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

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.

arXiv commentIEEE Trans. on Signal Processing

References in corpus (10)

• Message Passing Algorithms for Compressed Sensing
• Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Sky Maps, Systematic Errors, and Basic Results
• Explicit constructions of RIP matrices and related problems
• A novel sampling theorem on the sphere
• Fast approximation of matrix coherence and statistical leverage
• Certifying the restricted isometry property is hard
• Sparse recovery for spherical harmonic expansions
• Weighted eigenfunction estimates with applications to compressed sensing
• Coherence Optimization and Best Complex Antipodal Spherical Codes
• Sparse image reconstruction on the sphere: implications of a new sampling theorem

Cited by in corpus (4)

• Optimizing Sensing Matrices for Spherical Near-Field Antenna Measurements
• Compressive Sensing with Wigner $D$-functions on Subsets of the Sphere
• On Grid Compressive Sampling for Spherical Field Measurements in Acoustics
• Tight bounds on the mutual coherence of sensing matrices for Wigner D-functions on regular grids