Greedy Algorithms for Hybrid Compressed Sensing

Compressed sensing (CS) is a technique which uses fewer measurements than dictated by the Nyquist sampling theorem. The traditional CS with linear measurements achieves effective recovery, but it suffers from large bit consumption due to the precision required by those measurements. Then, the one-bit CS with binary measurements is proposed to save the bit budget, but it is infeasible when the energy information of signals is not available as a prior knowledge. Subsequently, the hybrid CS which combines traditional CS and one-bit CS appears, striking a balance between the pros and cons of both types of CS. Given that one-bit CS is optimal for the direction estimation of signals under noise with a fixed bit budget and that traditional CS is able to provide residue information and estimated signals, we focus on the design of greedy algorithms, which consist of the main steps of support detection and recovered signal updates, for hybrid CS in this paper. We propose two greedy algorithms for hybrid CS, with traditional CS offering signal estimates and updated residues, which help one-bit CS detect the support iteratively. Then, we provide a theoretical analysis of the error bound between the normalized original signal and the normalized estimated signal. Numerical results demonstrate the efficacy of the proposed greedy algorithms for hybrid CS in noisy environments.