Non-uniform recovery guarantees for binary measurements and infinite-dimensional compressed sensing

TL;DR

This paper establishes non-uniform recovery guarantees for infinite-dimensional compressed sensing using Walsh measurements, demonstrating effectiveness comparable to Fourier-based methods when sampling strategies align with signal sparsity.

Contribution

It provides the first theoretical non-uniform recovery guarantees for Walsh transform-based compressed sensing with wavelet reconstruction, bridging a significant gap in the field.

Findings

Walsh measurements can achieve similar recovery guarantees as Fourier measurements with structured sampling.

The effectiveness depends on the structured sparsity of the signal and sampling strategy.

Differences in asymptotic behavior relate to wavelet properties, not sampling patterns.

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 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.