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

@article{Bangun2020SensingMD, title={Sensing Matrix Design and Sparse Recovery on the Sphere and the Rotation Group}, author={Arya Bangun and Arash Behboodi and Rudolf Mathar}, journal={IEEE Transactions on Signal Processing}, year={2020}, volume={68}, pages={1439-1454} }

In this article, 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…

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## References

SHOWING 1-10 OF 70 REFERENCES

Sparse signal recovery on the sphere: Optimizing the sensing matrix through sampling

- Computer Science2012 6th International Conference on Signal Processing and Communication Systems
- 2012

This work proposes a method for constructing a spherical harmonic sensing matrix that can be used to effectively recover a sparse signal on the sphere from limited measurements and shows that the success rate in near exact recovery of a sparse coefficient vector with the spiral scheme is superior to that of the regular scheme over a range of SNRs.

Coherence Bounds for Sensing Matrices in Spherical Harmonics Expansion

- Mathematics2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
- 2018

It will be shown that for a class of sampling patterns, the mutual coherence would be at its maximum, yielding the worst performance, and the sampling strategy is proposed to achieve the derived lower bound.

On the Minimum Number of Samples for Sparse Recovery in Spherical Antenna Near-Field Measurements

- Computer ScienceIEEE Transactions on Antennas and Propagation
- 2019

The resulting modified phase transition diagrams (PTDs) show that a reconstruction by the quadratically constraint basis pursuit strategy is sufficiently stable and robust for practical purposes to reduce the number of measurement samples with predictable accuracy, when the sparsity level is known.

Compressed Sensing Matrices From Fourier Matrices

- Computer Science, MathematicsIEEE Transactions on Information Theory
- 2015

Using Katz' character sum estimation, a deterministic procedure to select rows from a Fourier matrix to form a good CS matrix for sparse recovery is designed, which yields an approximately mutually unbiased bases from Fourier matrices which is of particular interest to quantum information theory.

Sparse recovery on sphere via probabilistic compressed sensing

- Computer Science2014 IEEE Workshop on Statistical Signal Processing (SSP)
- 2014

This paper incorporates a preconditioning technique into the probabilistic approach to derive a slightly improved bound on the order of measurements for accurate recovery of spherical harmonic expansions.

The Computational Complexity of the Restricted Isometry Property, the Nullspace Property, and Related Concepts in Compressed Sensing

- Computer ScienceIEEE Transactions on Information Theory
- 2014

It is confirmed by showing that for a given matrix A and positive integer k, computing the best constants for which the RIP or NSP hold is, in general, NP-hard.

On sparse reconstruction from Fourier and Gaussian measurements

- Computer Science, Mathematics
- 2008

This paper improves upon best‐known guarantees for exact reconstruction of a sparse signal f from a small universal sample of Fourier measurements by showing that there exists a set of frequencies Ω such that one can exactly reconstruct every r‐sparse signal f of length n from its frequencies in Ω, using the convex relaxation.

Optimized Projections for Compressed Sensing

- Computer ScienceIEEE Transactions on Signal Processing
- 2007

This paper considers the optimization of compressed sensing projections, and targets an average measure of the mutual coherence of the effective dictionary, and shows that this leads to better CS reconstruction performance.

Coherence Optimization and Best Complex Antipodal Spherical Codes

- Computer ScienceIEEE Transactions on Signal Processing
- 2015

Using methods used to find best spherical codes in the real-valued Euclidean space, the proposed approach aims to find BCASCs, and thereby, a complex-valued vector set with minimal coherence.

Message-passing algorithms for compressed sensing

- Computer ScienceProceedings of the National Academy of Sciences
- 2009

A simple costless modification to iterative thresholding is introduced making the sparsity–undersampling tradeoff of the new algorithms equivalent to that of the corresponding convex optimization procedures, inspired by belief propagation in graphical models.