Sliced and Radon Wasserstein Barycenters of Measures

This article details two approaches to compute barycenters of measures using 1-D Wasserstein distances along radial projections of the input measures. The first method makes use of the Radon transform of the measures, and the second is the solution of a convex optimization problem over the space of...

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Bibliographic Details
Published inJournal of mathematical imaging and vision Vol. 51; no. 1; pp. 22 - 45
Main Authors Bonneel, Nicolas, Rabin, Julien, Peyré, Gabriel, Pfister, Hanspeter
Format Journal Article
LanguageEnglish
Published Boston Springer US 01.01.2015
Springer Verlag
Subjects
Applications of Mathematics
Computer Science
Engineering Sciences
Image Processing and Computer Vision
Mathematical Methods in Physics
Mathematics
Numerical Analysis
Signal and Image Processing
Signal,Image and Speech Processing
Radon transform
Wasserstein distance
Barycenter of measures
Optimal transport
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Summary:This article details two approaches to compute barycenters of measures using 1-D Wasserstein distances along radial projections of the input measures. The first method makes use of the Radon transform of the measures, and the second is the solution of a convex optimization problem over the space of measures. We show several properties of these barycenters and explain their relationship. We show numerical approximation schemes based on a discrete Radon transform and on the resolution of a non-convex optimization problem. We explore the respective merits and drawbacks of each approach on applications to two image processing problems: color transfer and texture mixing.
ISSN:0924-9907
1573-7683
DOI:10.1007/s10851-014-0506-3