Geometry, Imaging and Computing

Volume 2 (2015)

Number 4

Surface-based shape classification using Wasserstein distance

Pages: 237 – 255

DOI: http://dx.doi.org/10.4310/GIC.2015.v2.n4.a1

Authors

Ming Ma (Department of Computer Science, Stony Brook University, Stony Brook, New York, N.Y., U.S.A.)

Na Lei (Dalian University of Technology, Dalian, Liaoning, China)

Kehua Su (State Key Laboratory of Software Engineering, Wuhan University, Wuhan, Huibei, China)

Junwei Zhang (Stony Brook University, Stony Brook, New York, N.Y., U.S.A.)

Chengfeng Wen (Department of Computer Science, Stony Brook University, Stony Brook, New York, N.Y., U.S.A.)

Li Cui (School of Mathematical Sciences, Beijing Normal University, Beijing, China)

Xin Fan (Dalian University of Technology, Dalian, Liaoning, China)

Xianfeng Gu (Department of Computer Science, Stony Brook University, Stony Brook, New York, N.Y., U.S.A.)

Abstract

Surface based shape analysis plays a fundamental role in computer vision and medical imaging. In this work, we proposes a novel method for shape classification of brain’s hippocampus using Wasserstein distance based on optimal mass transport theory. In comparison with the conventional method based on Monge–Kantorovich theory, our proposed method employs Monge–Brenier theory for the computation of the optimal mass transport map, which remarkably ameliorates the efficiency by reducing computational complexity from $O(n^2)$ to $O(n)$. Using the conformal mapping, our method maps the metric surface with disk topology to the unit planar disk, which pushes the area element on the surface to the disk and incurs the area distortion. A probability measure is then determined by this area distortion. Given any two probability measures on two surfaces, our method is capable of obtaining a unique optimal mass transport map between them. The transportation cost of this optimal mass transport defines the Wasserstein distance between two surfaces, which intrinsically measures the dissimilarities between surface based shapes and thus can be used for shape classification. Experimental results on surface based hippocampal shape analysis demonstrates the efficiency and efficacy of our proposed method.

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