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We introduce a new paradigm, $\textit{measure synchronization}$, forsynchronizing graphs with measure-valued edges. We formulate this problem asmaximization of the cycle-consistency in the space of probability measures overrelative rotations. In particular, we aim at estimating marginal distributionsof absolute orientations by synchronizing the $\textit{conditional}$ ones,which are defined on the Riemannian manifold of quaternions. Such graphoptimization on distributions-on-manifolds enables a natural treatment ofmultimodal hypotheses, ambiguities and uncertainties arising in many computervision applications such as SLAM, SfM, and object pose estimation. We firstformally define the problem as a generalization of the classical rotation graphsynchronization, where in our case the vertices denote probability measuresover rotations. We then measure the quality of the synchronization by usingSinkhorn divergences, which reduces to other popular metrics such asWasserstein distance or the maximum mean discrepancy as limit cases. We proposea nonparametric Riemannian particle optimization approach to solve the problem.Even though the problem is non-convex, by drawing a connection to the recentlyproposed sparse optimization methods, we show that the proposed algorithmconverges to the global optimum in a special case of the problem under certainconditions. Our qualitative and quantitative experiments show the validity ofour approach and we bring in new perspectives to the study of synchronization.

Synchronizing Probability Measures on Rotations via Optimal Transport