Off-the-Grid Low-Rank Matrix Recovery and Seismic Data Reconstruction

• Published in: IEEE journal of selected topics in signal processing Vol. 10; no. 4; pp. 658 - 671
• Main Authors: Lopez, Oscar, Kumar, Rajiv, Yilmaz, Ozgur, Herrmann, Felix J.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.06.2016; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Bar codes; Compressed sensing; data regularization; Discrete Fourier transforms; Interpolation; matrix completion; Matrix sensing; Noise measurement; non-uniform discrete Fourier transform; nuclearnorm relaxation; Periodic structures; Robustness; seismic data; seismic trace interpolation; Sensors; data regularization; seismic trace interpolation; matrix completion; nonuniform discrete Fourier transform; Matrix sensing; nuclear-norm relaxation; seismic data
• ISSN: 1932-4553; 1941-0484
• DOI: 10.1109/JSTSP.2016.2555482

Matrix sensing problems capitalize on the knowledge that a data matrix of interest exhibits low rank properties. This low dimensional structure often arises because the data matrix is obtained by sampling a smooth function on a regular (or structured) grid. However, in many practical situations the measurements are taken on an irregular grid (that is accurately known). This results in an "unstructured data matrix" that is less fit for the low rank model in comparison to its regular counterpart and therefore subject to degraded reconstruction via rank penalization techniques. In this paper, we propose and analyze a modified low-rank matrix recovery work-flow that admits unstructured observations. By incorporating a regularization operator which accurately maps structured data to unstructured data, into the nuclear-norm minimization problem, we are able to compensate for data irregularity. Furthermore, by construction our formulation yields output that is supported on a structured grid. We establish recovery error bounds for our methodology and offer matrix sensing and matrix completion numerical experiments including applications to seismic trace interpolation to demonstrate the potential of the approach.