Journal article
Off-the-Grid Recovery of Piecewise Constant Images from Few Fourier Samples
SIAM journal on imaging sciences, Vol.9(3), pp.1004-1041
2016
DOI: 10.1137/15M1042280
PMCID: PMC6028195
PMID: 29973971
Abstract
We introduce a method to recover a continuous domain representation of a piecewise constant two-dimensional image from few low-pass Fourier samples. Assuming the edge set of the image is localized to the zero set of a trigonometric polynomial, we show the Fourier coefficients of the partial derivatives of the image satisfy a linear annihilation relation. We present necessary and sufficient conditions for unique recovery of the image from finite low-pass Fourier samples using the annihilation relation. We also propose a practical two-stage recovery algorithm which is robust to model-mismatch and noise. In the first stage we estimate a continuous domain representation of the edge set of the image. In the second stage we perform an extrapolation in Fourier domain by a least squares two-dimensional linear prediction, which recovers the exact Fourier coefficients of the underlying image. We demonstrate our algorithm on the super-resolution recovery of MRI phantoms and real MRI data from low-pass Fourier samples, which shows benefits over standard approaches for single-image super-resolution MRI.
Details
- Title: Subtitle
- Off-the-Grid Recovery of Piecewise Constant Images from Few Fourier Samples
- Creators
- Greg Ongie - Department of Mathematics, University of Iowa, Iowa City, IowaMathews Jacob - Department of Electrical and Computer Engineering, University of Iowa, Iowa City, Iowa
- Resource Type
- Journal article
- Publication Details
- SIAM journal on imaging sciences, Vol.9(3), pp.1004-1041
- DOI
- 10.1137/15M1042280
- PMID
- 29973971
- PMCID
- PMC6028195
- NLM abbreviation
- SIAM J Imaging Sci
- ISSN
- 1936-4954
- eISSN
- 1936-4954
- Publisher
- United States
- Grant note
- DOI: 10.13039/100000048, name: American Cancer Society, award: RSG-11-267-01-CCE; DOI: 10.13039/100000006, name: Office of Naval Research, award: N000141310202; DOI: 10.13039/100000002, name: National Institutes of Health, award: 1R21HL109710-01A1, 1R01EB019961-01A1; DOI: 10.13039/100000001, name: National Science Foundation, award: CCF-0844812, CCF-1116067
- Language
- English
- Date published
- 2016
- Academic Unit
- Roy J. Carver Department of Biomedical Engineering; Radiology; Electrical and Computer Engineering; Iowa Neuroscience Institute; Radiation Oncology
- Record Identifier
- 9984070999802771
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