Skip to main content
Log in

Three deterministic constructions of compressed sensing matrices with low coherence

  • Published:
Cryptography and Communications Aims and scope Submit manuscript

Abstract

Matrices with low coherence have applications in compressed sensing and some other areas. In this paper, we present three deterministic constructions of compressed sensing matrices by using algebraic and combinatorial methods. We show that our results outperform Gaussian random matrices. Moreover, some of our matrices are binary entries, and thus can be used in the embedding operations to get more matrices with low coherence recursively.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

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

    MathSciNet  MATH  Google Scholar 

  2. Baraniuk, R., Devenport, M., DeVore, R., Wakin, M.B.: A simple proof of the resrticted property for random matrices. Construct. Approx. 28(3), 253–263 (2008)

    MATH  Google Scholar 

  3. Candès, E.: The restricted isometry property and its implications for compressed sensing. C. R. Math. Acad. Sci. Paris 346(9), 589–592 (2008)

    MathSciNet  MATH  Google Scholar 

  4. Candès, E., Tao, T.: Decoding by linear programming. IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  6. Calderbank, R., Jafarpour, S.: Reed Muller sensing matrices and the LASSO. In: Carlet, C, Pott, A (eds.) Sequences and their applications–SETA 2010. SETA 2010. Lecture Notes in Computer Science, vol. 6338, pp 442–463. Springer, Berlin (2010)

  7. Colbourn, C.J., Dinitz, J.H.: Handbook of Combinatorical Designs, 2nd edn. Chapman and Hall/CRC, Taylor and Francis Group, New York (2007)

  8. Cao, X., Chou, W.-S., Gu, J.: On the number of solutions of certain diagonal equations over finite fields. Finite Fields Appl. 42, 225–252 (2016)

    MathSciNet  MATH  Google Scholar 

  9. Ding, C., Feng, T.: A generic construction of complex codebooks meeting the Welch bound. IEEE Trans. Inf. Theory 53(11), 4245–4250 (2007)

    MathSciNet  MATH  Google Scholar 

  10. Donoho, D.: Comprssed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006)

    MATH  Google Scholar 

  11. Fickus, M., Mixon, D. G., Tremain, J. C.: Steiner equiangular tight frames. Linear Algebra Appl. 436(5), 1014–1027 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Godsil, C., Roy, A.: Equiangular lines, mutually unbiased bases and spin models. Eur. J. Comb. 30, 246–262 (2009)

    MathSciNet  MATH  Google Scholar 

  13. Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn, p 320. Johns Hopkins University Press, Baltimore (1996)

    Google Scholar 

  14. Howard, S. D., Calderbank, A. R., Searle, S. J.: A fast reconstruction algorithm for deterministic compressive sensing using second order Reed-Muller codes, CISS (2008)

  15. Helleseth, T., Kholosha, A.: On the dual of monomial quadratic p-ary bent functions. In: Golomb, S.W., et al. (eds.) SSC 2007, LNCS 4803, pp 50–61 (2007)

  16. Jarfarpour, S., Duarte, M. F., Calderbank, R.: Beyond worst-case reconstruction in deterministic compressed sensing. In: ISIT (2012)

  17. Li, S., Ge, G.: Deterministic sensing matrices arising from near orthogonal systems. IEEE Trans. Inf. Theory 60(4), 2291–2302 (2014)

    MathSciNet  MATH  Google Scholar 

  18. Li, S., Ge, G.: Deterministic comstruction of sparse sensing matrices via finite geometry. IEEE Trans. Inf. Theory 62(11), 2850–2859 (2014)

    MATH  Google Scholar 

  19. Kunis, S., Rauhut, H.: Random sampling of sparse trigonometric polynomials, II. Orthogonal matching pursuit versus basis pursuit. Found. Comput. Math. 8(6), 737–763 (2008)

    MathSciNet  MATH  Google Scholar 

  20. Lidl, R., Niederreiter, H.: Finite Fields, Encycl. Math. Appl., vol. 20. Addison-Wesley Publishing Company, London (1983)

    Google Scholar 

  21. Ma, S.L.: A survey on partial difference sets. Des. Codes Cryptogr. 4, 221–261 (1994)

    MathSciNet  MATH  Google Scholar 

  22. Strohmer, T., Heath, R. W. Jr: Grassmannian frames with applications to coding and communication. Appl. Comput. Harmonic Anal. 14(3), 257–275 (2003)

    MathSciNet  MATH  Google Scholar 

  23. Welch, L.: Lower bounds on the maximum cross correlation of signals. IEEE Trans. Inf. Theory 20(3), 397–399 (1974)

    MATH  Google Scholar 

  24. Xia, P., Zhou, S., Giannakis, G. B.: Achieving the Welch bound with difference sets. IEEE Trans. Inf. Theory 51(5), 1900–1907 (2005)

    MathSciNet  MATH  Google Scholar 

  25. Yu, N. Y.: Additive character sequences with small alphabets for compressed sensing matrices. In: ICASSP (2011)

  26. Yu, N.Y., Zhao, N.: Deterministic construction of real-valued ternary sensing matrices using optimal orthogonal codes. IEEE Signal Process Lett. 20(11), 1106–1109 (2013)

    Google Scholar 

  27. Zhou, Z., Tang, X.: New nearly optimal codebooks from relative difference sets. Adv. Math. Commun. 5(3), 521–527 (2011)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Xiwang Cao.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the Topical Collection on Special Issue on Sequences and Their Applications

First Author is supported by the NNSF of China (11771007, 61572027).

Third Author is supported by the NNSF of China (Grant No. 11601177), the Funding for Outstanding Doctoral Dissertation in NUAA (Grant No. BCXJ16-08), Anhui Provincial Natural Science Foundation (1608085QA05), the Foundation for Distinguished Young Talents in Higher Education of Anhui Province of China (gxyqZD2016258)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cao, X., Luo, G. & Xu, G. Three deterministic constructions of compressed sensing matrices with low coherence. Cryptogr. Commun. 12, 547–558 (2020). https://doi.org/10.1007/s12095-019-00375-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12095-019-00375-5

Keywords

Mathematics Subject Classification (2010)

Navigation