混合压缩感知的贪婪算法
Greedy Algorithms for Hybrid Compressed Sensing
AI总结：
本文针对混合压缩感知设计两种贪婪算法，利用一比特 CS 检测支撑集、传统 CS 更新残差与信号估计，并通过定理证明和仿真验证其在有限比特预算噪声环境中的有效性。
摘要：
压缩感知（CS）是一种使用比 Nyquist 采样定理所要求数量更少的测量值的技术。采用线性测量的传统 CS 可实现高效的恢复性能，但由于这些测量值占用巨大存储空间，其面临比特消耗较大的问题。随后，采用二元测量的一比特 CS 被提出以节省比特预算，但当信号的能量信息无法作为先验知识获得时，该方法并不可行。之后，结合传统 CS 与一比特 CS 的混合 CS 出现，在两类 CS 的优缺点之间取得平衡。考虑到在固定比特预算下，一比特 CS 对于噪声环境中信号的方向估计是最优的，而传统 CS 能够提供残差信息和估计信号，本文聚焦于为混合 CS 设计贪婪算法，这些算法主要由支撑集检测和恢复信号更新两个步骤组成。我们首先提出一个关于稀疏信号随机均匀剖分的定理，以进一步研究一比特 CS 的性质。随后，我们为混合 CS 提出两种贪婪算法，其中一比特 CS 负责支撑集检测，传统 CS 提供更新后的残差和信号估计。对于所提出的每种算法，我们都给出相应定理及证明，以从理论上分析其能力。仿真结果证明了所提出的贪婪算法在噪声环境和有限比特预算下的有效性。
英文摘要：
Compressed sensing (CS) is a technique which uses fewer measurements than dictated by the Nyquist sampling theorem. The traditional CS with linear measurements achieves efficient recovery performances, but it suffers from the large bit consumption due to the huge storage occupied by those measurements. Then, the one-bit CS with binary measurements is proposed and saves 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 the traditional CS and one-bit CS appears, striking a balance between the pros and cons of both types of CS. Considering the fact that the one-bit CS is optimal for the direction estimation of signals under noise with a fixed bit budget and that the 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 update, for the hybrid CS in this paper. We first propose a theorem on the random uniform tessellations for sparse signals to further investigate the properties of one-bit CS. Afterwards, we propose two greedy algorithms for the hybrid CS, with the one-bit CS responsible for support detection and traditional CS offering updated residues and signal estimates. For each of the proposed algorithms, we provide the corresponding theorem with proof to analyze their capabilities theoretically. Simulation results have demonstrated the efficacy of the proposed greedy algorithms under a limited bit budget in noisy environments.
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