Simultaneously Sparse Solutions to Linear Inverse Problems with Multiple System Matrices and a Single Observation Vector

A problem that arises in slice-selective magnetic resonance imaging (MRI) radio-frequency (RF) excitation pulse design is abstracted as a novel linear inverse problem with a simultaneous sparsity constraint. Multiple unknown signal vectors are to be determined, where each passes through a different...

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Bibliographic Details
Published in:SIAM journal on scientific computing Vol. 31; no. 6; pp. 4533 - 4579
Main Authors: Zelinski, Adam C., Goyal, Vivek K., Adalsteinsson, Elfar
Format: Journal Article
Language:English
Published: Philadelphia, PA Society for Industrial and Applied Mathematics 20.01.2010
Subjects:
Algebra
Algebraic geometry
Algorithms
Approximation
Exact sciences and technology
Experiments
Inverse problems
Magnetic resonance imaging
Mathematics
Methods of scientific computing (including symbolic computation, algebraic computation)
Numerical analysis
Numerical analysis in abstract spaces
Numerical analysis. Scientific computation
Numerical linear algebra
Performance evaluation
Sciences and techniques of general use
Sparsity
iteratively reweighted least squares
34A55
Programming
Iterative method
matching pursuit
Direct method
Least squares method
simultaneous sparse approximation
94A12
46N10
second-order cone programming
Approximation
Numerical linear algebra
magnetic resonance imaging excitation pulse design
Inverse problem
15A29
iterative shrinkage
Numerical analysis
Linear system
Scientific computation
Matrix inversion
Sparse set
Greedy algorithm
sparse approximation
multiple measurement vectors
Resonance
ISSN:1064-8275, 1095-7197
Online Access:Get full text
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Summary:A problem that arises in slice-selective magnetic resonance imaging (MRI) radio-frequency (RF) excitation pulse design is abstracted as a novel linear inverse problem with a simultaneous sparsity constraint. Multiple unknown signal vectors are to be determined, where each passes through a different system matrix and the results are added to yield a single observation vector. Given the matrices and lone observation, the objective is to find a simultaneously sparse set of unknown vectors that approximately solves the system. We refer to this as the multiple-system single-output (MSSO) simultaneous sparse approximation problem. This manuscript contrasts the MSSO problem with other simultaneous sparsity problems and conducts an initial exploration of algorithms with which to solve it. Greedy algorithms and techniques based on convex relaxation are derived and compared empirically. Experiments involve sparsity pattern recovery in noiseless and noisy settings and MRI RF pulse design.
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ISSN:1064-8275
1095-7197
DOI:10.1137/080730822