Compressed Sensing and Parallel Acquisition

TL;DR

This paper explores the combination of compressed sensing with parallel multi-sensor acquisition, providing theoretical guarantees that reduce measurements per sensor and demonstrating these results through numerical phase transition experiments.

Contribution

It establishes new recovery guarantees for parallel compressed sensing systems, including both distinct and identical sampling scenarios, with broader conditions for the sparse in levels model.

Findings

Recovery guarantees show measurement reduction proportional to the number of sensors.

Broader conditions on sensor profile matrices enable optimal recovery for complex signal models.

Numerical phase transition results validate theoretical predictions across various sensor environments.

Abstract

Parallel acquisition systems arise in various applications in order to moderate problems caused by insufficient measurements in single-sensor systems. These systems allow simultaneous data acquisition in multiple sensors, thus alleviating such problems by providing more overall measurements. In this work we consider the combination of compressed sensing with parallel acquisition. We establish the theoretical improvements of such systems by providing recovery guarantees for which, subject to appropriate conditions, the number of measurements required per sensor decreases linearly with the total number of sensors. Throughout, we consider two different sampling scenarios -- distinct (corresponding to independent sampling in each sensor) and identical (corresponding to dependent sampling between sensors) -- and a general mathematical framework that allows for a wide range of sensing matrices (e.g., subgaussian random matrices, subsampled isometries, random convolutions and random Toeplitz matrices). We also consider not just the standard sparse signal model, but also the so-called sparse in levels signal model. This model includes both sparse and distributed signals and clustered sparse signals. As our results show, optimal recovery guarantees for both distinct and identical sampling are possible under much broader conditions on the so-called sensor profile matrices (which characterize environmental conditions between a source and the sensors) for the sparse in levels model than for the sparse model. To verify our recovery guarantees we provide numerical results showing phase transitions for a number of different multi-sensor environments.