压缩感知及应用

Theory and Applicaton of Compressive Sensing

传统的信号采样必须遵循香农采样定理,产生的大量数据造成了存储空间的浪费。压缩感知(CS)提出一种新的采样理论,它能够以远低于奈奎斯特采样速率采样信号。压缩感知的基本论点是如果信号具有稀疏性,可投影到一个与变换基不相关的随机矩阵并获得远少于信号长度的测量值,再通过求解优化问题,精确重构信号。本文详述了压缩感知的基本理论,压缩感知适用的基本条件:稀疏性和非相干性,测量矩阵设计要求,及重构算法的RIP准则,并介绍了压缩感知的应用及仿真。仿真结果表明当采样个数大于K×log(N/K),就能将N维信号稳定地重建出来。

Conventional approaches to sampling signals follow Shannon principle. It take great costs on data storage. In this paper, the theory of Compressive sensing is introduced. Compressive sensing provides a new sampling theory to sample signal below the Nyquist rate. If signal or image is sparse in some orthonormal basis , signal or image can be recovered from small number of measurement using an optimization process. The structure of the signal is preserved in the measurement and the measure matrix is incoherent with the or- thonormal basis. CS relies on two principles ： sparsity and incoherence. RI Pprinciple is the precondiction of designing reconstruction algorithm. The application of CS theory are introduced and the simulation is illustrated in details. The simulation show that the signal can be reconstructed stablely when the number of samples is larger than K × log（N/K）.