基于混沌序列和RIPless理论的循环压缩测量矩阵的构造

郭静波; 汪韧

doi:10.7498/aps.63.198402

摘要

压缩测量矩阵的构造是压缩感知的核心工作之一. 循环矩阵由于其对应离散卷积且具有快速算法被广泛应用于压缩测量矩阵. 本文力图将混沌的优点和循环矩阵的优点相结合，提出基于混沌序列的循环压缩测量矩阵. 混沌循环测量矩阵元素的产生仅需要利用混沌的内在确定性，即利用混沌映射公式、初始值以及一定的采样间隔就可以产生独立同分布的随机序列；同时混沌序列的外在随机性可以满足压缩测量矩阵对随机性的要求. 本文给出了使用Cat混沌映射时混沌循环测量矩阵的构造方法，以及该矩阵RIPless特性的检验. 研究了采用构造的混沌循环测量矩阵对一维和二维信号进行压缩测量的效果，并与采用传统的循环测量矩阵的效果进行了比较. 结果表明，混沌循环测量矩阵对于一维和二维信号都具有很好的恢复效果，且对二维信号的恢复性能要优于已有的循环矩阵. 从相图角度分析了混沌循环测量矩阵优于已有的循环矩阵的机理，指出混沌的内在确定性和外在随机性的有机结合是所构造的混沌循环测量矩阵性能优于传统的循环矩阵的本质性原因.

关键词:

Abstract

Construction of a compressive measurement matrix is one of the key technologies of compressive sensing. A circulant matrix corresponds to the discrete convolutions with a high-speed algorithm, which has been widely used in compressive sensing. This paper combines the advantages of chaotic sequence with circulant matrix to propose a circulant compressive measurement matrix based on the chaotic sequence. The elements of a chaotic circulant measurement matrix are generated by taking advantage of the chaotic internal certainty, i.e. the independent identically distributed randomness sequence can be produced by the chaotic mapping formula using the initial value and a certain sampling distance. At the same time, the external randomness of chaotic sequence can satisfy the stochastic requirements of compressive measurement matrix. This paper presents the method of constructing chaotic circulant measurement matrix using a Cat chaotic map and the test method for RIPless property of the matrix. Measurement results of one-dimensional and two-dimensional signals using the chaotic circulant measurement matrix are studied and are compared with the results of conventional circulant measurement matrix. It can be shown that the chaotic circulant measurement matrix has good recovery results for both one-dimensional and two-dimensional signals. Moreover, it may get better results than the traditional matrix for the two-dimensional signal. From the point of view of phase diagram, the essential reason of chaotic circulant measurement matrix outperforms the conventional one is its integration of internal certainty with the external randomness of the chaotic sequence.

Keywords:

作者及机构信息

郭静波,
汪韧

1.
清华大学电机工程与应用电子技术系, 电力系统国家重点实验室, 北京 100084

基金项目: 国家自然科学基金（批准号：51277100）资助的课题.

Authors and contacts

1.
State Key Lab, Department of Electrical Engineering, Tsinghua University, Beijing 100084, China

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 51277100).

参考文献

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施引文献

[1]	Donoho D L 2006 IEEE Trans. Inform. Theory 52 1289
[2]	Candès E J, Romberg J, Tao T 2006 IEEE Trans. Inform. Theory 52 489
[3]	Gribonval R, Nielsen M 2003 IEEE Trans. Inform. Theory 49 3320
[4]	Candès E J, Wakin M B 2008 IEEE Sig. Proc. Mag. 25 21
[5]	Candès E J 2008 Comptes. Rendus Math. 346 589
[6]	Baraniuk R, Davenport M, DeVore R, Wakin M 2008 Constructive Approx. 23 253
[7]	Candès E J, Plan Y 2011 IEEE Trans. Inform. Theory 57 7235
[8]	Chen S S, Donoho D L, Saunders M A 2001 SIAM 43 129
[9]	Needell D 2009 Topics in compressed sensing Ph. D. Dissertation (California: University of California)
[10]	Zhang H M, Wang L Y, Yan B, Li L, Xi X Q, Lu L Z 2013 Chin. Phys. B 22 078701
[11]	Feng B C, Fang S, Zhang L G, Li H, Tong J J, Li W Q 2013 Acta Phys. Sin. 62 112901(in Chinese) [冯丙辰, 方晟, 张立国, 李红, 童节娟, 李文茜 2013 62 112901]
[12]	Yin W, Morgan S, Yang J F, Zhang Y 2010 Visual Communications and Image Processing (International Society for Optics and Photonics) p77440K
[13]	Romberg J 2009 SIAM. J. Imaging Sci. 2 1098
[14]	Guo J B, Xu X Z, Shi Q H, Hu T H 2013 Acta Phys. Sin. 62 110508(in Chinese) [郭静波, 徐新智, 史启航, 胡铁华 2013 62 110508]
[15]	Yang J, Zhang Y 2011 SIAM. J. Sci. Comp. 33 250
[16]	Chen G, Mao Y, Chui C K 2004 Chaos Solitons & Fractals 21 749
[17]	Badea B, Vlad A 2006 Computational Science and Its Applications (Berlin: Springer Berlin Heidelberg) p1166
[18]	Hoeffding W 1963 J. American Stat. Assoc. 58 13

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