期刊
SIAM JOURNAL ON IMAGING SCIENCES
卷 9, 期 3, 页码 1004-1041出版社
SIAM PUBLICATIONS
DOI: 10.1137/15M1042280
关键词
finite-rate-of-innovation; off-the-grid; parametric image models; Prony's method; trigonometric curves; annihilating filter; Fourier extrapolation; superresolution; MRI
类别
资金
- NSF [CCF-0844812, CCF-1116067]
- NIH [1R21HL109710-01A1, 1R01EB019961-01A1]
- ACS [RSG-11-267-01-CCE]
- ONR [N000141310202]
- NATIONAL HEART, LUNG, AND BLOOD INSTITUTE [R21HL109710] Funding Source: NIH RePORTER
- NATIONAL INSTITUTE OF BIOMEDICAL IMAGING AND BIOENGINEERING [R01EB019961] Funding Source: NIH RePORTER
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 that 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 that 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 superresolution recovery of MRI phantoms and real MRI data from low-pass Fourier samples, which shows benefits over standard approaches for single-image superresolution MRI.
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