GBCS: A two-step compressive sensing reconstruction based on group testing and basis pursuit

Compressive sensing (CS) reconstruction algorithms can recover a signal from its undersampled random projections given that the signal is sparse or has a sparse representation in some appropriate transform domain. These algorithms are grouped into three main categories: group-testing based, greedy methods, and linear programming based. The two former have lower complexities and lower reconstruction performance compared to the latter. In this paper, we propose group testing basis pursuit CS (GBCS), which exploits the low complexity of the former category and the accuracy of the latter. First, we design an efficient group-testing based CS reconstruction algorithm and then propose to integrate it with a regular basis pursuit (BP) CS reconstruction. We design and analyze GBCS and show that it surpasses existing algorithms for noiseless measurements and absolutely sparse signals. Further, we show that if the number of random projections is large enough our designed group-testing reconstruction fully recovers the signal and the need for BP is eliminated.