Provable orbit recovery over SO(3) from the non-uniform second moment

TL;DR

This paper presents a method for recovering 3D signals from noisy, randomly rotated observations with non-uniform rotation distribution, improving noise robustness and providing a provable, efficient reconstruction algorithm.

Contribution

It introduces a joint identifiability framework for signal and rotation distribution from moments and develops a computationally efficient reconstruction algorithm.

Findings

Quadratic noise scaling in sample complexity for non-uniform rotations

Successful recovery of 3D signals using linear systems approach

Validated algorithm through extensive numerical experiments

Abstract

We study the recovery of an unknown three-dimensional band-limited signal from multiple noisy observations that are randomly rotated by latent elements of SO(3), where the rotations are drawn from an unknown, non-uniform distribution. Because the rotations are unobserved, only the signal orbit under the rotation group can be recovered. We show that the signal orbit and the rotation distribution are jointly identifiable from the first and second moments. This yields an improved high-noise sample complexity that scales quadratically with the noise variance, rather than cubically as in the uniform-rotation case. We further develop a provable, computationally efficient reconstruction algorithm that recovers the 3-D signal by successively solving a sequence of well-conditioned linear systems. The algorithm is validated through extensive numerical experiments. Our results provide a principled and tractable framework for high-noise 3-D orbit recovery, with potential relevance to cryo-electron microscopy and cryo-electron tomography modeling, where molecules are observed in unknown orientations.