Convex Recovery From Interferometric Measurements
This paper discusses some questions that arise when a linear inverse problem involving Ax = b is reformulated in the interferometric framework, where quadratic combinations of b are considered as data in place of b. First, we show a deterministic recovery result for vectors x from measurements of th...
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| Format: | Article |
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Institute of Electrical and Electronics Engineers (IEEE)
2018
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| Online Access: | http://hdl.handle.net/1721.1/116246 https://orcid.org/0000-0001-7052-5097 |
| _version_ | 1826207620845797376 |
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| author | Demanet, Laurent Jugnon, Vincent |
| author2 | Massachusetts Institute of Technology. Department of Earth, Atmospheric, and Planetary Sciences |
| author_facet | Massachusetts Institute of Technology. Department of Earth, Atmospheric, and Planetary Sciences Demanet, Laurent Jugnon, Vincent |
| author_sort | Demanet, Laurent |
| collection | MIT |
| description | This paper discusses some questions that arise when a linear inverse problem involving Ax = b is reformulated in the interferometric framework, where quadratic combinations of b are considered as data in place of b. First, we show a deterministic recovery result for vectors x from measurements of the form (Ax)[subscript i] [bar over (Ax)[subscript j]] for some left-invertible A. Recovery is exact, or stable in the noisy case, when the couples (i, j) are chosen as edges of a well-connected graph. One possible way of obtaining the solution is as a feasible point of a simple semidefinite program. Furthermore, we show how the proportionality constant in the error estimate depends on the spectral gap of a data-weighted graph Laplacian. Second, we present a new application of this formulation to interferometric waveform inversion, where products of the form (Ax)[subscript i] [bar over (Ax)[subscript j]] in frequency encode generalized cross correlations in time. We present numerical evidence that interferometric inversion does not suffer from the loss of resolution generally associated with interferometric imaging, and can provide added robustness with respect to specific kinds of kinematic uncertainties in the forward model A. |
| first_indexed | 2024-09-23T13:52:42Z |
| format | Article |
| id | mit-1721.1/116246 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T13:52:42Z |
| publishDate | 2018 |
| publisher | Institute of Electrical and Electronics Engineers (IEEE) |
| record_format | dspace |
| spelling | mit-1721.1/1162462022-10-01T17:42:09Z Convex Recovery From Interferometric Measurements Demanet, Laurent Jugnon, Vincent Massachusetts Institute of Technology. Department of Earth, Atmospheric, and Planetary Sciences Massachusetts Institute of Technology. Department of Mathematics Demanet, Laurent Jugnon, Vincent This paper discusses some questions that arise when a linear inverse problem involving Ax = b is reformulated in the interferometric framework, where quadratic combinations of b are considered as data in place of b. First, we show a deterministic recovery result for vectors x from measurements of the form (Ax)[subscript i] [bar over (Ax)[subscript j]] for some left-invertible A. Recovery is exact, or stable in the noisy case, when the couples (i, j) are chosen as edges of a well-connected graph. One possible way of obtaining the solution is as a feasible point of a simple semidefinite program. Furthermore, we show how the proportionality constant in the error estimate depends on the spectral gap of a data-weighted graph Laplacian. Second, we present a new application of this formulation to interferometric waveform inversion, where products of the form (Ax)[subscript i] [bar over (Ax)[subscript j]] in frequency encode generalized cross correlations in time. We present numerical evidence that interferometric inversion does not suffer from the loss of resolution generally associated with interferometric imaging, and can provide added robustness with respect to specific kinds of kinematic uncertainties in the forward model A. United States. Air Force. Office of Scientific Research (Grant FA9550-12-1-0328) United States. Air Force. Office of Scientific Research (Grant FA9550-15-1-0078) United States. Office of Naval Research (Grant N00014-16-1-2122) National Science Foundation (U.S.) (Grant DMS-1255203) Alfred P. Sloan Foundation Massachusetts Institute of Technology. Earth Resources Laboratory TOTAL (Firm) 2018-06-12T14:42:47Z 2018-06-12T14:42:47Z 2017-04 2018-05-17T17:18:35Z Article http://purl.org/eprint/type/JournalArticle 2333-9403 2334-0118 http://hdl.handle.net/1721.1/116246 Demanet, Laurent, and Vincent Jugnon. “Convex Recovery From Interferometric Measurements.” IEEE Transactions on Computational Imaging, vol. 3, no. 2, June 2017, pp. 282–95. https://orcid.org/0000-0001-7052-5097 http://dx.doi.org/10.1109/TCI.2017.2688923 IEEE Transactions on Computational Imaging Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Institute of Electrical and Electronics Engineers (IEEE) MIT Web Domain |
| spellingShingle | Demanet, Laurent Jugnon, Vincent Convex Recovery From Interferometric Measurements |
| title | Convex Recovery From Interferometric Measurements |
| title_full | Convex Recovery From Interferometric Measurements |
| title_fullStr | Convex Recovery From Interferometric Measurements |
| title_full_unstemmed | Convex Recovery From Interferometric Measurements |
| title_short | Convex Recovery From Interferometric Measurements |
| title_sort | convex recovery from interferometric measurements |
| url | http://hdl.handle.net/1721.1/116246 https://orcid.org/0000-0001-7052-5097 |
| work_keys_str_mv | AT demanetlaurent convexrecoveryfrominterferometricmeasurements AT jugnonvincent convexrecoveryfrominterferometricmeasurements |