Wasserstein Barycenter and Its Application to Texture Mixing

• Published in: Scale Space and Variational Methods in Computer Vision pp. 435 - 446
• Main Authors: Rabin, Julien, Peyré, Gabriel, Delon, Julie, Bernot, Marc
• Format: Book Chapter
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg
• Series: Lecture Notes in Computer Science
• Subjects: barycenter of distributions; Optimal transport; texture synthesis and mixing
• ISBN: 9783642247842; 3642247849
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-642-24785-9_37

This paper proposes a new definition of the averaging of discrete probability distributions as a barycenter over the Monge-Kantorovich optimal transport space. To overcome the time complexity involved by the numerical solving of such problem, the original Wasserstein metric is replaced by a sliced approximation over 1D distributions. This enables us to introduce a new fast gradient descent algorithm to compute Wasserstein barycenters of point clouds. This new notion of barycenter of probabilities is likely to find applications in computer vision where one wants to average features defined as distributions. We show an application to texture synthesis and mixing, where a texture is characterized by the distribution of the response to a multi-scale oriented filter bank. This leads to a simple way to navigate over a convex domain of color textures.