Compressed Sensing and Parallel Acquisition

Parallel acquisition systems arise in various applications 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 paper, we consider the combination of compressed sensing with parallel acquisition. We establish the theoretical improvements of such systems by providing nonuniform 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 (i.e., independent sampling in each sensor) and identical (i.e., dependent sampling between sensors)-and a general mathematical framework that allows for a wide range of sensing matrices. We also consider not just the standard sparse signal model, but also the so-called sparse in levels signal model. 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 different multi-sensor environments.