Functional Multi-Reference Alignment via Deconvolution

TL;DR

This paper introduces a novel approach to multi-reference alignment by connecting it with deconvolution, utilizing second-order statistics and Kotlarski's formula for improved signal estimation.

Contribution

It extends Kotlarski's deconvolution formula to higher dimensions and applies it to MRA, providing both theoretical insights and numerical validation.

Findings

Successful extension of Kotlarski's formula to general dimensions.

Effective deconvolution-based algorithms for MRA demonstrated through experiments.

Theoretical validation of the proposed deconvolution approach.

Abstract

This paper studies the multi-reference alignment (MRA) problem of estimating a signal function from shifted, noisy observations. Our functional formulation reveals a new connection between MRA and deconvolution: the signal can be estimated from second-order statistics via Kotlarski's formula, an important identification result in deconvolution with replicated measurements. To design our MRA algorithms, we extend Kotlarski's formula to general dimension and study the estimation of signals with vanishing Fourier transform, thus also contributing to the deconvolution literature. We validate our deconvolution approach to MRA through both theory and numerical experiments.