We introduce a new paradigm, $\textit{measure synchronization}$, for
synchronizing graphs with measure-valued edges. We formulate this problem as
maximization of the cycle-consistency in the space of probability measures over
relative rotations. In particular, we aim at estimating marginal distributions
of absolute orientations by synchronizing the $\textit{conditional}$ ones,
which are defined on the Riemannian manifold of quaternions. Such graph
optimization on distributions-on-manifolds enables a natural treatment of
multimodal hypotheses, ambiguities and uncertainties arising in many computer
vision applications such as SLAM, SfM, and object pose estimation. We first
formally define the problem as a generalization of the classical rotation graph
synchronization, where in our case the vertices denote probability measures
over rotations. We then measure the quality of the synchronization by using
Sinkhorn divergences, which reduces to other popular metrics such as
Wasserstein distance or the maximum mean discrepancy as limit cases. We propose
a nonparametric Riemannian particle optimization approach to solve the problem.
Even though the problem is non-convex, by drawing a connection to the recently
proposed sparse optimization methods, we show that the proposed algorithm
converges to the global optimum in a special case of the problem under certain
conditions. Our qualitative and quantitative experiments show the validity of
our approach and we bring in new perspectives to the study of synchronization.

提出一种新的度量同步的方法 —— 度量同步，将这个问题视为优化概率旋转测度的循环一致性，从而达到通过同步条件概率测度来估计绝对方向边际分布的目的，在计算机视觉应用中得到了有力的应用，提出了非参数黎曼粒子优化方法来解决问题。