Sparse signal recovery from modulo observations

Viraj Shah,Chinmay Hegde

doi:10.1186/s13634-021-00722-w

Abstract

We consider the problem of reconstructing a signal from under-determined modulo observations (or measurements). This observation model is inspired by a relatively new imaging mechanism called modulo imaging, which can be used to extend the dynamic range of imaging systems; variations of this model have also been studied under the category of phase unwrapping. Signal reconstruction in the under-determined regime with modulo observations is a challenging ill-posed problem, and existing reconstruction methods cannot be used directly. In this paper, we propose a novel approach to solving the signal recovery problem under sparsity constraints for the special case to modulo folding limited to two periods. We show that given a sufficient number of measurements, our algorithm perfectly recovers the underlying signal. We also provide experiments validating our approach on toy signal and image data and demonstrate its promising performance.

Highlights

IntroductionThe problem of reconstructing a signal (or image) from (possibly) nonlinear observations is a principal challenge in signal acquisition and imaging systems
The problem of reconstructing a signal from nonlinear observations is a principal challenge in signal acquisition and imaging systems
Our focus in this paper is the problem of signal reconstruction from modulo measurements, where the modulo operation with respect to a positive real valued parameter R returns the remainder after division by R

Summary

Introduction

The problem of reconstructing a signal (or image) from (possibly) nonlinear observations is a principal challenge in signal acquisition and imaging systems. This does not reflect practice, but we will see that such a variation of the modulo function already inherits much of the challenging aspects of the original recovery problem This simplification requires that the value of dynamic range parameter R should be finite, but large enough that most of the measurements fall within the interval [ −R, R]. Cucuringu and Tyagi [16] formulate and solve a QCQP problem with non-convex constraints for denoising the modulo-1 samples of the unknown function along with providing a least-square-based modulo recovery algorithm Both these methods rely on the smoothness of the band-limited function as a prior structure on the signal, and as such, it is unclear how to extend their use to more complex modeling priors (such as sparsity in a given basis). We vary the sampling setup along both the amplitude and time dimensions by incorporating sparsity in our model, which enables us to work with a non-uniform sampling grid (random measurements) and achieve a provable sub-Nyquist sample complexity

Preliminaries

Alternating minimization

Conclusions