Sampling theorems for inverse problems on Riemannian manifolds

TL;DR

This paper develops sampling theorems for inverse problems on Riemannian manifolds, providing explicit error bounds for reconstructing signals from pointwise samples, especially focusing on convolutions on spheres.

Contribution

It introduces new sampling theorems with explicit error bounds for inverse problems on Riemannian manifolds, including the case of convolutions on spheres.

Findings

Derived explicit bounds on reconstruction error depending on sample size and smoothness.

Established sampling theorems for convolutions on compact two-point homogeneous spaces.

Applied results to terrestrial and celestial measurement scenarios.

Abstract

We consider inverse problems consisting of the reconstruction of an unknown signal $f$ from noisy measurements $y=Ff+\text{noise}$, where $Ff$ is a function on a Riemannian manifold without boundary $\mathcal M$. We consider the case when only pointwise samples are available, namely $y_j = (Ff)(x_j)+\eta_j$, where $\{x_j\}_{j=1}^n\subseteq\mathcal M$ is a Marcinkiewicz-Zygmund family. We derive sampling theorems providing explicit bounds on the reconstruction error depending on $n$, the smoothness of $f$ and the properties of $F$. We study in detail the case when $F$ is a convolution on a compact two-point homogeneous space. As a corollary, we state a sampling theorem for convolutions on the two-dimensional sphere, and discuss four relevant examples related to terrestrial and celestial measurements.