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Chaotic Compressive Sampling Matrix: Where Sensing Architecture Meets Sinusoidal Iterator

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Abstract

A key issue in compressive sampling is to construct a sensing matrix that is stable and easy to design in hardware. In this paper, a novel chaotic sensing matrix is proposed by establishing a connection between sensing architecture and chaotic sinusoidal iterator. Based on a chaining method with the volumetric argument, we show that the proposed sinusoidal chaotic sensing matrix yields the well-known restricted isometry property. Numerical experiments with time-sparse signals and natural images are reported to compare the sampling performance between chaotic sensing constructions and random ones. The obtained results indicate that the proposed chaotic matrix is always able to exhibit similar or superior performance with respect to its counterparts. Moreover, the proposed construction can be easily implemented on the field-programmable gate array environment. Our framework may guide other researchers to extend the proposed method for other chaotic systems as well.

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Notes

  1. This equation is only a simplified version of \(w_{i+1}= \alpha {w_{i}}^2 \sin (\pi w_{i})\). More information can be found in [32].

References

  1. A. Amini, F. Marvasti, Deterministic construction of binary, bipolar, and ternary compressed sensing matrices. IEEE Trans. Inf. Theory 57(4), 2360–2370 (2011)

    MathSciNet  MATH  Google Scholar 

  2. R. Baraniuk, M. Davenport, R. DeVore, M. Wakin, A simple proof of the restricted isometry property for random matrices. Construct. Approx. 28(3), 253–263 (2008)

    MathSciNet  MATH  Google Scholar 

  3. G.D. Birkhoff, Dynamical Systems (American Mathematical Society, New York, 1960)

    Google Scholar 

  4. A. Boyarsky, P. Gora, Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension (Springer, New York, 2012)

    MATH  Google Scholar 

  5. R. Calderbank, S. Howard, S. Jafarpour, Construction of a large class of deterministic sensing matrices that satisfy a statistical isometry property. IEEE J. Select. Top. Signal Process. 4(2), 358–374 (2010)

    Google Scholar 

  6. E.J. Candès, The restricted isometry property and its implications for compressed sensing. C.R. Math. 346(9–10), 589–592 (2008)

    MathSciNet  MATH  Google Scholar 

  7. E.J. Candès, J. Romberg, Sparsity and incoherence in compressive sampling. Inverse Probl. 23(3), 969 (2007)

    MathSciNet  MATH  Google Scholar 

  8. E.J. Candès, J. Romberg, T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(2), 489–509 (2006)

    MathSciNet  MATH  Google Scholar 

  9. R. Caponetto, L. Fortuna, S. Fazzino, M.G. Xibilia, Chaotic sequences to improve the performance of evolutionary algorithms. IEEE Trans. Evol. Comput. 7(3), 289–304 (2003)

    Google Scholar 

  10. S.S. Chen, D.L. Donoho, M.A. Saunders, Atomic decomposition by basis pursuit. SIAM Rev. 43(1), 129–159 (2001)

    MathSciNet  MATH  Google Scholar 

  11. L. Chuang, S. Moi, Y. Lin, C. Yang, A comparative analysis of chaotic particle swarm optimizations for detecting single nucleotide polymorphism barcodes. Artif. Intell. Med. 73, 23–33 (2016)

    Google Scholar 

  12. A. Cohen, W. Dahmen, R. DeVore, Compressed sensing and best k-term approximation. J. Am. Math. Soc. 22(1), 211–231 (2009)

    MathSciNet  MATH  Google Scholar 

  13. T.T. Do, L. Gan, N.H. Nguyen, T.D. Tran, Fast and efficient compressive sensing using structurally random matrices. IEEE Trans. Signal Process. 60(1), 139–154 (2012)

    MathSciNet  MATH  Google Scholar 

  14. D.L. Donoho, Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006)

    MathSciNet  MATH  Google Scholar 

  15. H. Gan, Z. Li, J. Li, X. Wang, Z. Cheng, Compressive sensing using chaotic sequence based on Chebyshev map. Nonlinear Dyn. 78(4), 2429–2438 (2014)

    MathSciNet  MATH  Google Scholar 

  16. H. Gan, S. Xiao, Y. Zhao, A large class of chaotic sensing matrices for compressed sensing. Signal Process. 149, 193–203 (2018)

    Google Scholar 

  17. H. Gan, S. Xiao, Y. Zhao, X. Xue, Construction of efficient and structural chaotic sensing matrix for compressive sensing. Signal Process. Image Commun. 68, 129–137 (2018)

    Google Scholar 

  18. J. Gao, Recurrence time statistics for chaotic systems and their applications. Phys. Rev. Lett. 83(16), 3178 (1999)

    Google Scholar 

  19. P.R. Gill, A. Wang, A. Molnar, The in-crowd algorithm for fast basis pursuit denoising. IEEE Trans. Signal Process. 59(10), 4595–4605 (2011)

    MathSciNet  MATH  Google Scholar 

  20. S.D. Howard, A.R. Calderbank, S.J. Searle, A fast reconstruction algorithm for deterministic compressive sensing using second order Reed-Muller codes, in 2008 42nd Annual Conference on Information Sciences and Systems (IEEE, 2008), pp. 11–15

  21. V. Kafedziski, T. Stojanovski, Compressive sampling with chaotic dynamical systems. in Telecommunications Forum (TELFOR), 2011 19th (IEEE, 2011), pp. 695–698

  22. S.H. Kamel, M.B. Abd-el Malek, S.E. El-Khamy, Compressive spectrum sensing using chaotic matrices for cognitive radio networks. Int. J. Commun. Syst. 32, e3899 (2019)

    Google Scholar 

  23. R. Kang, G. Qu, B. Wang, Two effective strategies for complex domain compressive sensing. Circuits Syst. Signal Process. 35(9), 3380–3392 (2016)

    MathSciNet  Google Scholar 

  24. T. Kohda, Information sources using chaotic dynamics. Proc. IEEE 90(5), 641–661 (2002)

    Google Scholar 

  25. F. Krahmer, S. Mendelson, H. Rauhut, Suprema of chaos processes and the restricted isometry property. Commun. Pure Appl. Math. 67(11), 1877–1904 (2014)

    MathSciNet  MATH  Google Scholar 

  26. S. Li, G. Ge, Deterministic construction of sparse sensing matrices via finite geometry. IEEE Trans. Signal Process. 62(11), 2850–2859 (2014)

    MathSciNet  MATH  Google Scholar 

  27. X. Li, C. Wang, Z. Yang, L. Yan, S. Han, Energy-efficient and secure transmission scheme based on chaotic compressive sensing in underwater wireless sensor networks. Digit. Signal Proc. 81, 129–137 (2018)

    Google Scholar 

  28. N. Linh Trung, T. Minh Chinh, T. Tran Duc, H.V. Le, M.N. Do, Chaotic compressed sensing and its application to magnetic resonance imaging. REV J. Electron. Commun. 3(3–4), 84–92 (2014)

    Google Scholar 

  29. H. Ma, J. Jiao, J. Chen, X. Li, A fast channel state information feedback scheme based on compressive sensing for underwater acoustic MIMO systems. in 2016 35th Chinese Control Conference (CCC) (IEEE, 2016), pp. 5003–5007

  30. S. Mendelson, A. Pajor, N. Tomczak-Jaegermann, Uniform uncertainty principle for Bernoulli and sub-Gaussian ensembles. Construct. Approx. 28(3), 277–289 (2008)

    MATH  Google Scholar 

  31. G.H. Mohimani, M. Babaie-Zadeh, C. Jutten, Fast sparse representation based on smoothed \(l_0\) norm. in International Conference on Independent Component Analysis and Signal Separation (Springer, 2007), pp. 389–396

  32. H.O. Peitgen, H. Jürgens, D. Saupe, Chaos and Fractals: New Frontiers of Science (Springer, New York, 2006)

    MATH  Google Scholar 

  33. H. Peng, Y. Tian, J. Kurths, L. Li, Y. Yang, D. Wang, Secure and energy-efficient data transmission system based on chaotic compressive sensing in body-to-body networks. IEEE Trans. Biomed. Circuits Syst. 11(3), 558–573 (2017)

    Google Scholar 

  34. D. Rontani, D. Choi, C.Y. Chang, A. Locquet, D. Citrin, Compressive sensing with optical chaos. Sci. Rep. 6, 35206 (2016)

    Google Scholar 

  35. B. Saha, S.T. Malasani, J. Seventline, Application of modified chaotic sine map in secure communication. Int. J. Comput. Appl. 11313, 9–13 (2015)

    Google Scholar 

  36. M. Talagrand, The Generic Chaining: Upper and Lower Bounds of Stochastic Processes (Springer, New York, 2006)

    MATH  Google Scholar 

  37. J.A. Tropp, A.C. Gilbert, Signal recovery from random measurements via orthogonal matching pursuit. IEEE Trans. Inf. Theory 53(12), 4655–4666 (2007)

    MathSciNet  MATH  Google Scholar 

  38. T. Tuncer, E. Avaroglu, M. Türk, A.B. Ozer, Implementation of non-periodic sampling true random number generator on FPGA. Inf. MIDEM 44(4), 296–302 (2015)

    Google Scholar 

  39. L. Wan, G. Han, L. Shu, N. Feng, The critical patients localization algorithm using sparse representation for mixed signals in emergency healthcare system. IEEE Syst. J. 12(1), 52–63 (2018)

    Google Scholar 

  40. K. Wang, W. Pei, H. Xia, M.G. Nustes, J. Gonzalez, Statistical independence in nonlinear maps coupled to non-invertible transformations. Phys. Lett. A 372(44), 6593–6601 (2008)

    MathSciNet  MATH  Google Scholar 

  41. D. Xie, H. Peng, L. Li, Y. Yang, An efficient privacy-preserving scheme for secure network coding based on compressed sensing. AEU Int. J. Electron. Commun. 79, 33–42 (2017)

    Google Scholar 

  42. J. Xu, Y. Qiao, Z. Fu, Q. Wen, Image block compressive sensing reconstruction via group-based sparse representation and nonlocal total variation. Circuits Syst. Signal Process. 38(1), 304–328 (2018)

    Google Scholar 

  43. L. Yu, J.P. Barbot, G. Zheng, H. Sun, Compressive sensing with chaotic sequence. IEEE Signal Process. Lett. 17(8), 731–734 (2010)

    Google Scholar 

  44. L. Zeng, X. Zhang, L. Chen, T. Cao, J. Yang, Deterministic construction of toeplitzed structurally chaotic matrix for compressed sensing. Circuits Syst. Signal Process. 34(3), 797–813 (2015)

    Google Scholar 

  45. L. Zhang, K. Wong, Y. Zhang, J. Zhou, Bi-level protected compressive sampling. IEEE Trans. Multimed. 18(9), 1720–1732 (2016)

    Google Scholar 

  46. Y. Zhang, L. Zhang, J. Zhou, L. Liu, F. Chen, X. He, A review of compressive sensing in information security field. IEEE Access 4, 2507–2519 (2016)

    Google Scholar 

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Acknowledgements

This work was supported by the Chongqing Municipal Education Commission under Grant KJ1501105, in part by the National Natural Science Foundation of China under Grant 61372069, in part by the “111” Project of China Grant B08038, and in part by the SRF for ROCS, SEM under Grant JY0600090102. The authors would like to thank the anonymous reviewers and the editorial board for their suggestions and comments which significantly improve the quality of the paper.

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Authors and Affiliations

  1. State Key Lab of Integrated Services Networks, Xidian University, Xi’an, 710071, China

    Hongping Gan & Song Xiao

  2. School of Information Technology and Electrical Engineering, The University of Queensland, Brisbane, QLD, 4072, Australia

    Hongping Gan, Shanshan Shan & Yang Gao

  3. School of Control Science and Engineering, Shandong University, Jinan, 250100, China

    Zhimin Zhang

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  1. Hongping Gan
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Correspondence to Song Xiao.

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Gan, H., Xiao, S., Zhang, Z. et al. Chaotic Compressive Sampling Matrix: Where Sensing Architecture Meets Sinusoidal Iterator. Circuits Syst Signal Process 39, 1581–1602 (2020). https://doi.org/10.1007/s00034-019-01223-w

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