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We consider the problem of graph-matching on a network of 3D shapes withuncertainty quantification. We assume that the pairwise shape correspondencesare efficiently represented as \emph{functional maps}, that match real-valuedfunctions defined over pairs of shapes. By modeling functional maps betweennearly isometric shapes as elements of the Lie group $SO(n)$, we employ\emph{synchronization} to enforce cycle consistency of the collection offunctional maps over the graph, hereby enhancing the accuracy of the individualmaps. We further introduce a tempered Bayesian probabilistic inferenceframework on $SO(n)$. Our framework enables: (i) synchronization of functionalmaps as maximum-a-posteriori estimation on the Riemannian manifold offunctional maps, (ii) sampling the solution space in our energy based model soas to quantify uncertainty in the synchronization problem. We dub the latter\emph{Riemannian Langevin Functional Map (RLFM) Sampler}. Our experimentsdemonstrate that constraining the synchronization on the Riemannian manifold$SO(n)$ improves the estimation of the functional maps, while our RLFM samplerprovides for the first time an uncertainty quantification of the results.

Riemannian Functional Map Synchronization for Probabilistic Partial Correspondence in Shape Networks