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...
| Main Authors: | , , |
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| Other Authors: | |
| Format: | Article |
| Language: | en_US |
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Society for Industrial and Applied Mathematics
2010
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| Online Access: | http://hdl.handle.net/1721.1/57584 https://orcid.org/0000-0002-7637-2914 |
| _version_ | 1811086788833312768 |
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| author | Adalsteinsson, Elfar Zelinski, Adam C. Goyal, Vivek K. |
| author2 | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science |
| author_facet | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Adalsteinsson, Elfar Zelinski, Adam C. Goyal, Vivek K. |
| author_sort | Adalsteinsson, Elfar |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T13:34:45Z |
| format | Article |
| id | mit-1721.1/57584 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T13:34:45Z |
| publishDate | 2010 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | mit-1721.1/575842022-09-28T14:47:40Z Simultaneously Sparse Solutions to Linear Inverse Problems with Multiple System Matrices and a Single Observation Vector Adalsteinsson, Elfar Zelinski, Adam C. Goyal, Vivek K. Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology. Research Laboratory of Electronics Adalsteinsson, Elfar Adalsteinsson, Elfar Zelinski, Adam C. Goyal, Vivek K. 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. National Institutes of Health (grants 1P41RR14075, 1R01EB000790, 1R01EB006847, 1R01EB007942) National Science Foundation (CAREER Grant 0643836) United States Department of Defense. National Defense Science and Engineering Graduate Fellowship (F49620-02- C-0041) Mind Research Institute Athinoula A. Martinos Center for Biomedical Imaging Siemens Medical Solutions R. J. Shillman’s Career Development Award 2010-08-27T16:01:01Z 2010-08-27T16:01:01Z 2010-01 2009-07 Article http://purl.org/eprint/type/JournalArticle 1064-8275 1095-7197 http://hdl.handle.net/1721.1/57584 Zelinski, Adam C., Vivek K. Goyal, and Elfar Adalsteinsson. “Simultaneously Sparse Solutions to Linear Inverse Problems with Multiple System Matrices and a Single Observation Vector.” SIAM Journal on Scientific Computing 31.6 (2010): 4533-4579. ©2010 Society for Industrial and Applied Mathematics. https://orcid.org/0000-0002-7637-2914 en_US http://dx.doi.org/10.1137/080730822 SIAM Journal on Scientific Computing Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf Society for Industrial and Applied Mathematics SIAM |
| spellingShingle | Adalsteinsson, Elfar Zelinski, Adam C. Goyal, Vivek K. Simultaneously Sparse Solutions to Linear Inverse Problems with Multiple System Matrices and a Single Observation Vector |
| title | Simultaneously Sparse Solutions to Linear Inverse Problems with Multiple System Matrices and a Single Observation Vector |
| title_full | Simultaneously Sparse Solutions to Linear Inverse Problems with Multiple System Matrices and a Single Observation Vector |
| title_fullStr | Simultaneously Sparse Solutions to Linear Inverse Problems with Multiple System Matrices and a Single Observation Vector |
| title_full_unstemmed | Simultaneously Sparse Solutions to Linear Inverse Problems with Multiple System Matrices and a Single Observation Vector |
| title_short | Simultaneously Sparse Solutions to Linear Inverse Problems with Multiple System Matrices and a Single Observation Vector |
| title_sort | simultaneously sparse solutions to linear inverse problems with multiple system matrices and a single observation vector |
| url | http://hdl.handle.net/1721.1/57584 https://orcid.org/0000-0002-7637-2914 |
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