Abstract
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.
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References
Amini A, Marvasti F (2011) Deterministic construction of binary, bipolar and ternary compressed sensing matrices. IEEE Trans Inf Theory 57(4):2360–2370
Amini A, Montazerhodjat V, Marvasti F (2012) Matrices with small coherence using p-ary block codes. IEEE Trans Signal Process 60(1):172–181
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):283–290
Baraniuk R (2007) Compressive sensing. IEEE Signal Process Mag 24:118–121
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 798–805
Bourgain J, Dilworth S, Ford K, Konyagin S, Kutzarova D (2011) Explicit constructions of RIP matrices and related problems. Duke Math J 159(1):145–185
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):358–374
Candes E (2006) Compressive sampling. Proceedings of the International Congress of Mathematicians, pp 1433–1452
Candes E (2008) The restricted isometry property and its implications for compressed sensing. Comptes Rendus Math, Acad Sci Paris 346(9–10):589–592
Candes E, Tao T (2005) Decoding by linear programming. IEEE Trans Inf Theory 51(12):4203–4215
Candes E, Tao T (2006) Near-optimal signal recovery from random projections: Universal encoding strategies. IEEE Trans Inf Theory 52(12):5406–5425
Candes E, Romberg J, Tao T (2006) Stable signal recovery from incomplete and inaccurate measurement. Commun Pure Appl Math 59(8):1207–1223
Candes E, Romberg J, Tao T (2006) Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans Inf Theory 52(2):489–509
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):996–1002
Davenport M, Wakin M (2009) Analysis of orthogonal matching pursuit using the restricted isometry property. IEEE Trans Inf Theory 56(9):4395–4401
DeVore R (2007) Deterministic constructions of compressed sensing matrices. J Complex 23(4–6):918–925
Donoho D (2006) Compressed sensing. IEEE Trans Inf Theory 52(4):1289–1306
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 71–83
Ionin Y, Kaharaghani H (2007) The CRC handbook of combinatorial designs. CRC Press, Boca Raton, pp 306–313
Jie Y, Guo C, Fu Z (2017) Newly deterministic construction of compressed sensing matrices via singular linear spaces over finite fields. J Comb Optim 34:245–256
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):48–51
Li X (2010) Research on measurement matrix based on compressed sensing. Beijing Jiaotong University, thesis of Master Degree, pp 25–31
Li S, Ge G (2014) Deterministic sensing matrices arising from near orthogonal systems. IEEE Trans Inf Theory 60(4):2291–2302
Li S, Ge G (2014) Deterministic construction of sparse sensing matrices via finite geometry. IEEE Trans Signal Process 62(11):2850–2859
Li S, Gao F, Ge G, Zhang S (2012) Deterministic construction of compressed sensing matrices via algebraic curves. IEEE Trans Inf Theory 58(8):5035–5041
Li S (2016) Combinatorial configurations, exponential sums and their applications in signal processing and the design of codes. Dissertation, Zhejiang University
Liu X, Jie Y (2017) Deterministic construction of compressed sensing matrices over finite sets. J Comb Math Comb Comput 100:255–267
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 1617–1623
Liu Y, Nie L, Liu L, Rosenblum DS (2016) From action to activity: Sensor-based activity recognition. Neurocomputing 181:108–115
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 1266–1272
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 201–207
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 2576–2681
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):10701–10719
Macula AJ (1996) A simple construction of d-disjunct matrices with certain constant weights. Discret Math 162:311–312
Natarajan B (1995) Sparse approximate solutions to linear systems. SIAM J Comput 24(2):227–234
Needell D, Tropp J (2009) CoSaMP: iterative signal recovery from incomplete and inaccurate samples. Appl Comput Harmon Anal 26(3):301–321
Ramu Naidu R, Jampana P, Sastry C (2016) Deterministic compressed sensing matrices: construction via euler squares and applications. IEEE Trans Signal Process 64(14):3566–3575
Rao KR, Yip PC (2000) The transform and data compression handbook, 1st edn. CRC, New York
Strohmer T, Heath R (2003) Grassmannian frames with applications to coding and communication. Appl Comput Harmon Anal 14(3):257–275
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):258–264
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 34:461–475
Troop J (2004) Greed is good: algorithmic result for sparse approximation. IEEE Trans Inf Theory 50(10):2231–2242
Tropp J, Gilbert A (2007) Signal recovery from random measurements via orthogonal matching pursuit. IEEE Trans Inf Theory 53(12):4655–4666
Tsaig Y, Donoho D (2006) Extensions of compressed sensing. Signal Process 86(3):549–571
Wan Z (2002) Geometry of classical groups over finite fields, 2nd edn. Science, Beijing
Wang K, Guo J, Li F (2011) Singular linear space and its applications. Finite Fields Appl 17:395–406
Wootters W, Fields B (1989) Optimal state determination by mutually unbiased measurements. Ann Phys 191(2):363–381
Wu H, Zhang X, Chen W (2012) Measurement matrices in compressed sensing theory. Journal of Military Communication Technology 33(1):90–94
Xu L, Chen H (2015) Deterministic construction of RIP matrices in compressed sensing from constant weight codes. Science WISE arXiv:1506.02568
Zhang J, Han G, Fang Y (2015) Deterministic construction of compressed sensing matrices from protograph LDPC codes. IEEE Trans Signal Processing Letters 22(11):1960–1964
Acknowledgements
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.
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Jie, Y., Guo, C., Li, M. et al. Construction of compressed sensing matrices for signal processing. Multimed Tools Appl 77, 30551–30574 (2018). https://doi.org/10.1007/s11042-018-6120-4
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DOI: https://doi.org/10.1007/s11042-018-6120-4