# Sensing Matrix Design and Sparse Recovery on the Sphere and the Rotation Group

## TL;DR

This paper develops deterministic sampling patterns on the sphere and rotation group for sparse recovery, demonstrating improved coherence and phase transition performance over traditional regular sampling methods.

## Contribution

It introduces a novel approach to design low-coherence deterministic sensing matrices using angular momentum analysis, outperforming existing regular sampling patterns.

## Key findings

- Random sensing matrices satisfy RIP with proper preconditioning.
- Regular sampling patterns often have high mutual coherence, making them undesirable.
- Proposed deterministic matrices achieve lower coherence and better phase transition performance.

## Abstract

In this paper, {the goal is to design deterministic sampling patterns on the sphere and the rotation group} and, thereby, construct sensing matrices for sparse recovery of band-limited functions. It is first shown that random sensing matrices, which consists of random samples of Wigner D-functions, satisfy the Restricted Isometry Property (RIP) with proper preconditioning and can be used for sparse recovery on the rotation group. The mutual coherence, however, is used to assess the performance of deterministic and regular sensing matrices. We show that many of widely used regular sampling patterns yield sensing matrices with the worst possible mutual coherence, and therefore are undesirable for sparse recovery. Using tools from angular momentum analysis in quantum mechanics, we provide a new expression for the mutual coherence, which encourages the use of regular elevation samples. We construct low coherence deterministic matrices by fixing the regular samples on the elevation and minimizing the mutual coherence over the azimuth-polarization choice. It is shown that once the elevation sampling is fixed, the mutual coherence has a lower bound that depends only on the elevation samples. This lower bound, however, can be achieved for spherical harmonics, which leads to new sensing matrices with better coherence than other representative regular sampling patterns. This is reflected as well in our numerical experiments where our proposed sampling patterns perfectly match the phase transition of random sampling patterns.