Geometry, Imaging and Computing
Volume 2 (2015)
Surface-based shape classification using Wasserstein distance
Pages: 237 – 255
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.