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Non-uniform Recovery Guarantees for Binary Measurements and Infinite-Dimensional Compressed Sensing

Published version
Peer-reviewed

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Authors

Thesing, L. 
Hansen, A. C. 

Abstract

Abstract: Due to the many applications in Magnetic Resonance Imaging (MRI), Nuclear Magnetic Resonance (NMR), radio interferometry, helium atom scattering etc., the theory of compressed sensing with Fourier transform measurements has reached a mature level. However, for binary measurements via the Walsh transform, the theory has long been merely non-existent, despite the large number of applications such as fluorescence microscopy, single pixel cameras, lensless cameras, compressive holography, laser-based failure-analysis etc. Binary measurements are a mainstay in signal and image processing and can be modelled by the Walsh transform and Walsh series that are binary cousins of the respective Fourier counterparts. We help bridging the theoretical gap by providing non-uniform recovery guarantees for infinite-dimensional compressed sensing with Walsh samples and wavelet reconstruction. The theoretical results demonstrate that compressed sensing with Walsh samples, as long as the sampling strategy is highly structured and follows the structured sparsity of the signal, is as effective as in the Fourier case. However, there is a fundamental difference in the asymptotic results when the smoothness and vanishing moments of the wavelet increase. In the Fourier case, this changes the optimal sampling patterns, whereas this is not the case in the Walsh setting.

Description

Keywords

Article, Sampling theory, Compressed sensing, Structured sparsity, Wavelets, Walsh functions, Hadamard transform, 42C10, 42C40, 94A12, 94A20

Journal Title

Journal of Fourier Analysis and Applications

Conference Name

Journal ISSN

1069-5869
1531-5851

Volume Title

27

Publisher

Springer US