Skip to main content
SpringerLink
Log in

Sensing Matrices in Compressed Sensing

  • Conference paper
  • First Online:
Computing in Engineering and Technology

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 1025))

Abstract

One of the most important aspects of compressed sensing (CS) theory is an efficient design of sensing matrices. These sensing matrices are accountable for the required signal compression at the encoder end and its exact or approximate reconstruction at the decoder end. This paper presents an in-depth review of a variety of compressed sensing matrices such as random matrices, deterministic matrices, structural matrices, and optimized sensing matrices used in compressed sensing. Moreover, this paper presents insights into different research gaps which will provide the direction for further research in compressed sensing area.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  MathSciNet  Google Scholar 

  2. Candes, E.J., Tao, T.: Near-optimal signal recovery from random projections: universal encoding strategies. IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Lu, W., Li, W., Kpalma, K., Ronsin, J.: Compressed sensing performance of random Bernoulli matrices with high compression ratio. IEEE Signal Process. Lett. 22(8) (2015)

    Google Scholar 

  5. Gilbert, A., Indyk, P.: Sparse recovery using sparse matrices. Proc. IEEE 98(6), 937–947 (2010). https://doi.org/10.1109/JPROC.2010.2045092

    Article  Google Scholar 

  6. Mamaghanian, H., Khaled, N., Atienza, D., Vandergheynst, P.: Compressed sensing for real-time energy efficient ECG compression on wireless body sensor nodes. IEEE Trans. Biomed. Eng. 58(9), 2456–2466 (2011). https://doi.org/10.1109/tbme.2011.2156795

  7. Zhang, J., Gu, Z., Yu, Z., Li, Y.: Energy efficient ECG compression on wireless biosensors via minimal coherence sensing and weighted l1 minimization reconstruction. IEEE J. Biomed. Health Inform. 19(2), 520–528 (2015). https://doi.org/10.1109/JBHI.2014.2312374

    Article  Google Scholar 

  8. Mitra, U., Emken, A., Lee, S., Li, M., Rozgic, V., Thatte, G., Vathsangam, H., Zois, D.S., Annavaram, M., Narayanan, S., Levorato, M., Spruijt-Metz, D., Sukhatme, G.S.: KNOWME: a case study in wireless body area sensor network design. IEEE Commun. Mag. 50(5), 116–125 (2012)

    Article  Google Scholar 

  9. Zhang, X., Li, S.: Compressed sensing via dual frame based l1-analysis with Weibull matrices. IEEE Signal Process. Lett. 20(3), 265–268 (2013)

    Google Scholar 

  10. DeVore, Ronald A.: Deterministic constructions of compressed sensing matrices. J. Complex. 23, 918–925 (2007). https://doi.org/10.1016/j.jco.2007.04.002

    Article  MathSciNet  MATH  Google Scholar 

  11. Applebaum, L., Howard, S.D., Searle, S., Calderbank, R.: Chirp sensing codes: deterministic compressed sensing measurements for fast recovery. Appl. Comput. Harmon. Anal. 26(2), 283–290 (2009). https://doi.org/10.1016/j.acha.2008.08.002

    Article  MathSciNet  MATH  Google Scholar 

  12. Howard, S.D., Calderbank, A.R., Searle, S.J.: A fast reconstruction algorithm for deterministic compressive sensing using second order reed-muller codes. In: 42nd Annual Conference on Information Sciences and Systems (CISS 2008). IEEE, USA, pp. 11–15 (2008). https://doi.org/10.1109/ciss.2008.4558486

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

    Article  Google Scholar 

  14. Amini, A., Marvasti, F.: Deterministic construction of binary, bipolar and ternary compressed sensing matrices. IEEE Trans. Inf. Theory 57(4), 2360–2370 (2011). https://doi.org/10.1109/tit.2011.2111670

  15. Dimakis, A.G., Smarandache, R., Vontobel, P.O.: LDPC codes for compressed sensing. IEEE Trans. Inf. Theory 58(5), 3093–3114 (2012)

    Article  MathSciNet  Google Scholar 

  16. Zang, J., Han, G., Fang, Y.: Deterministic construction of compressed sensing matrices from protograph LDPC codes. IEEE Signal Process. Lett. 22(11), 1960–1964 (2015)

    Article  Google Scholar 

  17. Amini, A., Montazerhodjat, V., Marvasti, F.: Matrices with small coherence using p-Ary block codes. IEEE Trans. Signal Process. 60(1), 172–180 (2012)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  19. Xia, S.-T., Liu, X.-J., Jiang, Y., Zheng, H.-T.: Deterministic constructions of binary measurement matrices from finite geometry. IEEE Trans. Signal Process. 63(4), 1017–1029 (2015)

    Article  MathSciNet  Google Scholar 

  20. Xu, G., Xu, Z.: Compressed sensing matrices from Fourier matrices. IEEE Trans. Inf. Theory 61(1), 469–477 (2015)

    Article  MathSciNet  Google Scholar 

  21. Indyk, P.: Explicit constructions for compressed sensing matrices. In: Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, California, pp. 30–33 (2008)

    Google Scholar 

  22. Li, S., Gao, F., Ge, G., Zhang, S.: Deterministic construction of compressed sensing matrices via algebraic curves. IEEE Trans. Inf. Theory 58(8), 5035–5041 (2012)

    Article  MathSciNet  Google Scholar 

  23. Do, T.T., Gan, L., Nguyen, N.H.: Fast and efficient compressive sensing using structurally random matrices. IEEE Trans. Signal Process. 60(1), 139–154 (2012). https://doi.org/10.1109/TSP.2011.2170977

    Article  MathSciNet  MATH  Google Scholar 

  24. Bajwa, W.U., Haupt, J.D., Raz, G.M., Wright, S.J., Nowak, R.D.: Toeplitz-structured compressed sensing matrices. In: Proceedings of 14th IEEE/SP Workshop on Statistical Signal Processing (SSP 2007), Madison, WI, USA, pp. 294–298 (2007). https://doi.org/10.1109/ssp.2007.4301266

  25. Haupt, J., Bajwa, W.U., Raz, G., Nowak, R.: Toeplitz compressed sensing matrices with applications to sparse channel estimation. IEEE Trans. Inf. Theory 56(11), 5862–5875 (2010)

    Article  MathSciNet  Google Scholar 

  26. Yin, W., Morgan, S., Yang, J., Zhang, Y.: Practical compressive sensing with Toeplitz and Circulant matrices. In: Proceedings of SPIE 7744, Visual Communications and Image Processing 2010, 77440 K (15 July 2010), Huangshan, China (2010). https://doi.org/10.1117/12.863527

  27. Xu, Y., Yin, W., Osher, S.: Learning circulant sensing kernels. Inverse Problems Imaging 8(3), 901–923 (2014). https://doi.org/10.3934/ipi.2014.8.901

    Article  MathSciNet  MATH  Google Scholar 

  28. Li, K., Ling, C., Gan, L.: Deterministic compressed-sensing matrices: where Toeplitz meets Golay. In: International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, Prague, Czech Republic, pp. 3748–3751 (2011)

    Google Scholar 

  29. Li, K., Gan, L., Ling, C.: Convolutional compressed sensing using deterministic sequences. IEEE Trans. Signal Process. 61(3), 740–752 (2013)

    Article  MathSciNet  Google Scholar 

  30. Sun, J., Wang, S., Dong, Y.: Sparse block circulant matrices for compressed sensing. IET Commun. 7(13), 1412–1418 (2013)

    Article  Google Scholar 

  31. Li, K., Shuang, C.: State of the art and prospects of structured sensing matrices in compressed sensing. Front. Comput. Sci. 9(5), 665–677 (2015). https://doi.org/10.1007/s11704-015-3326-8

  32. Lee, D., Sasaki, T., Yamada, T., Akabane, K., Yamaguchi, Y., Uehara, K.: Spectrum sensing for networked system using 1-bit compressed sensing with partial random circulant measurement matrices. In: 75th Vehicular Technology Conference (VTC Spring). IEEE, Yokohama, Japan, pp. 1–5 (2012). https://doi.org/10.1109/vetecs.2012.6240259

  33. Salahdine, F., Kaabouch, N., El Ghazi, H.: Bayesian compressive sensing with circulant matrix for spectrum sensing in cognitive radio networks. In: 7th Annual Ubiquitous Computing, Electronics & Mobile Communication Conference (UEMCON). IEEE, New York, NY, USA, pp. 1–6 (2016)

    Google Scholar 

  34. Romberg, J.: Compressive sensing by random convolution. SIAM J. Imaging Sci. 2(4), 1098–1128 (2009). https://doi.org/10.1137/08072975X

    Article  MathSciNet  MATH  Google Scholar 

  35. Li, K., Ling, C., Gan, L.: Deterministic compressed sensing matrices: where Toeplitz meets Golay. In: International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, Prague, Czech Republic, pp. 3748–3751 (2011)

    Google Scholar 

  36. Li, K., Gao, S., Ling, C., Gan, L.: Wyner-ziv coding for distributed compressive sensing. In: Proceedings of Sensor Signal Processing for Defence (SSPD 2011), IET, London, UK, pp. 1–5 (2011)

    Google Scholar 

  37. Gan, L., Li, K., Ling, C.: Novel Toeplitz sensing matrices for compressive radar imaging. In: International Workshop on Compressed Sensing Applied to Radar (CoSeRa), Bonn, Germany (2012). https://doi.org/10.13140/2.1.1023.2964

  38. Gan, L., Li, K., Ling, C.: Golay meets Hadamard: Golay-paired Hadamard matrices for fast compressed sensing. In: IEEE-Information Theory Workshop (ITW), Lausanne, Switzerland, pp. 637–641 (2012). https://doi.org/10.1109/itw.2012.6404755

  39. Sun, R., Zhao, H., Xu, H.: The application of improved Hadamard measurement matrix in compressed sensing. In: International Conference on Systems and Informatics (ICSAI 2012), Yantai, China, 1994–1997 (2012)

    Google Scholar 

  40. Ma, J., Yuan, X., Ping, L.: Turbo compressed sensing with partial DFT sensing matrix. IEEE Signal Process. Lett. 22(2), 158–161 (2015)

    Article  Google Scholar 

  41. Elad, M.: Optimized projections for compressed sensing. IEEE Trans. Signal Process. 55(12), 5695–5702 (2007). https://doi.org/10.1109/TSP.2007.900760

    Article  MathSciNet  MATH  Google Scholar 

  42. Abolghasemi, V., Jarchi, D., Sanei, S.: A robust approach for optimization of the measurement matrix in compressed sensing. In: 2nd International Workshop on Cognitive Information Processing. IEEE, Elba, Italy, pp. 388–392 (2010). https://doi.org/10.1109/cip.2010.5604134

  43. Duarte-carvajalino, J.M., Sapiro, G.: Learning to sense sparse signals: simultaneous sensing matrix and sparsifying dictionary optimization. IEEE Trans. Image Process. 18(7), 1395–1408 (2009)

    Article  MathSciNet  Google Scholar 

  44. Pan, J., Qiu, Y.: An orthogonal method for measurement matrix optimization. Circuits Syst. Signal Process. 35(3), 837–849 (2016)

    Google Scholar 

  45. Chen, W., Rodrigues, M.R.D.: Dictionary learning with optimized projection design for compressive sensing applications. IEEE Signal Process. Lett. 20(10), 992–995 (2013)

    Google Scholar 

  46. Pereira, M.P., Lovisolo, L., da EAB, Silva, de Campos, M.L.R.: On the design of maximally incoherent sensing matrices for compressed sensing using orthogonal bases and its extension for biorthogonal bases case. Digit. Signal Process. 27, 12–22 (2014). https://doi.org/10.1016/j.dsp.2014.01.006. Elsevier

    Article  Google Scholar 

  47. Rosenblum, K., Zelnik-Manor, L., Eldar, Y.C.: Sensing matrix optimization for block-sparse decoding. IEEE Trans. Signal Process. 59(9), 4300–4312 (2011). https://doi.org/10.1109/TSP.2011.2159211

    Article  MathSciNet  MATH  Google Scholar 

  48. Bao, G., Ye, Z., Xu, X., Zhou, Y.: A compressed sensing approach to blind separation of speech mixture based on a two-layer sparsity model. IEEE Trans. Audio Speech Lang. Process. 21(5), 899–906 (2013)

    Article  Google Scholar 

  49. Defraene, B., Mansour, N., De Hertogh, S., van Waterschoot, T., Diehl, M., Moonen, M.: Declipping of audio signals using perceptual compressed sensing. IEEE Trans. Audio Speech Lang. Process. 21(12), 2627–2637 (2013). https://doi.org/10.1109/tasl.2013.2281570

  50. Giacobello, D., Christensen, M.G., Murthi, M.N., Jensen, S.H., Moonen, M.: Retrieving sparse patterns using a compressed sensing framework: applications to speech coding based on sparse linear prediction. IEEE Signal Process. Lett. 17(1), 103–106 (2010). https://doi.org/10.1109/LSP.2009.2034560

    Article  Google Scholar 

  51. Giacobello, D., Christensen, M.G., Murthi, M.N., Jensen, S.H., Moonen, M.: Sparse linear prediction and its applications to speech processing. IEEE Trans. Audio Speech Lang. Process. 20(5), 1644–1657 (2012). https://doi.org/10.1109/TASL.2012.2186807

    Article  Google Scholar 

  52. Wu, D., Zhu, W.-P., Swamy, M.N.S.: The theory of compressive sensing matching pursuit considering time-domain noise with application to speech enhancement. IEEE/ACM Trans. Audio Speech Lang. Process. 22(3), 682–696 (2014)

    Article  Google Scholar 

  53. Griffin, A., Hirvonen, T., Tzagkarakis, C., Mouchtaris, A., Tsakalides, P.: Single-channel and multi-channel sinusoidal audio coding using compressed sensing. IEEE Trans. Audio Speech Lang. Process. 19(5), 1382–1395 (2011). https://doi.org/10.1109/TASL.2010.2090656

    Article  Google Scholar 

  54. Gemmeke, J.F., Hamme, H.V., Cranen, B., Boves, L.: Compressive sensing for missing data imputation in noise robust speech recognition. IEEE J. Sel. Top. Signal Process 4, 272–287 (2010). https://doi.org/10.1109/JSTSP.2009.2039171

    Article  Google Scholar 

  55. Abrol, V., Sharma, P., Sao, A.K.: Voiced/nonvoiced detection in compressively sensed speech signals. Speech Commun. 72, 194–207 (2015). Elsevier

    Google Scholar 

  56. Abrol, V., Sharma, P., Budhiraja, S.: Deterministic compressed-sensing matrix from Grassmannian matrix: application to speech processing. IEEE, Ghaziabad, India, pp. 1165–1170 (2012). https://doi.org/10.1109/iadcc.2013.6514392

  57. Abrol, V., Sharma, P., Budhiraja, S.: Evaluating performance of compressed sensing for speech signals. In: 3rd International Advance Computing Conference (IACC), Ghaziabad. IEEE, India, pp. 1159–1164 (2013)

    Google Scholar 

  58. Bhadoria, B.S., Shukla, U., Joshi, A.M.: Comparative analysis of basis & measurement matrices for non-speech audio signal using compressive sensing. In: International Conference on Computational Intelligence and Computing Research (ICCIC). IEEE, Coimbatore, India. pp. 1–5 (2014)

    Google Scholar 

  59. Savic, T., Albijanic, R.: CS reconstruction of the speech and musical signals. In: 4th Mediterranean Conference on Embedded Computing (MECO). IEEE, Budva, Montenegro, pp. 299–302 (2015). https://doi.org/10.1109/meco.2015.7181927

  60. Joshi, A.M., Upadhyaya, V.: Analysis of compressive sensing for non stationary music signal. In: International Conference on Advances in Computing, Communications and Informatics (ICACCI). IEEE, Jaipur, India. pp. 1172–1176 (2016)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electronics and Telecommunication Engineering, Dr. Babasaheb Ambedkar Technological University (DBATU), Lonere, Raigad, Maharashtra, India

    Yuvraj V. Parkale & Sanjay L. Nalbalwar

Authors
  1. Yuvraj V. Parkale
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sanjay L. Nalbalwar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuvraj V. Parkale .

Editor information

Editors and Affiliations

  1. Department of Electronics and Telecommunication Engineering, Dr. Babasaheb Ambedkar Technological University, Lonere, Maharashtra, India

    Brijesh Iyer

  2. Department of Computer Engineering, Dr. Babasaheb Ambedkar Technological University, Lonere, Maharashtra, India

    P. S. Deshpande

  3. Department of Electronics and Computer Engineering, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, India

    S. C. Sharma

  4. Deogiri Institute of Engineering and Management Studies, Aurangabad, Maharashtra, India

    Ulhas Shiurkar

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Parkale, Y.V., Nalbalwar, S.L. (2020). Sensing Matrices in Compressed Sensing. In: Iyer, B., Deshpande, P., Sharma, S., Shiurkar, U. (eds) Computing in Engineering and Technology. Advances in Intelligent Systems and Computing, vol 1025. Springer, Singapore. https://doi.org/10.1007/978-981-32-9515-5_11

Download citation

Publish with us

Policies and ethics

Search

Navigation