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Wasserstein Barycenter and Its Application to Texture Mixing

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Scale Space and Variational Methods in Computer Vision (SSVM 2011)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 6667))

Abstract

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.

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Author information

Authors and Affiliations

  1. CMLA, ENS de Cachan, France

    Julien Rabin

  2. Ceremade, Univ. Paris-Dauphine, France

    Gabriel Peyré

  3. LTCI, Telecom ParisTech, France

    Julie Delon

  4. Thales Alenia Space, France

    Marc Bernot

Authors
  1. Julien Rabin
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  2. Gabriel Peyré
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  3. Julie Delon
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  4. Marc Bernot
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Editor information

Editors and Affiliations

  1. Computer Science Department, Technion — Israel Institute of Technology, 718 Taub Building Technion, 32000, Haifa, Israel

    Alfred M. Bruckstein

  2. Department of Biomedical Engineering, Biomedical Image Analysis and Interpretation, Eindhoven University of Technology, Den Dolech 2 – WH 2.101, 5600 MB, Eindhoven, The Netherlands

    Bart M. ter Haar Romeny

  3. Faculaty of Engineering, School of Electrical Engineering, Tel Aviv University, Ramat Aviv, 69978, Israel

    Alexander M. Bronstein

  4. Institute of Computationa Science, Faculty of Informatics SI-109, Università della Svizzera Italiana, Via Giuseppe Buffi 13, 6904, Lugano, Switzerland

    Michael M. Bronstein

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© 2012 Springer-Verlag Berlin Heidelberg

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Rabin, J., Peyré, G., Delon, J., Bernot, M. (2012). Wasserstein Barycenter and Its Application to Texture Mixing. In: Bruckstein, A.M., ter Haar Romeny, B.M., Bronstein, A.M., Bronstein, M.M. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2011. Lecture Notes in Computer Science, vol 6667. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24785-9_37

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  • DOI: https://doi.org/10.1007/978-3-642-24785-9_37

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-24784-2

  • Online ISBN: 978-3-642-24785-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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