Coherence Bounds for Sensing Matrices in Spherical Harmonics Expansion

@article{Bangun2018CoherenceBF,
  title={Coherence Bounds for Sensing Matrices in Spherical Harmonics Expansion},
  author={Arya Bangun and Arash Behboodi and Rudolf Mathar},
  journal={2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)},
  year={2018},
  pages={4634-4638}
}
The mutual coherence provides a basis for deriving recovery guarantees in compressed sensing. In this paper, the mutual coherence of spherical harmonics sensing matrices is examined for a class of sensing patterns common in practice and is used as a figure of merit for designing sensing matrices. We will show that for each sampling pattern, the coherence is lower bounded by the inner product of two Legendre polynomials with different degrees. In some practical situation, it is desirable to have… 

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References

SHOWING 1-10 OF 30 REFERENCES
Optimized Projections for Compressed Sensing
  • Michael Elad
  • Computer Science
    IEEE Transactions on Signal Processing
  • 2007
TLDR
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.
Designing Incoherent Frames Through Convex Techniques for Optimized Compressed Sensing
TLDR
This paper gives a detailed analysis of the optimization problem at the heart of this approach and proposes a new method that substantially outperforms the initial approach and all current methods in the literature for all types of frames, with low and high redundancy.
Weighted Eigenfunction Estimates with Applications to Compressed Sensing
TLDR
Using tools from semiclassical analysis, weighted L^\infty estimates for eigenfunctions of strictly convex surfaces of revolution are given and imply that any function having an s-sparse expansion in the first N spherical harmonics can be efficiently recovered from its values at m > s N^(1/6) log^4(N) sampling points.
The Computational Complexity of the Restricted Isometry Property, the Nullspace Property, and Related Concepts in Compressed Sensing
TLDR
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.
Coherence Optimization and Best Complex Antipodal Spherical Codes
TLDR
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.
Fast Antenna Far-Field Characterization via Sparse Spherical Harmonic Expansion
A procedure is proposed to significantly reduce the amount of time to characterize 3-D antenna far-field patterns. The measured far field is expanded into spherical harmonics, and a sparse recovery
Analysis and design of spherical microphone arrays
  • B. Rafaely
  • Physics
    IEEE Transactions on Speech and Audio Processing
  • 2005
TLDR
Alternative spatial sampling schemes for the positioning of microphones on a sphere are presented, and the errors introduced by finite number of microphones, spatial aliasing, inaccuracies in microphone positioning, and measurement noise are investigated both theoretically and by using simulations.
Grassmannian beamforming for multiple-input multiple-output wireless systems
TLDR
A codebook design method for quantized versions of maximum ratio transmission, equal gain transmission, and generalized selection diversity with maximum ratio combining at the receiver is presented and systems using the beamforming codebooks are shown to have a diversity order of the product of the number of transmit and thenumber of receive antennas.
A Mathematical Introduction to Compressive Sensing
TLDR
A Mathematical Introduction to Compressive Sensing gives a detailed account of the core theory upon which the field is build and serves as a reliable resource for practitioners and researchers in these disciplines who want to acquire a careful understanding of the subject.
Can matrix coherence be efficiently and accurately estimated?
TLDR
A novel sampling-based algorithm for estimating coherence is introduced, associated estimation guarantees are presented and the results of extensive experiments for coherence estimation are reported, revealing the extent to which coherence assumptions made in a number of recent machine learning publications are testable.
...
...