Bending Graphs: Hierarchical Shape Matching using Gated Optimal Transport

Shape matching has been a long-studied problem for the computer graphics and vision community. The objective is to predict a dense correspondence between meshes that have a certain degree of deformation. Existing methods either consider the local description of sampled points or discover correspondences based on global shape information. In this work, we investigate a hierarchical learning design, to which we incorporate local patch-level information and global shape-level structures. This flexible representation enables correspondence prediction and provides rich features for the matching stage. Finally, we propose a novel optimal transport solver by recurrently updating features on non-confident nodes to learn globally consistent correspondences between the shapes. Our results on publicly available datasets suggest robust performance in presence of severe deformations without the need of extensive training or refinement.

1. Introduction

Deformable surfaces have been studied extensively by both computer graphics and computer vision communities. Dense correspondence estimation is closely linked to applications such as reconstruction and human pose estimation. In a standard formulation, we want to find a function where is a mapping from a shape to another shape . A shape is usually discretized in a triangle mesh that can be expressed in the form of a graph constituted by vertices with associated 3D coordinates and edges . Among the most successful optimization methods to find correspondences between deformable shapes, there is the Functional Map framework [41]. The pointwise correspondence between two shapes is expressed as a linear map between the functional bases (i.e. eigen-functions of the Laplace Beltrami Operator) defined on the shapes. Figure 1.

We process deformable shape pairs by two levels of hierarchy, local graphs and shape graphs. Such hierarchical structure offers a flexible and holistic shape representation that enables correspondence matching and provides rich features for an optimal transport matching stage.