Sparse image reconstruction on the sphere: a general approach with uncertainty quantification

TL;DR

This paper introduces a flexible, scalable framework for inverse problems on the sphere, emphasizing uncertainty quantification using Bayesian methods and demonstrating efficiency and applicability across various spherical inverse problems.

Contribution

It provides a general, adaptable approach for spherical inverse problems with a focus on Bayesian uncertainty quantification and computational efficiency, including analytic solutions.

Findings

Efficient Bayesian UQ methods for spherical inverse problems.

Trade-offs between problem formulation and setting are quantified.

Open-source code in C++ and Python is provided.

Abstract

Inverse problems defined naturally on the sphere are becoming increasingly of interest. In this article we provide a general framework for evaluation of inverse problems on the sphere, with a strong emphasis on flexibility and scalability. We consider flexibility with respect to the prior selection (regularization), the problem definition - specifically the problem formulation (constrained/unconstrained) and problem setting (analysis/synthesis) - and optimization adopted to solve the problem. We discuss and quantify the trade-offs between problem formulation and setting. Crucially, we consider the Bayesian interpretation of the unconstrained problem which, combined with recent developments in probability density theory, permits rapid, statistically principled uncertainty quantification (UQ) in the spherical setting. Linearity is exploited to significantly increase the computational efficiency of such UQ techniques, which in some cases are shown to permit analytic solutions. We showcase this reconstruction framework and UQ techniques on a variety of spherical inverse problems. The code discussed throughout is provided under a GNU general public license, in both C++ and Python.