IJBTT

Cosparse Analysis Model-Based Compressive Sensing With Optimized Projection Matrix

Cosparse Analysis ModelBased Compressive Sensing With Optimized Projection Matrix
	© 2021 by IJETT Journal
	Volume-69 Issue-11
	Year of Publication : 2021
	Authors : Endra Oey, Dadang Gunawan, Dodi Sudiana
	DOI :  10.14445/22315381/IJETT-V69I11P214

How to Cite?

Endra Oey, Dadang Gunawan, Dodi Sudiana, "Cosparse Analysis ModelBased Compressive Sensing With Optimized Projection Matrix," International Journal of Engineering Trends and Technology, vol. 69, no. 11, pp. 113-121, 2021. Crossref, https://doi.org/10.14445/22315381/IJETT-V69I11P214

Abstract
The Compressive Sensing (CS) technique provides a signal acquisition dimensional reduction by multiplying a projection matrix with the signal. Until now, the projection matrix optimization is commonly performed using the Sparse Synthesis Model-Based (SSMB), where it takes a linear combination of a few atoms in a synthesis dictionary to form a signal. The Cosparse Analysis Model-Based (CAMB) provides an alternative model where the multiplication of the signal with an analysis dictionary (operator) produces a cosparse coefficient. The CAMB-CS method is proposed in this paper by taking into account the amplified Cosparse Representation Error (CSRE) parameter and the relative amplified CSRE optimize the projection matrix, in addition to the mutual coherence parameter. The optimized projection matrix in CAMB-CS is obtained using an alternating minimization algorithm and nonlinear conjugation gradient method. In the optimization algorithm, the Gaussian random matrix is used as the initial projection matrix. The simulation results showed an increase in the Peak Signal to Noise Ratio (PSNR) and Structural Similarity Index Measure (SSIM) of the reconstructed image in the CAMB-CS system up to 10.23% and 8.46%, respectively, compared to the Gaussian random matrix. Compared to the SSMB-CS optimized projection matrix, the developed method increases the PSNR and SSIM of the recovered image up to 21.21% and 17.11%, respectively.

Keywords
Compressive Sensing, Cosparse Analysis Model, Cosparse Representation Error, Projection Matrix Optimization.

Reference
[1] M. Unser, .,Sampling—50 years after Shannon,., Proc. IEEE, 88 (2000) 569–587.
[2] D. L. Donoho, .,Compressed sensing,., IEEE Trans. Inf. Theory, 52(4) (2006) 1289–1306.
[3] E. J. Candès, J. Romberg, and T. Tao, .,Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,., IEEE Trans. Inf. Theory, 52( 2) (2006) 489–509.
[4] E. J. Candes, J. K. Romberg, and T. Tao, .,Stable signal recovery from incomplete and inaccurate measurements,., Commun. Pure Appl. Math. A J. Issued by Courant Inst. Math. Sci., 59( 8) (2006) 1207–1223.
[5] E. J. Candes and T. Tao, .,Near-optimal signal recovery from random projections: Universal encoding strategies?,., IEEE Trans. Inf. Theory, 52(12) (2006) 5406–5425.
[6] M. F. Duarte and Y. C. Eldar, .,Structured compressed sensing: From theory to applications,., IEEE Trans. signal Process., 59 (9) (2011) 4053–4085.
[7] T. Strohmer, .,Measure what should be measured: progress and challenges in compressive sensing,., IEEE Signal Process. Lett., 19(12) (2012) 887–893.
[8] M. Elad, Sparse and redundant representations: from theory to applications in signal and image processing. Springer Science & Business Media, (2010).
[9] S. Qaisar, R. M. Bilal, W. Iqbal, M. Naureen, and S. Lee, .,Compressive sensing: From theory to applications, a survey,., J. Commun. Networks, 15(5) (2013) 443–456,.
[10] M. Rani, S. B. Dhok, and R. B. Deshmukh, .,A systematic review of compressive sensing: Concepts, implementations, and applications,., IEEE Access, 6 (2018) 4875–4894.
[11] A. M. Bruckstein, D. L. Donoho, and M. Elad, .,From sparse solutions of systems of equations to sparse modeling of signals and images,., SIAM Rev., 51(1) (2009) 34–8.
[12] M. Elad, P. Milanfar, and R. Rubinstein, .,Analysis versus synthesis in signal priors,., Inverse Probl., vol. 23(3) (2007) 947.
[13] S. Nam, M. E. Davies, M. Elad, and R. Gribonval, .,Cosparse analysis modeling-uniqueness and algorithms,., in (2011) IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), (2011) 5804–5807.
[14] S. Nam, M. E. Davies, M. Elad, and R. Gribonval, .,The cosparse analysis model and algorithms?,., Appl. Comput. Harmon. Anal, 34 (2013) 30–56.
[15] R. Giryes, .,Sampling in the analysis transform domain,., Appl. Comput. Harmon. Anal., 40( 1) (2016) 172–185. doi: https://doi.org/10.1016/j.acha.2015.04.004.
[16] J. Wörmann, S. Hawe, and M. Kleinsteuber, .,Analysis based blind compressive sensing,., IEEE Signal Process. Lett., 20(5) (2013) 491–494.
[17] S. Ravishankar and Y. Bresler, .,Blind compressed sensing using sparsifying transforms,., in International Conference on Sampling Theory and Applications (SampTA), (2015) 513–517.
[18] O. Endra and D. Gunawan, .,Comparison of synthesis-based and analysis-based compressive sensing,., in International Conference on Quality in Research (QiR), (2015) 167–170.
[19] S. Ravishankar and Y. Bresler, .,Data-driven learning of a union of sparsifying transforms model for blind compressed sensing,., IEEE Trans. Comput. Imaging, 2 (3) (2016) 294–309.
[20] K. Kreutz-Delgado, J. F. Murray, B. D. Rao, K. Engan, T.-W. Lee, and T. J. Sejnowski, .,Dictionary learning algorithms for sparse representation,., Neural Comput., 15(2) (2003) 349–39.
[21] K. Engan, S. O. Aase, and J. H. Husoy, .,Method of optimal directions for frame design,., in 1999 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings. ICASSP99 (Cat. No. 99CH36258), 5 (1999) 2443–2446.
[22] M. Aharon, M. Elad, and A. Bruckstein, .,K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation,., IEEE Trans. signal Process., vol. 54 (11) (2006) 4311–4322 .
[23] I. Kviatkovsky, M. Gabel, E. Rivlin, and I. Shimshoni, .,On the Equivalence of the LC-KSVD and the D-KSVD Algorithms.,., IEEE Trans. Pattern Anal. Mach. Intell., 39(2) (2017) 411–416.
[24] B. Ophir, M. Elad, N. Bertin, and M. D. Plumbley, .,Sequential minimal eigenvalues-an approach to analysis dictionary learning,., in 2011 19th European Signal Processing Conference, (2011) 1465–1469.
[25] M. Yaghoobi, S. Nam, R. Gribonval, and M. E. Davies, .,Analysis operator learning for overcomplete cosparse representations,., in 2011 19th European Signal Processing Conference, (2011) 1470–1474.
[26] M. Yaghoobi, S. Nam, R. Gribonval, and M. E. Davies, .,Noise aware analysis operator learning for approximately cosparse signals,., in 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), (2012) 5409–5412.
[27] M. Yaghoobi, S. Nam, R. Gribonval, and M. E. Davies, .,Constrained overcomplete analysis operator learning for cosparse signal modelling,., IEEE Trans. Signal Process., 61(9) (2013) 2341–2355.
[28] R. Rubinstein, T. Peleg, and M. Elad, .,Analysis K-SVD: A Dictionary-Learning Algorithm for the Analysis Sparse Model,., IEEE Trans. Signal Process., 3(61) 661–677 (2013).
[29] S. Hawe, M. Kleinsteuber, and K. Diepold, .,Analysis operator learning and its application to image reconstruction,., IEEE Trans. Image Process., 22(6) (2013) 2138–2150,
[30] J. Dong, W. Wang, and W. Dai, .,Analysis SimCO: A new algorithm for analysis dictionary learning,., in 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), (2014) 7193–7197.
[31] S. Ravishankar and Y. Bresler, .,Learning Sparsifying Transforms,., IEEE Trans. Signal Process., 61(5) (2013) 1072–1086,.
[32] S. Ravishankar and Y. Bresler, .,Closed-form solutions within sparsifying transform learning,., in 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, (2013) 5378–5382.
[33] S. Ravishankar and Y. Bresler, .,Learning overcomplete sparsifying transforms for signal processing,., in 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, (2013) 3088–3092.
[34] S. Ravishankar, B. Wen, and Y. Bresler, .,Online sparsifying transform learning—Part I: Algorithms,., IEEE J. Sel. Top. Signal Process., 9(4) 625–636, 2015.
[35] S. Ravishankar and Y. Bresler, .,Online sparsifying transform learning—Part II: Convergence analysis,., IEEE J. Sel. Top. Signal Process., 9(4) (2015) 637–646 .
[36] B. Hou, Z. Zhu, G. Li, and A. Yu, .,An Efficient Algorithm for Overcomplete Sparsifying Transform Learning with Signal Denoising,., Math. Probl. Eng., 2016, (2016).
[37] C. Rusu and J. Thompson, .,Learning Fast Sparsifying Transforms,., IEEE Trans. Signal Process., 65(16) (2017) 4367–4378, doi: 10.1109/TSP.2017.2712120.
[38] Y. Arjoune, N. Kaabouch, H. El Ghazi, and A. Tamtaoui, .,Compressive sensing: Performance comparison of sparse recovery algorithms,., in 2017 IEEE 7th annual computing and communication workshop and conference (CCWC), (2017) 1–7.
[39] R. Giryes, S. Nam, M. Elad, R. Gribonval, and M. E. Davies, .,Greedy-like algorithms for the cosparse analysis model,., Linear Algebra Appl., 441 (2014) 22–60,.
[40] J. Li, Z. Liu, and W. Li, .,The reweighed greedy analysis pursuit algorithm for the cosparse analysis model,., in 2015 11th International Conference on Natural Computation (ICNC), (2015) 1018–1022.
[41] R. Giryes, .,A greedy algorithm for the analysis transform domain,., Neurocomputing, 173 (2016) 278–289,.
[42] E. Candes and J. Romberg, .,Sparsity and incoherence in compressive sampling,., Inverse Probl., 23( 3) (2007) 969,.
[43] W. Yin, S. Morgan, J. Yang, and Y. Zhang, .,Practical compressive sensing with Toeplitz and circulant matrices,., in Visual Communications and Image Processing, 7744 (2010) 77440K.
[44] T. T. Do, L. Gan, N. H. Nguyen, and T. D. Tran, .,Fast and efficient compressive sensing using structurally random matrices,., IEEE Trans. signal Process., 60(1) (2011) 139–154.
[45] T. L. N. Nguyen and Y. Shin, .,Deterministic sensing matrices in compressive sensing: a survey,., Sci. World J. ( 2013).
[46] S. Li and G. Ge, .,Deterministic construction of sparse sensing matrices via finite geometry,., IEEE Trans. Signal Process., 62(11) (2014) 2850–2859.
[47] S. Li and G. Ge, .,Deterministic sensing matrices were arising from near orthogonal systems,., IEEE Trans. Inf. Theory, 60(4) (2014) 2291–2302,
[48] A. Ravelomanantsoa, H. Rabah, and A. Rouane, .,Compressed sensing: A simple deterministic measurement matrix and a fast recovery algorithm,., IEEE Trans. Instrum. Meas., 64(12) (2015) 3405–3413,.
[49] R. R. Naidu, P. Jampana, and C. S. Sastry, .,Deterministic compressed sensing matrices: Construction via Euler squares and applications,., IEEE Trans. Signal Process., 64(14) (2016) 3566–3575 .
[50] R. R. Naidu and C. R. Murthy, .,Construction of binary sensing matrices using extremal set theory,., IEEE Signal Process. Lett., 24( 2) (2016) 211–215,.
[51] W. Lu, T. Dai, and S. Xia, .,Binary Matrices for Compressed Sensing,., IEEE Trans. Signal Process., vol. 66(1) (2018) 77–85, doi: 10.1109/TSP.2017.2757915.
[52] S.-H. Hsieh, C.-S. Lu, and S.-C. Pei, .,Compressive sensing matrix design for fast encoding and decoding via sparse FFT,., IEEE Signal Process. Lett., 25( 4) 591–595 (2018).
[53] M. Elad, .,Optimized projections for compressed sensing,., IEEE Trans. Signal Process., 55(12) (2007) 5695–5702,.
[54] W. Yan, Q. Wang, and Y. Shen, .,Shrinkage-based alternating projection algorithm for efficient measurement matrix construction in compressive sensing,., IEEE Trans. Instrum. Meas., 63(5) (2014) 1073–1084.
[55] J. M. Duarte-Carvajalino and G. Sapiro, .,Learning to sense sparse signals: Simultaneous sensing matrix and sparsifying dictionary optimization,., IEEE Trans. Image Process., 18 (7) (2009) 1395–1408 .
[56] V. Abolghasemi, S. Ferdowsi, B. Makkiabadi, and S. Sanei, .,On the optimization of the measurement matrix for compressive sensing,., in (2010) 18th European Signal Processing Conference, (2010) 427–431.
[57] L. Zelnik-Manor, K. Rosenblum, and Y. C. Eldar, .,Sensing Matrix Optimization for Block-Sparse Decoding,., IEEE Trans. Signal Process., 59(9) ( 2011) 4300–4312 , doi: 10.1109/TSP.2011.2159211.
[58] T. Strohmer and R. W. Heath Jr, .,Grassmannian frames with applications to coding and communication,., Appl. Comput. Harmon. Anal., 14( 3) (2003) 257–275.
[59] J. Xu, Y. Pi, and Z. Cao, .,Optimized projection matrix for compressive sensing,., EURASIP J. Adv. Signal Process., 2010(1) (2010) 560349.
[60] V. Abolghasemi, S. Ferdowsi, and S. Sanei, .,A gradient-based alternating minimization approach for optimization of the measurement matrix in compressive sensing,., Signal Processing, 92(4) 999–1009 (2010) 2012).
[61] W. Chen, M. R. D. Rodrigues, and I. J. Wassell, .,On the use of unit-norm tight frames to improve the average MSE performance in compressive sensing applications,., IEEE Signal Process. Lett., 19(1) (2011) 8–11.
[62] W. Chen, M. R. D. Rodrigues, and I. J. Wassell, .,Projection design for statistical compressive sensing: A tight frame-based approach,., IEEE Trans. signal Process., 61(8) (2013) 2016–2029.
[63] G. Li, Z. Zhu, D. Yang, L. Chang, and H. Bai, .,On projection matrix optimization for compressive sensing systems,., IEEE Trans. Signal Process., 61(11) (2013) 2887–2898.
[64] A. Yang, J. Zhang, and Z. Hou, .,Optimized sensing matrix design based on Parseval tight frame and matrix decomposition,., J. Commun., 8(7) (2013) 456–462.
[65] E. V Tsiligianni, L. P. Kondi, and A. K. Katsaggelos, .,Construction of incoherent unit norm tight frames with application to compressed sensing,., IEEE Trans. Inf. Theory, 60( 4) (2014) 2319–2330 .
[66] Q. Jiang, S. Li, H. Bai, R. C. de Lamare, and X. He, .,Gradient-based algorithm for designing sensing matrix considering real mutual coherence for compressed sensing systems,., IET Signal Process., 11( 4) (2017) 356–363.
[67] C. Rusu, .,Design of incoherent frames via convex optimization,., IEEE Signal Process. Lett., 20(7) (2013) 673–676.
[68] C. Rusu and N. González-Prelcic, .,Designing incoherent frames through convex techniques for optimized compressed sensing,., IEEE Trans. Signal Process., 64(9) (2016) 2334–2344.
[69] M. Sadeghi and M. Babaie-Zadeh, .,Incoherent unit-norm frame design via an alternating minimization penalty method,., IEEE Signal Process. Lett., 24(1) (2016) 32–36.
[70] R. Entezari and A. Rashidi, .,Measurement matrix optimization based on incoherent unit norm tight frame,., AEU-International J. Electron. Commun., 82 (2017) 321–326.
[71] N. Cleju, .,Optimized projections for compressed sensing via rank-constrained nearest correlation matrix,., Appl. Comput. Harmon. Anal., 36(3) (2014) 495–507.
[72] H. Bai, G. Li, S. Li, Q. Li, Q. Jiang, and L. Chang, .,Alternating optimization of sensing matrix and sparsifying dictionary for compressed sensing,., IEEE Trans. Signal Process., 63(6) (2015) 1581–1594 .
[73] G. Li, X. Li, S. Li, H. Bai, Q. Jiang, and X. He, .,Designing robust sensing matrix for image compression,., IEEE Trans. Image Process., 24(12) (2015) 5389–5400,.
[74] H. Bai, S. Li, and X. He, .,Sensing matrix optimization based on equiangular tight frames with consideration of sparse representation error,., IEEE Trans. Multimed., 18(10) (2016) 2040–2053.
[75] X. Li, H. Bai, and B. Hou, .,A gradient-based approach to optimization of compressed sensing systems,., Signal Processing, 139 (2017) 49–61.
[76] G. Li, Z. Zhu, X. Wu, and B. Hou, .,On joint optimization of sensing matrix and sparsifying dictionary for robust compressed sensing systems,., Digit. Signal Process., 73 (2017) 62–71, , doi: 10.1016/j.dsp.2017.10.023.
[77] T. Hong and Z. Zhu, .,Online Learning Sensing Matrix and Sparsifying Dictionary Simultaneously for Compressive Sensing,., Signal Processing, 153 (2018), doi: 10.1016/j.sigpro.2018.05.021.
[78] T. Hong and Z. Zhu, .,An efficient method for robust projection matrix design,., Signal Processing, 143 (2018) . 200–210.
[79] B. K. Natarajan, .,Sparse approximate solutions to linear systems,., SIAM J. Comput., 24(2) (1995) 227–234.
[80] J. A. Tropp and A. C. Gilbert, .,Signal recovery from random measurements via orthogonal matching pursuit,., IEEE Trans. Inf. Theory, 53( 12) (2007) 4655–4666.
[81] E. Candes and T. Tao, .,Decoding by linear programming,., arXiv Prepr. Math/0502327 (2005).
[82] D. L. Donoho and M. Elad, .,Optimally sparse representation in general (nonorthogonal) dictionaries via ?1 minimization,., Proc. Natl. Acad. Sci., 100(5) ( 2003) 2197–2202.
[83] J. A. Tropp, .,Greed is good: Algorithmic results for sparse approximation,., IEEE Trans. Inf. Theory, 50(10) (2004) 2231–2242.
[84] D. L. Donoho, M. Elad, and V. N. Temlyakov, .,Stable recovery of sparse overcomplete representations in the presence of noise,., IEEE Trans. Inf. Theory, 52( 1) (2005) 6–18,.
[85] R. Gribonval and M. Nielsen, .,Sparse representations in unions of bases,., Ieee Trans. Inf. Theory, vol. 49(12) (2003) 3320–3325,.
[86] E. Oey, D. Gunawan, and D. Sudiana, .,Projection Matrix Design for Co-Sparse Analysis Model-Based Compressive Sensing,., in MATEC Web of Conferences, 159 (2018) 1061.
[87] R. Rubinstein, T. Peleg, and M. Elad, .,Analysis K-SVD: A dictionary-learning algorithm for the sparse analysis model,., IEEE Trans. Signal Process., 61(61) (2013) 661–677.
[88] Y. Huang and C. Liu, .,Dai-Kou type conjugate gradient methods with a line search only using a gradient,., J. inequalities Appl., 2017(1) (2017) 66.
[89] B. C. Russell, A. Torralba, K. P. Murphy, and W. T. Freeman, .,LabelMe: a database and web-based tool for image annotation,., Int. J. Comput. Vis., 77 1–3 (2008) 157–173.
[90] R. Uetz and S. Behnke, .,Large-scale object recognition with CUDA-accelerated hierarchical neural networks,., in 2009 IEEE international conference on intelligent computing and intelligent systems, 1 (2009) 536–541.
[91] Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, .,Image quality assessment: from error visibility to structural similarity,., IEEE Trans. image Process., 13(4) (2004) 600–612.