Séminaire Brillouin - Sciences Géométriques de l'Information, IRCAM, 27 Janvier 2012 : Transport Optimal

6 December 2011

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Catégorie : Journée d étude

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Séminaire Brillouin - Sciences Géométriques de l'Information

Lieu : IRCAM, 1 place Stravinsky, Paris (http://www.ircam.fr/contact.html)

Date : 27 janvier 2012

Programme

10h-12h : Gabriel Peyré (CNRS, CEREMADE, Université Paris-Dauphine), Wasserstein Methods in Imaging

12h-14h : Pause 14h-16h : Guillaume Carlier (Professeur, CEREMADE, Université Paris-Dauphine), Barycenters in the Wasserstein Space

Gabriel Peyré. Wasserstein Methods in Imaging.

In this talk I will review the use of optimal transport methods to tackle various imaging problems such as texture synthesis and mixing, color transfer, and shape retrieval. Representing texture variations as well as shapes geometry can be achieved by recording histograms of high dimensional feature distributions. I will present a fast approximate Wasserstein distance to achieve fast optimal transport manipulations of these high dimensional histograms. The resulting approximate distance can be optimized using standard first order optimization schemes to perform color equalization and texture synthesis. It is also possible to use this optimal transport as a data fidelity term in standard inverse problems regularization. One can try online several ideas related to Wasserstein imaging (as many other imaging methods) by visiting www.numerical-tours.com (computer graphics section). This is a joint work with Julien Rabin, Julie Delon and Marc Bernot.

Guillaume Carlier. Barycenters in the Wasserstein Space.

In the first part of the talk, after recalling some basic facts from optimal transport theory, we will explain how some matching problems arising in mathematical economics are intimately related to optimal transport problems. In the second part of the talk, focusing on the quadratic case, we will relate the problem to a notion of barycenters that generalizes the McCann interpolation to the case of more than two marginals. We will give existence, characterization,uniqueness and regularity results for these barycenters and will consider some examples.

This talk will be based on joint works with Martial Agueh and Ivar Ekeland.