Functional maps are efficient representations of shape correspondences, that provide matching of real-valued functions between pairs of shapes. Functional maps can be modelled as elements of the Lie group $SO(n)$ for nearly isometric shapes. Synchronization can subsequently be employed to enforce cycle consistency between functional maps computed on a set of shapes, hereby enhancing the accuracy of the individual maps. There is an interest in developing synchronization methods that respect the geometric structure of $SO(n)$, while introducing a probabilistic framework to quantify the uncertainty associated with the synchronization results. This paper introduces a Bayesian probabilistic inference framework on $SO(n)$ for Riemannian synchronization of functional maps, performs a maximum-a-posteriori estimation of functional maps through synchronization and further deploys a Riemannian Markov-Chain Monte Carlo sampler for uncertainty quantification. Our experiments demonstrate that constraining the synchronization on the Riemannian manifold $SO(n)$ improves the estimation of the functional maps, while our Riemannian MCMC sampler provides for the first time an uncertainty quantification of the results.

翻译：功能性地图是形状对应物的有效表示,能够匹配形状对等体之间的实际价值功能。功能性地图可以仿照接近等度形状的Lie Group $SO(n)的元素。随后可以使用同步化来强制按一组形状计算功能性地图之间的周期一致性,从而提高单个地图的准确性。人们有兴趣制定尊重美元SO(n)的几何结构的同步方法,同时引入一个概率框架来量化同步结果的不确定性。本文介绍了Bayesian 美元SO(n)的概率性推论框架,用于功能性地图的Riemannian同步,通过同步和进一步部署Riemannian Markov-Chain Monte Carlo取样员来测量不确定性,对功能性地图进行最大程度的超前估计。我们的实验表明,限制Riemannian $(n) $SO(n) 的同步性方法可以改进功能性地图的估算,而我们的Riemannian MC取样员则首次提供了结果的不确定性量化。