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Circulant measurement matrix has been widely used in compressive sensing because of its high-speed discrete convolution algorithm. The typical work of optimizing circulant measurement matrix was introduced by Wotao Yin in Reference [16]. Motivated by his work, the construction of circulant measurement matrix in this paper is explored from the view point of generating elements' amplitudes and phases; and the optimal construction procedures are proposed based on alternately optimizing amplitudes in conjunction with chaotic stochastic phases of the matrix generating elements. The main idea of this paper is based on two innovations: The first one is to reduce the mutual coherence between column vectors of equivalent dictionary by alternately optimizing the generating elements' amplitudes, thus improving the recovery performance of the circulant measurement matrix. From the different expressions of the circulant matrixes of one-dimensional and two-dimensional signal, by setting the Welch bound for the coefficient of mutual coherence between the column vectors of equivalent dictionary as the approximation objective, two novel unified mathematical models are derived from the optimizing function for generating elements' amplitudes of the two different matrixes. Optimal solutions for generating elements' amplitudes are gained by alternately optimizing method. The second innovation is to construct the generating elements' phases of circulant measurement matrix by utilization of a chaotic sequence with independent property. The chaotic stochastic phase of the circulant measurement matrix generating elements are generated by taking advantage of the chaotic internal certainty, which means an independent identically-distributed randomness sequence can be produced by the chaotic map with the initial value at certain sampling distance. At the same time, the external randomness of chaotic sequence can satisfy the stochastic requirement of circulant measurement matrix. This paper presents the method of constructing chaotic stochastic phase using Cat chaotic map. Experimental results of one-dimensional and two-dimensional signals in the optimized circulant measurement matrix are studied in this paper, which has a better performance as compared with the results of conventional circulant measurement matrixes, such as Gaussian circulant matrix and optimized circulant matrix proposed by Wotao Yin. The column vectors of equivalent dictionary in the optimized circulant measurement matrix have lower mutual coherence, this is the essence of the superiority of the optimized circulant matrix.
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Keywords:
- compressive sensing /
- circulant matrix /
- alternating optimizing /
- chaotic sequence
[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] Chen S S, Donoho D L, Saunders M A 2001 SIAM 43 129
[4] Needell D 2009 Ph. D. Dissertation (California: University of California)
[5] Zhang H M, Wang L Y, Yan B, Li L, Xi X Q, Lu L Z 2013 Chin. Phys. B 22 078701
[6] 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]
[7] Yin W, Morgan S, Yang J F, Zhang Y 2010 Visual Communications and Image Processing (International Society for Optics and Photonics) p77440K
[8] Donoho D L, Elad M 2003 Proc. Natl. Acad. Sci. USA 100 2197
[9] Duarte-Carvajalino J M, Sapiro G 2009 IEEE Trans. Image Process. 18 1395
[10] Abolghasemi V, Ferdowsi S, Sanei S 2012 Signal Process. 92 999
[11] Xu J P, Pi Y M, Cao Z J 2010 EURASIP J. Adv. Signal Process. 2010 43
[12] Li G, Zhu Z H, Yang D H, Chang L P, Bai H 2013 IEEE Trans. Signal Process. 6 2887
[13] Rauhut H, Romberg J, Tropp J A 2012 Applied and Computational Harmonic Analysis 32 242
[14] Romberg J 2009 16th IEEE International Conference on Digital Signal Processing p1
[15] Romberg J 2009 SIAM. J. Imaging Sci. 2 1098
[16] Yin W, Osher S, Xu Y 2014 to appear in Inverse Problems and Imaging
[17] 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]
[18] Candès E J 2008 Comptes. Rendus Math. 346 589
[19] Donoho D L, Elad M 2003 Proc. Natl. Acad. Sci. USA 100 2197
[20] Tropp J A 2004 IEEE Trans. Inform. Theory 50 2231
[21] Gribonval R, Nielsen M 2003 IEEE Trans. Inform. Theory 49 3320
[22] Strohmer T, Heath R W 2003 Appl. Comput. Harmon. Anal. 14 257
[23] Tropp J A, Dhillon I S, Heath R W, Strohmer T 2005 IEEE Trans. Inform. Theory 51 188
[24] Aharon M, Elad M, Bruckstein A 2006 IEEE Trans. Signal Process. 54 4311
[25] Arnold V I, Avez A 1968 Ergodic problems of classical mechanics (New York: Benjamin) pp5-7
[26] Chen G, Mao Y, Chui C K 2004 Chaos Solitons & Fractals 21 749
[27] Guo J B, Wang R 2014 Acta Phys. Sin. 63 198402 (in Chinese) [郭静波, 汪韧 2014 63 198402]
[28] Yang J, Zhang Y 2011 SIAM. J. Sci. Comp. 33 250
[29] Martin D, Fowlkes C, Tal D, Malik J 2001 Proceedings 8th IEEE International Conference on Computer Vision July 2001 2 p416
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[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] Chen S S, Donoho D L, Saunders M A 2001 SIAM 43 129
[4] Needell D 2009 Ph. D. Dissertation (California: University of California)
[5] Zhang H M, Wang L Y, Yan B, Li L, Xi X Q, Lu L Z 2013 Chin. Phys. B 22 078701
[6] 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]
[7] Yin W, Morgan S, Yang J F, Zhang Y 2010 Visual Communications and Image Processing (International Society for Optics and Photonics) p77440K
[8] Donoho D L, Elad M 2003 Proc. Natl. Acad. Sci. USA 100 2197
[9] Duarte-Carvajalino J M, Sapiro G 2009 IEEE Trans. Image Process. 18 1395
[10] Abolghasemi V, Ferdowsi S, Sanei S 2012 Signal Process. 92 999
[11] Xu J P, Pi Y M, Cao Z J 2010 EURASIP J. Adv. Signal Process. 2010 43
[12] Li G, Zhu Z H, Yang D H, Chang L P, Bai H 2013 IEEE Trans. Signal Process. 6 2887
[13] Rauhut H, Romberg J, Tropp J A 2012 Applied and Computational Harmonic Analysis 32 242
[14] Romberg J 2009 16th IEEE International Conference on Digital Signal Processing p1
[15] Romberg J 2009 SIAM. J. Imaging Sci. 2 1098
[16] Yin W, Osher S, Xu Y 2014 to appear in Inverse Problems and Imaging
[17] 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]
[18] Candès E J 2008 Comptes. Rendus Math. 346 589
[19] Donoho D L, Elad M 2003 Proc. Natl. Acad. Sci. USA 100 2197
[20] Tropp J A 2004 IEEE Trans. Inform. Theory 50 2231
[21] Gribonval R, Nielsen M 2003 IEEE Trans. Inform. Theory 49 3320
[22] Strohmer T, Heath R W 2003 Appl. Comput. Harmon. Anal. 14 257
[23] Tropp J A, Dhillon I S, Heath R W, Strohmer T 2005 IEEE Trans. Inform. Theory 51 188
[24] Aharon M, Elad M, Bruckstein A 2006 IEEE Trans. Signal Process. 54 4311
[25] Arnold V I, Avez A 1968 Ergodic problems of classical mechanics (New York: Benjamin) pp5-7
[26] Chen G, Mao Y, Chui C K 2004 Chaos Solitons & Fractals 21 749
[27] Guo J B, Wang R 2014 Acta Phys. Sin. 63 198402 (in Chinese) [郭静波, 汪韧 2014 63 198402]
[28] Yang J, Zhang Y 2011 SIAM. J. Sci. Comp. 33 250
[29] Martin D, Fowlkes C, Tal D, Malik J 2001 Proceedings 8th IEEE International Conference on Computer Vision July 2001 2 p416
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