American Institute of Mathematical Sciences

Advanced
November  2013, 7(4): 1379-1392. doi: 10.3934/ipi.2013.7.1379

Seismic data reconstruction via matrix completion

Yi Yang 1, , Jianwei Ma 2, and  Stanley Osher 3,

1. 

Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095, United States
2. 

Department of Mathematics, Harbin Institute of Technology, Harbin, China
3. 

Department of Mathematics, University of California, Los Angeles, CA 90095

Received  February 2012 Revised  February 2013 Published  November 2013

In seismic processing, one goal is to recover missing traces when the data is sparsely and incompletely sampled. We present a method which treats this reconstruction problem from a novel perspective. By utilizing its connection with the general matrix completion (MC) problem, we build an approximately low-rank matrix, which can be reconstructed through solving a proper nuclear norm minimization problem. Two efficient algorithms, accelerated proximal gradient method (APG) and low-rank matrix fitting (LMaFit) are discussed in this paper. The seismic data can then be recovered by the conversion of the completed matrix into the original signal space. Numerical experiments show the efficiency and high performance of data recovery for our model compared with other models.
Keywords: Principal component analysis, matrix completion, pattern estimation..
Mathematics Subject Classification: 15A29, 15A8.
Citation: Yi Yang, Jianwei Ma, Stanley Osher. Seismic data reconstruction via matrix completion. Inverse Problems and Imaging, 2013, 7 (4) : 1379-1392. doi: 10.3934/ipi.2013.7.1379
References:
[1]

M. Sacchi, T. Ulrych and C. Walker, Interpolation and extrapolation using a high-resolution discrete fourier transform, IEEE Transactions on Signal Processing, 46 (1998), 31-38. doi: 10.1109/78.651165.

[2]

A. Duijndam, M. Schonewille and C. Hindriks, Reconstruction of band-limited signals, irregularly sampled along one spatial direction, Geophysics, 64 (1999), 524-538. doi: 10.1190/1.1444559.

[3]

B. Liu and M. D. Sacchi, Minimum weighted norm interpolation of seismic records, Geophysics, 69 (2004), 1560-1568. doi: 10.1190/1.1836829.

[4]

S. Xu, Y. Zhang, D. L. Pham and G. Lambaré, Antileakage Fourier transform for seismic data regularization, Geophysics, 70 (2005), V87-V95. doi: 10.1190/1.1993713.

[5]

F. J. Herrmann and G. Hennenfent, Non-parametric seismic data recovery with curvelet frames, Geophysical Journal International, 173 (2008), 233-248. doi: 10.1111/j.1365-246X.2007.03698.x.

[6]

R. Shahidi, G. Tang, J. Ma and F. J. Herrmann, Application of randomized sampling schemes to curvelet-based sparsity-promoting seismic data recovery, Geophysical Prospecting, 61(2013), 973-997. doi: 10.1111/1365-2478.12050.

[7]

M. Naghizadeh and M. Sacchi, Beyond alias hierarchical scale curvelet interpolation of regularly and irregularly sampled seismic data, Geophysics, 75 (2010), WB189-WB202. doi: 10.1190/1.3509468.

[8]

S. Hauser and J. Ma, Seismic data reconstruction via directional weighted shearlet-regularized inpainting, Preprint, TUK, 2012.

[9]

S. Spitz, Seismic trace interpolation in the f-x domain, Geophysics, 56 (1991), 785-794.

[10]

S. Crawley, J. Claerbout and R. Clapp, Interpolation with smoothly nonstationary prediction-error filters, in 69th Annual International Meeting, SEG, Expanded Abstracts, 1913-1916, 1999. doi: 10.1190/1.1820707.

[11]

M. Porsani, Seismic trace interpolation using half-step prediction filters, Geophysics, 64 (1999), 1461-1467. doi: 10.1190/1.1444650.

[12]

Y. Liu and S. Fomel, Seismic data interpolation beyond aliasing using regularized nonstationary autoegression, Geophysics, 76 (2011), V69-V77. doi: 10.1190/geo2010-0231.1.

[13]

M. Naghizadeh and M. Sacchi, Seismic data reconstruction using multidimensional prediction filters, Geophysical Prospecting, 58 (2010), 157-173. doi: 10.1111/j.1365-2478.2009.00805.x.

[14]

S. Trickett, L. Burroughs, A. Milton, L. Walton and R. Dack, Rank-reduction-based trace interpolation, 80th Annual meeting, SEG, Expanded Abstracts, (2010). doi: 10.1190/1.3513645.

[15]

V. Oropeza and M. Sacchi, Simultaneous seismic data denoising and reconstruction via multichannel singular spectrum analysis, Geophysics, 76 (2011), V25-V32. doi: 10.1190/1.3552706.

[16]

R. Vautard and M. Ghil, Singular spectrum analysis in nonlinear dynamics, with applications to paleoclimatic time series, Physica D, 35 (1989), 395-424. doi: 10.1016/0167-2789(89)90077-8.

[17]

E. J. Candes and B. Recht, Exact matrix completion via convex optimization, Foundations of Computational Mathematics, 9 (2009), 717-772. doi: 10.1007/s10208-009-9045-5.

[18]

H. Ji, C. Liu, Z. Shen and Y. Xu, Robust video denoising using low rank matrix completion, in Computer Vision and Pattern Recognition (CVPR), 2010 IEEE Conference, June 2010. doi: 10.1109/CVPR.2010.5539849.

[19]

J.-F. Cai, E. J. Candes and Z. Shen, A singular value thresholding algorithm for matrix completion, SIAM Journal on Optimization, 20 (2008), 1956-1982. doi: 10.1137/080738970.

[20]

S. Ma, D. Goldfarb and L. Chen, Fixed point and bregman iterative methods for matrix rank minimization, Mathematical Programming: Series A and B, 128 (2011), 321-353. doi: 10.1007/s10107-009-0306-5.

[21]

J. Yang and X. Yuan, An inexact alternating direction method for trace norm regularized least squares problem, Optimization Online, (2010).

[22]

R. Keshavan, A. Montanari and S.Oh, Matrix completion from a few entries, IEEE Transactions on Information Theory, 56 (2010), 2980-2998. doi: 10.1109/TIT.2010.2046205.

[23]

Y. Nesterov, A method of solving a convex programming problem with convergence rate o($1/k^2$), Soviet Mathematics Doklady, 27 (1983), 372-376.

[24]

A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM Journal on Imaging Sciences, (2009), 183-202. doi: 10.1137/080716542.

[25]

K.-C. Toh and S. Yun, An advanced proximal gradient algorithm for nuclear norm regularized linear least squares problems, Pac. J. Optim., 6 (2010), 615-640.

[26]

Z. Wen, W. Yin and Y. Zhang, Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm, Mathematical Programming Computation, 4 (2012), 333-361. doi: 10.1007/s12532-012-0044-1.

[27]

H. Schaeffer and S. Osher, A low patch-rank interpretation of texture, SIAM J. Imaging Sci., 6 (2013), 226-262. doi: 10.1137/110854989.

[28]

Y. Hua, Estimating two-dimensional frequencies by matrix enhancement and matrix pencil, IEEE Transactions on Signal Processing, 40 (1992), 2267-2280.

show all references

References:
[1]

M. Sacchi, T. Ulrych and C. Walker, Interpolation and extrapolation using a high-resolution discrete fourier transform, IEEE Transactions on Signal Processing, 46 (1998), 31-38. doi: 10.1109/78.651165.

[2]

A. Duijndam, M. Schonewille and C. Hindriks, Reconstruction of band-limited signals, irregularly sampled along one spatial direction, Geophysics, 64 (1999), 524-538. doi: 10.1190/1.1444559.

[3]

B. Liu and M. D. Sacchi, Minimum weighted norm interpolation of seismic records, Geophysics, 69 (2004), 1560-1568. doi: 10.1190/1.1836829.

[4]

S. Xu, Y. Zhang, D. L. Pham and G. Lambaré, Antileakage Fourier transform for seismic data regularization, Geophysics, 70 (2005), V87-V95. doi: 10.1190/1.1993713.

[5]

F. J. Herrmann and G. Hennenfent, Non-parametric seismic data recovery with curvelet frames, Geophysical Journal International, 173 (2008), 233-248. doi: 10.1111/j.1365-246X.2007.03698.x.

[6]

R. Shahidi, G. Tang, J. Ma and F. J. Herrmann, Application of randomized sampling schemes to curvelet-based sparsity-promoting seismic data recovery, Geophysical Prospecting, 61(2013), 973-997. doi: 10.1111/1365-2478.12050.

[7]

M. Naghizadeh and M. Sacchi, Beyond alias hierarchical scale curvelet interpolation of regularly and irregularly sampled seismic data, Geophysics, 75 (2010), WB189-WB202. doi: 10.1190/1.3509468.

[8]

S. Hauser and J. Ma, Seismic data reconstruction via directional weighted shearlet-regularized inpainting, Preprint, TUK, 2012.

[9]

S. Spitz, Seismic trace interpolation in the f-x domain, Geophysics, 56 (1991), 785-794.

[10]

S. Crawley, J. Claerbout and R. Clapp, Interpolation with smoothly nonstationary prediction-error filters, in 69th Annual International Meeting, SEG, Expanded Abstracts, 1913-1916, 1999. doi: 10.1190/1.1820707.

[11]

M. Porsani, Seismic trace interpolation using half-step prediction filters, Geophysics, 64 (1999), 1461-1467. doi: 10.1190/1.1444650.

[12]

Y. Liu and S. Fomel, Seismic data interpolation beyond aliasing using regularized nonstationary autoegression, Geophysics, 76 (2011), V69-V77. doi: 10.1190/geo2010-0231.1.

[13]

M. Naghizadeh and M. Sacchi, Seismic data reconstruction using multidimensional prediction filters, Geophysical Prospecting, 58 (2010), 157-173. doi: 10.1111/j.1365-2478.2009.00805.x.

[14]

S. Trickett, L. Burroughs, A. Milton, L. Walton and R. Dack, Rank-reduction-based trace interpolation, 80th Annual meeting, SEG, Expanded Abstracts, (2010). doi: 10.1190/1.3513645.

[15]

V. Oropeza and M. Sacchi, Simultaneous seismic data denoising and reconstruction via multichannel singular spectrum analysis, Geophysics, 76 (2011), V25-V32. doi: 10.1190/1.3552706.

[16]

R. Vautard and M. Ghil, Singular spectrum analysis in nonlinear dynamics, with applications to paleoclimatic time series, Physica D, 35 (1989), 395-424. doi: 10.1016/0167-2789(89)90077-8.

[17]

E. J. Candes and B. Recht, Exact matrix completion via convex optimization, Foundations of Computational Mathematics, 9 (2009), 717-772. doi: 10.1007/s10208-009-9045-5.

[18]

H. Ji, C. Liu, Z. Shen and Y. Xu, Robust video denoising using low rank matrix completion, in Computer Vision and Pattern Recognition (CVPR), 2010 IEEE Conference, June 2010. doi: 10.1109/CVPR.2010.5539849.

[19]

J.-F. Cai, E. J. Candes and Z. Shen, A singular value thresholding algorithm for matrix completion, SIAM Journal on Optimization, 20 (2008), 1956-1982. doi: 10.1137/080738970.

[20]

S. Ma, D. Goldfarb and L. Chen, Fixed point and bregman iterative methods for matrix rank minimization, Mathematical Programming: Series A and B, 128 (2011), 321-353. doi: 10.1007/s10107-009-0306-5.

[21]

J. Yang and X. Yuan, An inexact alternating direction method for trace norm regularized least squares problem, Optimization Online, (2010).

[22]

R. Keshavan, A. Montanari and S.Oh, Matrix completion from a few entries, IEEE Transactions on Information Theory, 56 (2010), 2980-2998. doi: 10.1109/TIT.2010.2046205.

[23]

Y. Nesterov, A method of solving a convex programming problem with convergence rate o($1/k^2$), Soviet Mathematics Doklady, 27 (1983), 372-376.

[24]

A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM Journal on Imaging Sciences, (2009), 183-202. doi: 10.1137/080716542.

[25]

K.-C. Toh and S. Yun, An advanced proximal gradient algorithm for nuclear norm regularized linear least squares problems, Pac. J. Optim., 6 (2010), 615-640.

[26]

Z. Wen, W. Yin and Y. Zhang, Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm, Mathematical Programming Computation, 4 (2012), 333-361. doi: 10.1007/s12532-012-0044-1.

[27]

H. Schaeffer and S. Osher, A low patch-rank interpretation of texture, SIAM J. Imaging Sci., 6 (2013), 226-262. doi: 10.1137/110854989.

[28]

Y. Hua, Estimating two-dimensional frequencies by matrix enhancement and matrix pencil, IEEE Transactions on Signal Processing, 40 (1992), 2267-2280.

[1]

Hui Zhang, Jian-Feng Cai, Lizhi Cheng, Jubo Zhu. Strongly convex programming for exact matrix completion and robust principal component analysis. Inverse Problems and Imaging, 2012, 6 (2) : 357-372. doi: 10.3934/ipi.2012.6.357
[2]

Qingshan You, Qun Wan, Yipeng Liu. A short note on strongly convex programming for exact matrix completion and robust principal component analysis. Inverse Problems and Imaging, 2013, 7 (1) : 305-306. doi: 10.3934/ipi.2013.7.305
[3]

Yitong Guo, Bingo Wing-Kuen Ling. Principal component analysis with drop rank covariance matrix. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2345-2366. doi: 10.3934/jimo.2020072
[4]

Ningyu Sha, Lei Shi, Ming Yan. Fast algorithms for robust principal component analysis with an upper bound on the rank. Inverse Problems and Imaging, 2021, 15 (1) : 109-128. doi: 10.3934/ipi.2020067
[5]

Yuhong Dai, Nobuo Yamashita. Convergence analysis of sparse quasi-Newton updates with positive definite matrix completion for two-dimensional functions. Numerical Algebra, Control and Optimization, 2011, 1 (1) : 61-69. doi: 10.3934/naco.2011.1.61
[6]

Henry Adams, Lara Kassab, Deanna Needell. An adaptation for iterative structured matrix completion. Foundations of Data Science, 2021, 3 (4) : 769-791. doi: 10.3934/fods.2021028
[7]

Azam Moradi, Jafar Razmi, Reza Babazadeh, Ali Sabbaghnia. An integrated Principal Component Analysis and multi-objective mathematical programming approach to agile supply chain network design under uncertainty. Journal of Industrial and Management Optimization, 2019, 15 (2) : 855-879. doi: 10.3934/jimo.2018074
[8]

Yuan Shen, Xin Liu. An alternating minimization method for matrix completion problems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1757-1772. doi: 10.3934/dcdss.2020103
[9]

Ke Wei, Jian-Feng Cai, Tony F. Chan, Shingyu Leung. Guarantees of riemannian optimization for low rank matrix completion. Inverse Problems and Imaging, 2020, 14 (2) : 233-265. doi: 10.3934/ipi.2020011
[10]

Charles Curry, Stephen Marsland, Robert I McLachlan. Principal symmetric space analysis. Journal of Computational Dynamics, 2019, 6 (2) : 251-276. doi: 10.3934/jcd.2019013
[11]

Yangyang Xu, Ruru Hao, Wotao Yin, Zhixun Su. Parallel matrix factorization for low-rank tensor completion. Inverse Problems and Imaging, 2015, 9 (2) : 601-624. doi: 10.3934/ipi.2015.9.601
[12]

Daniel Wetzel. Pattern analysis in a benthic bacteria-nutrient system. Mathematical Biosciences & Engineering, 2016, 13 (2) : 303-332. doi: 10.3934/mbe.2015004
[13]

Fengqi Yi, Eamonn A. Gaffney, Sungrim Seirin-Lee. The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 647-668. doi: 10.3934/dcdsb.2017031
[14]

Kolade M. Owolabi. Numerical analysis and pattern formation process for space-fractional superdiffusive systems. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 543-566. doi: 10.3934/dcdss.2019036
[15]

Israa Mohammed Khudher, Yahya Ismail Ibrahim, Suhaib Abduljabbar Altamir. Individual biometrics pattern based artificial image analysis techniques. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 567-578. doi: 10.3934/naco.2020056
[16]

Dominique Duncan, Thomas Strohmer. Classification of Alzheimer's disease using unsupervised diffusion component analysis. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1119-1130. doi: 10.3934/mbe.2016033
[17]

Krzysztof Fujarewicz, Krzysztof Łakomiec. Parameter estimation of systems with delays via structural sensitivity analysis. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2521-2533. doi: 10.3934/dcdsb.2014.19.2521
[18]

Vassilios A. Tsachouridis, Georgios Giantamidis, Stylianos Basagiannis, Kostas Kouramas. Formal analysis of the Schulz matrix inversion algorithm: A paradigm towards computer aided verification of general matrix flow solvers. Numerical Algebra, Control and Optimization, 2020, 10 (2) : 177-206. doi: 10.3934/naco.2019047
[19]

Daniel Roggen, Martin Wirz, Gerhard Tröster, Dirk Helbing. Recognition of crowd behavior from mobile sensors with pattern analysis and graph clustering methods. Networks and Heterogeneous Media, 2011, 6 (3) : 521-544. doi: 10.3934/nhm.2011.6.521
[20]

Bilal Saad, Mazen Saad. Numerical analysis of a non equilibrium two-component two-compressible flow in porous media. Discrete and Continuous Dynamical Systems - S, 2014, 7 (2) : 317-346. doi: 10.3934/dcdss.2014.7.317

2020 Impact Factor: 1.639

Tools

Metrics

  • PDF downloads (275)
  • HTML views (0)
  • Cited by (21)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy