Fast Globally Optimal and Geometrically Consistent 3D Shape Matching
This addresses the challenge of achieving geometric consistency in 3D shape matching for downstream applications, offering a scalable solution without limiting assumptions like good initialization.
The paper tackles the problem of computing geometrically consistent 3D shape matchings, which are crucial for applications like texture transfer, by proposing a novel formalism based on cyclic paths and a hyper product graph, cast as a minimum-cost circulation flow problem, resulting in globally optimal solutions that are scalable and yield high-quality results.
Geometric consistency, i.e. the preservation of neighbourhoods, is a natural and strong prior in 3D shape matching. Geometrically consistent matchings are crucial for many downstream applications, such as texture transfer or statistical shape modelling. Yet, in practice, geometric consistency is often overlooked, or only achieved under severely limiting assumptions (e.g. a good initialisation). In this work, we propose a novel formalism for computing globally optimal and geometrically consistent matchings between 3D shapes which is scalable in practice. Our key idea is to represent the surface of the source shape as a collection of cyclic paths, which are then consistently matched to the target shape. Mathematically, we construct a hyper product graph (between source and target shape), and then cast 3D shape matching as a minimum-cost circulation flow problem in this hyper graph, which yields global geometrically consistent matchings between both shapes. We empirically show that our formalism is efficiently solvable and that it leads to high-quality results.