Optimal Mass Transport for Shape Matching and Comparison

Abstract

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.

Conformal invariants for multiply connected surfaces: Application to landmark curve-based brain morphometry analysis

“…In image analysis research, shape space was well studied for brain atlas estimation (Fletcher et al, 2009; Fletcher, 2013), shape analysis (Kurtek et al, 2011; Gutman et al, 2015; Su et al, 2015a), morphometry study (Younes et al, 2009; Boyer et al, 2011), and other applications. In a computational anatomy framework (Grenander and Miller, 1998), the space of diffeomorphisms was carefully studied (Miller et al, 2002; Miller and Younes, 2001; Trouvé, 1998; Younes, 2010).…”

Isometry Invariant Shape Descriptors for Abnormality Detection on Brain Surfaces Affected by Alzheimer’s Disease

“…The following theorem, has been proven using a variational principle in [19], plays a fundamental role in finding the optimal discrete transport plan.…”

“…Here we propose a novel shape descriptor, based on the area-preserving mapping [19] and Beltrami coefficients, to complement current isometric invariant methods. The area-preserving mapping, which preserves the local area of the surface, provides a globally optimal diffeomorphic mapping from cortical surface to the unit sphere.…”

Measure controllable volumetric mesh parameterization

“…Within the scope of this paper, we only consider the area induced measures. Recently, Su et al applied Brenier's approach for shape matching and comparison in computer vision field [28]. Similar method has been used for surface areapreserving parameterization in graphics/visualization field by Kaufman et al in [29], which focuses on the optimal mass transportation maps between 2D surfaces.…”

“…The theoretic deduction for optimal mass transportation map can be found in [28] and [4]. In order to be complete, we give all details of these algorithms.…”