Journal Article10.1190/GEO2014-0369.1
Efficient matrix completion for seismic data reconstruction
Rajiv Kumar,Curt Da Silva,Okan Akalin,Aleksandr Y. Aravkin,Hassan Mansour,Benjamin Recht,Felix J. Herrmann +6 more
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TL;DR: In this article, a low-rank optimization technique was proposed to recover the missing trace of seismic data from the source and receiver coordinates, where the original signal is low rank and the subsampling scheme increases the singular values of the matrix.
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Abstract: Despite recent developments in improved acquisition, seismic data often remain undersampled along source and receiver coordinates, resulting in incomplete data for key applications such as migration and multiple prediction. We have interpreted the missing-trace interpolation problem in the context of matrix completion (MC), and developed three practical principles for using low-rank optimization techniques to recover seismic data. Specifically, we strive for recovery scenarios wherein the original signal is low rank and the subsampling scheme increases the singular values of the matrix. We use an optimization program that restores this low-rank structure to recover the full volume. Omitting one or more of these principles can lead to poor interpolation results, as we found experimentally. In light of this theory, we compensate for the high-rank behavior of data in the source-receiver domain by using the midpoint-offset transformation for 2D data and a source-receiver permutation for 3D data to red...
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References
Exact Matrix Completion via Convex Optimization
TL;DR: It is proved that one can perfectly recover most low-rank matrices from what appears to be an incomplete set of entries, and that objects other than signals and images can be perfectly reconstructed from very limited information.
A Singular Value Thresholding Algorithm for Matrix Completion
TL;DR: This paper develops a simple first-order and easy-to-implement algorithm that is extremely efficient at addressing problems in which the optimal solution has low rank, and develops a framework in which one can understand these algorithms in terms of well-known Lagrange multiplier algorithms.
Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions
TL;DR: This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation, and presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions.
Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization
TL;DR: It is shown that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum-rank solution can be recovered by solving a convex optimization problem, namely, the minimization of the nuclear norm over the given affine space.
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•Posted Content
A Singular Value Thresholding Algorithm for Matrix Completion
TL;DR: In this article, a convex relaxation of a rank minimization problem is proposed to approximate the matrix with minimum nuclear norm among all matrices obeying a set of convex constraints.
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