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

L Thesing,A C Hansen

doi:10.1007/s00041-021-09813-6

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.

Highlights

Since Shannon’s classical sampling theorem [59,64], sampling theory has been a widely studied field in signal and image processing
The applications divide themselves in three different groups: those that are modelled by Fourier measurements, like Magnetic Resonance Imaging (MRI) [48], those that are based on the Radon transform, as in CT imaging [22,58], and those that are represented by binary measurements, which are named above
In this paper we focus on the reconstruction of one-dimensional signals from binary measurements, which can be modelled as inner products of the signal with functions that take only values in {0, 1}

Summary

Introduction

Since Shannon’s classical sampling theorem [59,64], sampling theory has been a widely studied field in signal and image processing. Infinite-dimensional compressed sensing [7,9,18,43,44,56,57] is part of this rich theory and offers a method that allows for infinite-dimensional signals to be recovered from undersampled linear measurements This gives a non-linear alternative to other methods like generalized sampling [3,4,6,8,39,41,49] and the Parametrized-Background Data-Weak (PBDW)-method [15,16,27,50,51,52] that reconstruct infinite-dimensional objects from linear measurement. In this paper we consider binary measurements and provide the first non-uniform recovery guarantees in one dimension for infinite-dimensional compressed sensing with the reconstruction with boundary corrected Daubechies wavelets

Objectives

Results

Conclusion