如何解决基不匹配问题:从原子范数到无网格压缩感知

压缩感知理论能够以远低于经典Nyquist速率进行采样, 采用非自适应线性投影获得了保留信号有用信息的少量观测点, 并通过求解最优化问题精确重构原始信号.压缩感知理论大大缓解了信号采样、存储和传输的巨大压力, 在计算机科学、电子工程和信号处理等领域具有广阔的应用前景.信号的稀疏表示是对信号进行压缩采样和重构的前提, 即假设信号在某个变换基(傅里叶基、小波基等)下是稀疏的, 这些基可以看作是用于描述信号参数空间的有限离散字典.然而在如雷达、阵列信号处理、通信等领域的应用中, 信号的参数空间是连续的, 在假定的离散变换基下并不稀疏, 这种基不匹配问题会严重影响信号重构精度.本文首先介绍了基不匹配产生的原因及其对重构精度的影响, 接着从原子范数出发, 综述了无网格压缩感知的理论框架和关键技术问题, 着重介绍了一维和多维无网格压缩感知的最新研究进展, 最后对其在信号处理等领域的应用进行了探讨.

How to Overcome Basis Mismatch: From Atomic Norm to Gridless Compressive Sensing

Compressive sensing (CS) presents a new method to capture and represent compressible signals at a rate significantly below the Nyquist rate which employs non-adaptive linear projections to preserve the structure of the signal. The nonlinear optimization process is then able to recover the signal from very few measurements. In recent years, CS has attracted considerable attention in areas of computer science, electrical engineering and signal processing since it may be possible to reduce the cost of sampling, storage and transmission. CS builds upon the fundamental fact that we can represent many signals using only a few nonzero coefficients in finite discrete bases or dictionaries. For many problems, signals are sparse in a known basis which is usually Fourier basis or wavelet basis. However, signals encountered in applications such as radar, array processing and communication are specified by parameters in a continuous domain and consequently the mismatch between the assumed and the actual bases for sparsity results in the reconstruction inaccuracy. In this paper, we begin with an introduction of basis mismatch and its impact on reconstruction. Then, we summarize the underlying framework as well as key techniques of gridless CS and provide an up-to-date review of the researches on D-dimensional (D≥2) gridless CS. We conclude with a discussion of applying gridless CS in the field of signal processing.