Cram\'er-Rao bounds for synchronization of rotations

TL;DR

This paper derives Cramér-Rao bounds for the estimation accuracy in rotation synchronization problems on Riemannian manifolds, accounting for various noise models and providing insights into measurement graph structures.

Contribution

It introduces a general framework for deriving bounds on rotation synchronization under diverse noise models, extending previous work beyond planar rotations and Gaussian noise.

Findings

Bounds depend on the pseudoinverse of the measurement graph Laplacian.

Bounds are interpretable via random walk models.

Visualization tools are provided for both anchored and anchor-free cases.

Abstract

Synchronization of rotations is the problem of estimating a set of rotations R_i in SO(n), i = 1, ..., N, based on noisy measurements of relative rotations R_i R_j^T. 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 Cram\'er-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 interpretation in terms of random walks and visualization tools for these bounds in both the anchored and anchor-free scenarios. Similar bounds previously established were limited to rotations in the plane and Gaussian-like noise.