Structured low-rank recovery of piecewise constant signals with performance guarantees

Greg Ongie; Sampurna Biswas; Mathews Jacob

doi:10.1109/ICIP.2016.7532500

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Conference proceeding

Structured low-rank recovery of piecewise constant signals with performance guarantees

Greg Ongie, Sampurna Biswas and Mathews Jacob

2016 IEEE International Conference on Image Processing (ICIP), Vol.2016-, pp.963-967

09/2016

DOI: 10.1109/ICIP.2016.7532500

PMID: 33762896

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https://arxiv.org/pdf/1604.04888View

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Abstract

We derive theoretical guarantees for the exact recovery of piecewise constant two-dimensional images from a minimal number of non-uniform Fourier samples using a convex matrix completion algorithm. We assume the discontinuities of the image are localized to the zero level-set of a bandlimited function, which induces certain linear dependencies in Fourier domain, such that a multifold Toeplitz matrix built from the Fourier data is known to be low-rank. The recovery algorithm arranges the known Fourier samples into the structured matrix then attempts recovery of the missing Fourier data by minimizing the nuclear norm subject to structure and data constraints. This work adapts results by Chen and Chi on the recovery of isolated Diracs via nuclear norm minimization of a similar multifold Hankel structure. We show that exact recovery is possible with high probability when the bandlimited function describing the edge set satisfies an incoherency property. Finally, we demonstrate the algorithm on the recovery of undersampled MRI data.

Jacobian matrices

Structured Low-Rank Matrix Completion

Convolution

Image edge detection

Magnetic resonance imaging

MRI

Compressed Sensing

Minimization

Finite Rate of Innovation

Indexes

Annihilating Filter Method

Details

Title: Subtitle: Structured low-rank recovery of piecewise constant signals with performance guarantees
Creators: Greg Ongie - Dept. of Math., Univ. of Iowa, Iowa City, IA, USA
Sampurna Biswas - Dept. of Electr. & Comput. Eng., Univ. of Iowa, Iowa City, IA, USA
Mathews Jacob - Dept. of Electr. & Comput. Eng., Univ. of Iowa, Iowa City, IA, USA
Resource Type: Conference proceeding
Publication Details: 2016 IEEE International Conference on Image Processing (ICIP), Vol.2016-, pp.963-967
DOI: 10.1109/ICIP.2016.7532500
PMID: 33762896
NLM abbreviation: Proc Int Conf Image Proc
ISSN: 1522-4880
eISSN: 2381-8549
Publisher: IEEE
Language: English
Date published: 09/2016
Academic Unit: Roy J. Carver Department of Biomedical Engineering; Radiology; Electrical and Computer Engineering; Iowa Neuroscience Institute; Radiation Oncology
Record Identifier: 9984070495802771

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