Riemannian Functional Map Synchronization for Probabilistic Partial Correspondence in Shape Networks

TL;DR

This paper introduces a probabilistic framework for graph-matching of 3D shapes using functional maps, employing Riemannian synchronization and Bayesian inference to improve accuracy and quantify uncertainty.

Contribution

It presents a novel Riemannian Bayesian inference framework for functional map synchronization, enabling uncertainty quantification in shape correspondence.

Findings

Riemannian synchronization improves functional map accuracy.

The RLFM sampler quantifies uncertainty in shape matching.

Cycle consistency enhances correspondence reliability.

Abstract

We consider the problem of graph-matching on a network of 3D shapes with uncertainty quantification. We assume that the pairwise shape correspondences are efficiently represented as \emph{functional maps}, that match real-valued functions defined over pairs of shapes. By modeling functional maps between nearly isometric shapes as elements of the Lie group $SO(n)$, we employ \emph{synchronization} to enforce cycle consistency of the collection of functional maps over the graph, hereby enhancing the accuracy of the individual maps. We further introduce a tempered Bayesian probabilistic inference framework on $SO(n)$. Our framework enables: (i) synchronization of functional maps as maximum-a-posteriori estimation on the Riemannian manifold of functional maps, (ii) sampling the solution space in our energy based model so as to quantify uncertainty in the synchronization problem. We dub the latter \emph{Riemannian Langevin Functional Map (RLFM) Sampler}. Our experiments demonstrate that constraining the synchronization on the Riemannian manifold $SO(n)$ improves the estimation of the functional maps, while our RLFM sampler provides for the first time an uncertainty quantification of the results.