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Circuits Syst Signal Process (2013) 32:23712383
DOI 10.1007/s00034-013-9574-7
Compressed Sensing for Sparse Error Correcting Model
Yuli Fu Qiheng Zhang Shengli Xie
Received: 22 February 2012 / Revised: 12 March 2013 / Published online: 27 March 2013 Springer Science+Business Media New York 2013
Abstract Compressed sensing (CS)-based cross-and-bouquet (CAB) model was proposed by J. Wright et al. to reduce the complexity of sparse error correcting. For the sake of leading to better performance of CS-based decoding for the CAB model, an algorithm is proposed in this paper for constructing a well-designed projection matrix to minimize the average measures of mutual coherence. One was proposed by M. Elad. Another is dened in this paper for higher dimensional cases. Using the equivalent dictionary, the dimensionality is reduced. Also high-dimensional singular value decomposition (SVD) is avoided in the procedure of constructing a well-designed projection matrix. The high-dimensional CAB model of sparse error correcting can be solved by the proposed algorithm without computational difculty. The validity of the proposed method is illustrated by decoding experiments in high-dimensional cases.
Keywords Compressed sensing Sparse error correcting Cross-and-bouquet
model Particle swarm optimization
1 Introduction
Modern data processing applications in signal/image processing, web search and ranking, and bioinformatics are increasingly characterized by large quantities of very
Y. Fu ( ) Q. Zhang
School of Electronic and Information Engineering, South China University of Technology, Guangzhou, 510640, Chinae-mail: mailto:[email protected]
Web End [email protected]
Q. Zhange-mail: mailto:[email protected]
Web End [email protected]
S. XieSchool of Automation, Guangdong University of Technology, Guangzhou, 510006, China e-mail: [email protected]
2372 Circuits Syst Signal Process (2013) 32:23712383
high-dimensional data. As datasets grow larger, however, methods of data collection have necessarily become less controlled, introducing corruption, outliers, and missing data [15]. High-dimensional error correction is one method to solve the problems.
Consider the error correcting problem in coding theory [3], a natural error correcting problem with real valued input/output. The problem is to recover an input vector Rk0 from following n-dimensional corrupted measure:y = A + e, (1.1)
where the coding matrix A Rnk0 satises n k0 and the error vector e Rn is
unknown. The constraint on the error usually is e 0 := |{i : ei = 0}| n for some
constant 1 > > 0. Obviously, the error e is a sparse vector...