Cramér-Rao bounds for synchronization of rotations

Abstract

Synchronization of rotations is the problem of estimating a set of rotations Ri ∈ SO(n), i = 1 · · · N, based on noisy measurements of relative rotations RiRj. This fundamental problem has found many recent applications, most importantly in structural biology. We provide a framework to study synchronization as estimation on Riemannian manifolds for arbitrary n under a large family of noise models. The noise models we address encompass zero-mean isotropic noise, and we develop tools for Gaussian-like as well as heavy-tail types of noise in particular. As a main contribution, we derive the Cramér-Rao bounds of synchronization, that is, lower bounds on the variance of unbiased estimators. We find that these bounds are structured by the pseudoinverse of the measurement graph Laplacian, where edge weights are proportional to measurement quality. We leverage this to provide visualization tools for these bounds and interpretation in terms of random walks in both the anchored and anchor-free scenarios. Similar bounds previously established were limited to rotations in the plane and Gaussian-like noise.