Skip to main content
SpringerLink
Log in

Non-Cartesian Spiral Binary Sensing Matrices

  • Short Paper
  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

Binary sensing matrices can offer rapid multiplier-less data acquisition, owing to their binarization structure and competitive sampling efficiency, which promise to promote compressive sensing from theory to application. However, the size of existing binary constructions is often limited, and the generating strategies require extensive computational complexity. In this work, we propose a trivial strategy to deterministically construct non-Cartesian spiral binary sensing matrices (SbMs) with arbitrary size for compressive sensing. The mutual coherence of SbMs is proven to satisfy the Welch bound, i.e., optimal theoretical guarantee. Moreover, the simulation results show that the proposed SbMs can not only outperform their counterparts in sampling performance but also facilitate reconstruction speed.

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

Access this article

Log in via an institution

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
Fig. 4

Similar content being viewed by others

Data availability

The data used to support the work are available from the corresponding author on reasonable request.

Notes

  1. The density of a binary matrix of size \(m \times n\) is the ratio of the total number of \(1's\) to \(m \times n\).

  2. In fact, we can set p to be 19, 23, 29, or perhaps a larger prime as \(p > \sqrt{300}=17.3205\).

References

  1. D. Achlioptas, Database-friendly random projections: Johnson-Lindenstrauss with binary coins. J. comput. Syst. Sci. 66(4), 671–687 (2003)

    Article  MathSciNet  Google Scholar 

  2. A. Amini, V. Montazerhodjat, F. Marvasti, Matrices with small coherence using p-ary block codes. IEEE Trans. Sig. Process. 60(1), 172 (2012)

    Article  MathSciNet  Google Scholar 

  3. D. Bryant, C.J. Colbourn, D. Horsley, P.Ó. Catháin, Compressed sensing with combinatorial designs: theory and simulations. IEEE Trans. Inform. Theory 63(8), 4850–4859 (2017)

    Article  MathSciNet  Google Scholar 

  4. T.T. Cai, G. Xu, J. Zhang, On recovery of sparse signals via \(\ell _0\) minimization. IEEE Trans. Inform. Theory 55(7), 3388–3397 (2009)

    Article  MathSciNet  Google Scholar 

  5. E.J. Candès, The restricted isometry property and its implications for compressed sensing. Comptes Rendus Mathematique 346(9–10), 589–592 (2008)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. R.A. DeVore, Deterministic constructions of compressed sensing matrices. J. Complex. 23(4–6), 918–925 (2007)

    Article  MathSciNet  Google Scholar 

  10. H. Gan, S. Xiao, T. Zhang, F. Liu, Bipolar measurement matrix using chaotic sequence. Commun. Nonlinear Sci. Num. Simul. 72, 139–151 (2019)

    Article  MathSciNet  Google Scholar 

  11. H. Gan, S. Xiao, Z. Zhang, S. Shan, Y. Gao, Chaotic compressive sampling matrix: where sensing architecture meets sinusoidal iterator. Circ. Syst. Sig. Process. 39(3), 1581–1602 (2020)

    Article  Google Scholar 

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

    Article  Google Scholar 

  13. S.A. Geršhgorin, Über die abgrenzung der eigenwerte einer matrix. Izv. Akad. Nauk SSSR Ser. Fiz. Mat. 6, 749–754 (1931)

    MATH  Google Scholar 

  14. S. Huang, H. Sun, L. Yu, H. Zhang, A class of deterministic sensing matrices and their application in harmonic detection. Circ. Syst. Sig. Process. 35(11), 4183–4194 (2016)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. X.J. Liu, S.T. Xia, T. Dai, Deterministic constructions of binary measurement matrices with various sizes. In: 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 3641–3645. IEEE (2015)

  19. M. Lotfi, M. Vidyasagar, Compressed sensing using binary matrices of nearly optimal dimensions. IEEE Trans. Sig. Process. 68, 3008–3021 (2020)

    Article  MathSciNet  Google Scholar 

  20. W. Lu, T. Dai, S.T. Xia, Binary matrices for compressed sensing. IEEE Trans. Sig. Process. 66(1), 77–85 (2018)

    Article  MathSciNet  Google Scholar 

  21. W. Lu, W. Li, W. Zhang, S.T. Xia, Expander recovery performance of bipartite graphs with girth greater than 4. IEEE Trans. Sig. Inform. Process. Netw. 5(3), 418–427 (2018)

    MathSciNet  Google Scholar 

  22. R.R. Naidu, P. Jampana, C.S. Sastry, Deterministic compressed sensing matrices: construction via euler squares and applications. IEEE Trans. Sig. Process. 64(14), 3566–3575 (2016)

    Article  MathSciNet  Google Scholar 

  23. P. Sasmal, C.R. Murthy, Incoherence is sufficient for statistical rip of unit norm tight frames: constructions and properties. IEEE Trans. Sig. Process. 69, 2343–2355 (2021)

    Article  MathSciNet  Google Scholar 

  24. P. Sasmal, R.R. Naidu, C.S. Sastry, P. Jampana, Composition of binary compressed sensing matrices. IEEE Sig. Process. Lett. 23(8), 1096–1100 (2016)

    Article  Google Scholar 

  25. J.D. Suever, G.J. Wehner, L. Jing, D.K. Powell, S.M. Hamlet, J.D. Grabau, D. Mojsejenko, K.N. Andres, C.M. Haggerty, B.K. Fornwalt, Right ventricular strain, torsion, and dyssynchrony in healthy subjects using 3D spiral cine DENSE magnetic resonance imaging. IEEE Trans. Med. Imag. 36(5), 1076–1085 (2016)

    Article  Google Scholar 

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

    Article  Google Scholar 

  27. J. Xu, Y. Qiao, Z. Fu, Q. Wen, Image block compressive sensing reconstruction via group-based sparse representation and nonlocal total variation. Circ. Syst. Sig. Process. 38(1), 304–328 (2019)

    Article  Google Scholar 

  28. L. Zeng, X. Zhang, L. Chen, T. Cao, J. Yang, Deterministic construction of Toeplitzed structurally chaotic matrix for compressed sensing. Circ. Syst. Sig. Process. 34(3), 797–813 (2015)

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the Basic Research Programs of Taicang under Grant TC2020JC07, in part by the National Natural Science Foundation of China under Grant 62101455, in part by the Fundamental Research Funds for the Central Universities under Grant D5000210693, in part by the Natural Science Basic Research Program of Shaanxi Province under Grants 2021JQ126 and 2021JQ099, in part by Guangdong Basic and Applied Basic Research Foundation under Grant 2020A1515111158.

Author information

Authors and Affiliations

  1. School of Software, Northwestern Polytechnical University, Xi’an, 710072, China

    Hongping Gan

  2. State Key Laboratory of Integrated Services Networks, Xidian University, Xi’an, 710071, China

    Hongping Gan

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

    Yang Gao

  4. Shanghai Key Laboratory of Intelligent Sensing and Recognition, Shanghai Jiao Tong University, Shanghai, 200240, China

    Tao Zhang

Authors
  1. Hongping Gan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yang Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tao Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hongping Gan.

Additional information

Publisher's Note

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

Appendix A

Appendix A

Theorem 2

(Geršhgorin circle theorem) [13] The eigenvalues of a matrix \(\varvec{M} \in \mathbb {R}^{n \times n}\) with elements \(M_{j,k}\), \(1\le j,k \le n\), lie in the union of n discs \(d_j=d_j(c_j,r_j)\) centered at \(c_j=M_{j,j}\) with radius \(r_j=\sum _{k\ne j}|M_{j,k}|\).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gan, H., Gao, Y. & Zhang, T. Non-Cartesian Spiral Binary Sensing Matrices. Circuits Syst Signal Process 41, 2934–2946 (2022). https://doi.org/10.1007/s00034-021-01899-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-021-01899-z

Keywords

Search

Navigation