Optimal Mass Transport for Shape Matching and Comparison

Zhengyu Su, Yalin Wang, Rui Shi, Wei Zeng, Jian Sun, Feng Luo, Xianfeng Gu

Research output: Contribution to journalArticlepeer-review

73 Scopus citations


Surface based 3D shape analysis plays a fundamental role in computer vision and medical imaging. This work proposes to use optimal mass transport map for shape matching and comparison, focusing on two important applications including surface registration and shape space. The computation of the optimal mass transport map is based on Monge-Brenier theory, in comparison to the conventional method based on Monge-Kantorovich theory, this method significantly improves the efficiency by reducing computational complexity from O(n2) to O(n). For surface registration problem, one commonly used approach is to use conformal map to convert the shapes into some canonical space. Although conformal mappings have small angle distortions, they may introduce large area distortions which are likely to cause numerical instability thus resulting failures of shape analysis. This work proposes to compose the conformal map with the optimal mass transport map to get the unique area-preserving map, which is intrinsic to the Riemannian metric, unique, and diffeomorphic. For shape space study, this work introduces a novel Riemannian framework, Conformal Wasserstein Shape Space, by combing conformal geometry and optimal mass transport theory. In our work, all metric surfaces with the disk topology are mapped to the unit planar disk by a conformal mapping, which pushes the area element on the surface to a probability measure on the disk. The optimal mass transport provides a map from the shape space of all topological disks with metrics to the Wasserstein space of the disk and the pullback Wasserstein metric equips the shape space with a Riemannian metric. We validate our work by numerous experiments and comparisons with prior approaches and the experimental results demonstrate the efficiency and efficacy of our proposed approach.

Original languageEnglish (US)
Article number7053911
Pages (from-to)2246-2259
Number of pages14
JournalIEEE Transactions on Pattern Analysis and Machine Intelligence
Issue number11
StatePublished - Nov 1 2015


  • optimal mass transport
  • shape representation
  • shape space
  • surface matching

ASJC Scopus subject areas

  • Software
  • Computer Vision and Pattern Recognition
  • Computational Theory and Mathematics
  • Artificial Intelligence
  • Applied Mathematics


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