and functional lifting which allow to computing globally optimal or near-optimal solutions to respective energy minimization problems. Experimental results demonstrate that these spatially continuous approaches provide numerous advantages over spatially discrete (graph cut) formulations, in particular they are easily parallelized (lower runtime) and they do not suffer from metrication errors (better accuracy).

On Optimization Algorithms in Imaging Sciences and Hamilton-Jacobi équations

The course will consist of two parts

1.   Total variation minimization and maximal flows in graphs

  • Applications to image processing
  • Anisotropic mean curvature flow

2.  Optimization in image processing, Hamilton-Jacobi equations, and optimal control

Computational optimal transport

Optimal transport has been used as a powerful theoretical tool to study partial differential equations, differential geometry and probability for a few decades. In comparison, its use in numerical applications is much more recent, not because of lack of interest but rather because of computational difficulties. The simplest discretization of the optimal transport problem lead to combinatorial optimization problems for which can only be solved with superquadratic cost. On the other hand, the partial differential equations arising from optimal transport are fully non-linear Monge-Ampère equations, for which there did not exist robust and efficient numerical solvers until recently. This course will present a variety of approaches to solve optimal transport and related problems, with applications in mind, such as:

  • Entropic penalization and Wasserstein barycenters
  • Benamou-Brenier algorithm, gradient flows and simulation of non-linear diffusion equations
  • Monge-Ampère equation, computational geometry and convexity constraints
  • Semi-discrete optimal transport and inverse problems in geometric optics
  • Measure-preserving maps, optimal quantization and Euler’s equation for incompressible fluids

 

 

 

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