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

Seismic data reconstruction via matrix completion

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.
Citation: Yi Yang, Jianwei Ma, Stanley Osher. Seismic data reconstruction via matrix completion. Inverse Problems & 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. 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. doi: 10.1190/1.1444559.

[3]

B. Liu and M. D. Sacchi, Minimum weighted norm interpolation of seismic records,, Geophysics, 69 (2004), 1560. 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). 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. 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, (2013), 973. 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). doi: 10.1190/1.3509468.

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

M. Naghizadeh and M. Sacchi, Seismic data reconstruction using multidimensional prediction filters,, Geophysical Prospecting, 58 (2010), 157. 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, (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). 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. 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. 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). 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. 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. 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. 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.

[24]

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

[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. 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. 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.

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. 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. doi: 10.1190/1.1444559.

[3]

B. Liu and M. D. Sacchi, Minimum weighted norm interpolation of seismic records,, Geophysics, 69 (2004), 1560. 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). 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. 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, (2013), 973. 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). doi: 10.1190/1.3509468.

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

M. Naghizadeh and M. Sacchi, Seismic data reconstruction using multidimensional prediction filters,, Geophysical Prospecting, 58 (2010), 157. 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, (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). 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. 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. 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). 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. 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. 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. 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.

[24]

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

[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. 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. 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.

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