Multimedia Tools and Applications

, Volume 77, Issue 23, pp 30551–30574 | Cite as

Construction of compressed sensing matrices for signal processing

  • Yingmo Jie
  • Cheng GuoEmail author
  • Mingchu Li
  • Bin Feng


To cope with the huge expenditure associated with the fast growing sampling rate, compressed sensing (CS) is proposed as an effective technique of signal processing. In this paper, first, we construct a type of CS matrix to process signals based on singular linear spaces over finite fields. Second, we analyze two kinds of attributes of sensing matrices. One is the recovery performance corresponding to compressing and recovering signals. In particular, we apply two types of criteria, error-correcting pooling designs (PD) and restricted isometry property (RIP), to investigate this attribute. Another is the sparsity corresponding to storage and transmission signals. Third, in order to improve the ability associated with our matrices, we use an embedding approach to merge our binary matrices with some other matrices owing low coherence. At last, we compare our matrices with other existing ones via numerical simulations and the results show that ours outperform others.


Compressed sensing matrices Signal processing Singular linear spaces Pooling design (PD) Restricted isometry property (RIP) Sparsity 



This paper is supported by the National Natural Science Foundation of China under grant No. 61501080, 61572095, and the Fundamental Research Funds for the Central Universities’ under No. DUT16QY09.


  1. 1.
    Amini A, Marvasti F (2011) Deterministic construction of binary, bipolar and ternary compressed sensing matrices. IEEE Trans Inf Theory 57(4):2360–2370MathSciNetCrossRefGoogle Scholar
  2. 2.
    Amini A, Montazerhodjat V, Marvasti F (2012) Matrices with small coherence using p-ary block codes. IEEE Trans Signal Process 60(1):172–181MathSciNetCrossRefGoogle Scholar
  3. 3.
    Applebaum L, Howard S, Searle S, Calderbank R (2009) Chirp sensing codes: deterministic compressed sensing measurements for fast recovery. Appl Comput Harmon Anal 26(2):283–290MathSciNetCrossRefGoogle Scholar
  4. 4.
    Baraniuk R (2007) Compressive sensing. IEEE Signal Process Mag 24:118–121CrossRefGoogle Scholar
  5. 5.
    Berinde R, Gilbert A, Indyk P, Karloff H, Strauss M (2014) Combining geometry and combinatorics: a unified approach to sparse signal recovery. In: Proc. 46th Annu. Allerton Conf. Commun., Control, Comput., pp 798–805Google Scholar
  6. 6.
    Bourgain J, Dilworth S, Ford K, Konyagin S, Kutzarova D (2011) Explicit constructions of RIP matrices and related problems. Duke Math J 159(1):145–185MathSciNetCrossRefGoogle Scholar
  7. 7.
    Calderbank R, Howard S, Jafarpour S (2010) Construction of a large class of deterministic sensing matrices that satisfy a statistical isometry property. IEEE Trans Inf Theory 4(2):358–374Google Scholar
  8. 8.
    Candes E (2006) Compressive sampling. Proceedings of the International Congress of Mathematicians, pp 1433–1452Google Scholar
  9. 9.
    Candes E (2008) The restricted isometry property and its implications for compressed sensing. Comptes Rendus Math, Acad Sci Paris 346(9–10):589–592MathSciNetCrossRefGoogle Scholar
  10. 10.
    Candes E, Tao T (2005) Decoding by linear programming. IEEE Trans Inf Theory 51(12):4203–4215MathSciNetCrossRefGoogle Scholar
  11. 11.
    Candes E, Tao T (2006) Near-optimal signal recovery from random projections: Universal encoding strategies. IEEE Trans Inf Theory 52(12):5406–5425MathSciNetCrossRefGoogle Scholar
  12. 12.
    Candes E, Romberg J, Tao T (2006) Stable signal recovery from incomplete and inaccurate measurement. Commun Pure Appl Math 59(8):1207–1223MathSciNetCrossRefGoogle Scholar
  13. 13.
    Candes E, Romberg J, Tao T (2006) Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans Inf Theory 52(2):489–509MathSciNetCrossRefGoogle Scholar
  14. 14.
    Cui J, Liu Y, Xu Y, Zhao H, Zha H (2013) Tracking generic human motion via fusion of low-and high-dimensional approaches. IEEE Trans Syst Man Cybern Syst Hum 43(4):996–1002CrossRefGoogle Scholar
  15. 15.
    Davenport M, Wakin M (2009) Analysis of orthogonal matching pursuit using the restricted isometry property. IEEE Trans Inf Theory 56(9):4395–4401MathSciNetCrossRefGoogle Scholar
  16. 16.
    DeVore R (2007) Deterministic constructions of compressed sensing matrices. J Complex 23(4–6):918–925MathSciNetCrossRefGoogle Scholar
  17. 17.
    Donoho D (2006) Compressed sensing. IEEE Trans Inf Theory 52(4):1289–1306MathSciNetCrossRefGoogle Scholar
  18. 18.
    Dyachkov AG, Macula AJ, Vilenkin PA (2007) Nonadaptive and trivial two-stage group testing with error-correcting d e-disjunct inclusion matrices, vol 16. Springer, Berlin, pp 71–83Google Scholar
  19. 19.
    Ionin Y, Kaharaghani H (2007) The CRC handbook of combinatorial designs. CRC Press, Boca Raton, pp 306–313Google Scholar
  20. 20.
    Jie Y, Guo C, Fu Z (2017) Newly deterministic construction of compressed sensing matrices via singular linear spaces over finite fields. J Comb Optim 34:245–256MathSciNetCrossRefGoogle Scholar
  21. 21.
    Karystinos G, Pados D (2003) New bounds on the total squared correlation and optimum design of DS-CDMA binary signature sets. IEEE Trans Inf Theory 51(1):48–51Google Scholar
  22. 22.
    Li X (2010) Research on measurement matrix based on compressed sensing. Beijing Jiaotong University, thesis of Master Degree, pp 25–31Google Scholar
  23. 23.
    Li S, Ge G (2014) Deterministic sensing matrices arising from near orthogonal systems. IEEE Trans Inf Theory 60(4):2291–2302MathSciNetCrossRefGoogle Scholar
  24. 24.
    Li S, Ge G (2014) Deterministic construction of sparse sensing matrices via finite geometry. IEEE Trans Signal Process 62(11):2850–2859MathSciNetCrossRefGoogle Scholar
  25. 25.
    Li S, Gao F, Ge G, Zhang S (2012) Deterministic construction of compressed sensing matrices via algebraic curves. IEEE Trans Inf Theory 58(8):5035–5041MathSciNetCrossRefGoogle Scholar
  26. 26.
    Li S (2016) Combinatorial configurations, exponential sums and their applications in signal processing and the design of codes. Dissertation, Zhejiang UniversityGoogle Scholar
  27. 27.
    Liu X, Jie Y (2017) Deterministic construction of compressed sensing matrices over finite sets. J Comb Math Comb Comput 100:255–267MathSciNetzbMATHGoogle Scholar
  28. 28.
    Liu Y, Nie L, Han L, Zhang L, Rosenblum DS (2015) Action2Activity: recognizing complex activities from sensor data. Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence, pp 1617–1623Google Scholar
  29. 29.
    Liu Y, Nie L, Liu L, Rosenblum DS (2016) From action to activity: Sensor-based activity recognition. Neurocomputing 181:108–115CrossRefGoogle Scholar
  30. 30.
    Liu L, Cheng L, Liu Y, Jia Y, Rosenblum DS (2016) Recognizing complex activities by a probabilistic interval-based model. Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence, pp 1266–1272Google Scholar
  31. 31.
    Liu Y, Zhang L, Nie L, Yan Y, Rosenblum DS (2016) Fortune teller: predicting your career path. Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence, pp 201–207Google Scholar
  32. 32.
    Liu Y, Zheng Y, Liang Y, Liu S, Rosenblum DS (2016) Urban water quality prediction based on multi-task multi-view learning. Proceedings of the 25th International Joint Conference on Artificial Intelligence, pp 2576–2681Google Scholar
  33. 33.
    Lu Y, Wei Y, Liu L, Zhong J, Sun L, Liu Y (2017) Towards unsupervised physical activity recognition using smartphone accelerometers. Multimedia Tools and Applications 76(8):10701–10719CrossRefGoogle Scholar
  34. 34.
    Macula AJ (1996) A simple construction of d-disjunct matrices with certain constant weights. Discret Math 162:311–312MathSciNetCrossRefGoogle Scholar
  35. 35.
    Natarajan B (1995) Sparse approximate solutions to linear systems. SIAM J Comput 24(2):227–234MathSciNetCrossRefGoogle Scholar
  36. 36.
    Needell D, Tropp J (2009) CoSaMP: iterative signal recovery from incomplete and inaccurate samples. Appl Comput Harmon Anal 26(3):301–321MathSciNetCrossRefGoogle Scholar
  37. 37.
    Ramu Naidu R, Jampana P, Sastry C (2016) Deterministic compressed sensing matrices: construction via euler squares and applications. IEEE Trans Signal Process 64(14):3566–3575MathSciNetCrossRefGoogle Scholar
  38. 38.
    Rao KR, Yip PC (2000) The transform and data compression handbook, 1st edn. CRC, New YorkCrossRefGoogle Scholar
  39. 39.
    Strohmer T, Heath R (2003) Grassmannian frames with applications to coding and communication. Appl Comput Harmon Anal 14(3):257–275MathSciNetCrossRefGoogle Scholar
  40. 40.
    Sun Y, Gu F (2017) Compressive sensing of piezoelectric sensor reponse signal for phased array structural health monitoring. International Journal of Sensor Networks 23(4):258–264CrossRefGoogle Scholar
  41. 41.
    Sylvester JJ (1867) Thoughts on inverse orthogonal matrices, simultaneous sign successions, and tessellated pavements in two or more colours, with applications to Newton's rule, ornamental tile-work, and the theory of numbers. Philos Mag 34:461–475CrossRefGoogle Scholar
  42. 42.
    Troop J (2004) Greed is good: algorithmic result for sparse approximation. IEEE Trans Inf Theory 50(10):2231–2242MathSciNetCrossRefGoogle Scholar
  43. 43.
    Tropp J, Gilbert A (2007) Signal recovery from random measurements via orthogonal matching pursuit. IEEE Trans Inf Theory 53(12):4655–4666MathSciNetCrossRefGoogle Scholar
  44. 44.
    Tsaig Y, Donoho D (2006) Extensions of compressed sensing. Signal Process 86(3):549–571CrossRefGoogle Scholar
  45. 45.
    Wan Z (2002) Geometry of classical groups over finite fields, 2nd edn. Science, BeijingGoogle Scholar
  46. 46.
    Wang K, Guo J, Li F (2011) Singular linear space and its applications. Finite Fields Appl 17:395–406MathSciNetCrossRefGoogle Scholar
  47. 47.
    Wootters W, Fields B (1989) Optimal state determination by mutually unbiased measurements. Ann Phys 191(2):363–381MathSciNetCrossRefGoogle Scholar
  48. 48.
    Wu H, Zhang X, Chen W (2012) Measurement matrices in compressed sensing theory. Journal of Military Communication Technology 33(1):90–94Google Scholar
  49. 49.
    Xu L, Chen H (2015) Deterministic construction of RIP matrices in compressed sensing from constant weight codes. Science WISE arXiv:1506.02568Google Scholar
  50. 50.
    Zhang J, Han G, Fang Y (2015) Deterministic construction of compressed sensing matrices from protograph LDPC codes. IEEE Trans Signal Processing Letters 22(11):1960–1964CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesDalian University of TechnologyDalianChina
  2. 2.School of Software TechnologyDalian University of TechnologyDalianPeople’s Republic of China
  3. 3.Key Laboratory for Ubiquitous Network and Service Software of Liaoning ProvinceDalianChina

Personalised recommendations