Regularized Discrete Optimal Transport

Sira Ferradans (1) , Nicolas Papadakis (2, 3) , Julien Rabin (4) , Gabriel Peyré (1) , Jean-François Aujol (3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 MOISE - Modelling, Observations, Identification for Environmental Sciences
3 IMB - Institut de Mathématiques de Bordeaux
4 Equipe Image - Laboratoire GREYC - UMR6072

This article introduces a generalization of discrete Optimal Transport that includes a regularity penalty and a relaxation of the bijectivity constraint. The corresponding transport plan is solved by minimizing an energy which is a convexification of an integer optimization problem. We propose to use a proximal splitting scheme to perform the minimization on large scale imaging problems. For un-regularized relaxed transport, we show that the relaxation is tight and that the transport plan is an assignment. In the general case, the regularization prevents the solution from being an assignment, but we show that the corresponding map can be used to solve imaging problems. We show an illustrative application of this discrete regularized transport to color transfer between images. This imaging problem cannot be solved in a satisfying manner without relaxing the bijective assignment constraint because of mass variation across image color palettes. Furthermore, the regularization of the transport plan helps remove colorization artifacts due to noise amplification.

Mots clés: Optimal Transport; manifold learning; proximal splitting; convex optimization; variational regularization; color transfer

Domaines: Traitement des images [eess.IV]; Traitement du signal et de l'image [eess.SP]

Sira Ferradans, Nicolas Papadakis, Julien Rabin, Gabriel Peyré, Jean-François Aujol. Regularized Discrete Optimal Transport. International Conference on Scale Space and Variational Methods in Computer Vision (SSVM'13), Jun 2013, Schloss Seggau, Leibnitz, Austria. pp.428-439, ⟨10.1007/978-3-642-38267-3_36⟩. ⟨hal-00797078⟩