Off-the-Grid Recovery of Piecewise Constant Images from Few Fourier Samples

Ongie, Greg; Jacob, Mathews

doi:10.1137/15m1042280

Cited by 114 publications

(156 citation statements)

References 69 publications

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“…For better readability, the theory here is outlined by assuming 1-D signals, but the principle can be extended for multidimensional signals [22].…”

Section: B Transform Domain Sparsity and Low-rankness In Weighted K-mentioning

confidence: 99%

“…H (l ĝ i ) ∈ C (n −d+1)×d are used in (53) instead of H c (l ĝ i )), then due to the special structure of the Hankel matrix, Y v becomes the transpose of Y h ; accordingly, by interchanging the role of n − d + 1 and d, one can make the rank of Y v equal to that of Y h . However, the theoretical analysis of the concatenated Hankel matrices without wrap around property turns out to be quite involved due to the boundary condition and data truncation; and even more, it is even not necessary in accelerated MR except for the super-resolution imaging (see [22]), because the image should be recovered on a fixed grid rather than on a continuum which makes the cardinal spline model more appropriate. Under this condition, the performance of the resulting rank minimization for the vertical stacking approach becomes deteriorated as will be shown in Discussion (see Supplement Material).…”

Section: ) Low-rank Matrix Completionmentioning

confidence: 99%

“…we solve (12) by applyingl(ω x ) first, which is followed by solving (12) withl(ω y ). However, simultaneous weighting would be possible as demonstrated in a recent work for off-the-grid recovery of piecewise constant image [22]. For the case of dynamic imaging in Fig.…”

Section: Reconstruction Flowmentioning

confidence: 99%

See 2 more Smart Citations

A General Framework for Compressed Sensing and Parallel MRI Using Annihilating Filter Based Low-Rank Hankel Matrix

Jin

Lee

2016

IEEE Trans. Comput. Imaging

220

325

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Abstract-Parallel MRI (pMRI) and compressed sensing MRI (CS-MRI) have been considered as two distinct reconstruction problems. Inspired by recent k-space interpolation methods, an annihilating filter-based low-rank Hankel matrix approach is proposed as a general framework for sparsity-driven k-space interpolation method which unifies pMRI and CS-MRI. Specifically, our framework is based on a novel observation that the transform domain sparsity in the primary space implies the low-rankness of weighted Hankel matrix in the reciprocal space. This converts pMRI and CS-MRI to a k-space interpolation problem using a structured matrix completion. Experimental results using in vivo data for single/multicoil imaging as well as dynamic imaging confirmed that the proposed method outperforms the state-of-the-art pMRI and CS-MRI.Index Terms-Annihilating filter, cardinal spline, compressed sensing, parallel MRI, pyramidal representation, structured low rank block Hankel matrix completion, wavelets.

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“…For better readability, the theory here is outlined by assuming 1-D signals, but the principle can be extended for multidimensional signals [22].…”

Section: B Transform Domain Sparsity and Low-rankness In Weighted K-mentioning

confidence: 99%

Section: ) Low-rank Matrix Completionmentioning

confidence: 99%

See 1 more Smart Citation

A General Framework for Compressed Sensing and Parallel MRI Using Annihilating Filter Based Low-Rank Hankel Matrix

Jin

Lee

2016

IEEE Trans. Comput. Imaging

220

325

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show abstract

“…The linear dependencies between the Fourier coefficients exploited the in structured low-rank matrix priors result from a variety of assumptions, including continuous domain analogs of sparsity [3], [6], [8], [9], correlations in the locations of the sparse coefficients in space [8], [9], multi-channel sampling [2], [10], [11], or smoothly varying complex phase [4]. For example, the LORAKS framework [4] capitalized on the sparsity and smooth phase of the continuous domain image using structured low-rank priors, which offers improved reconstructions over conventional ℓ 1 recovery.…”

Section: Introductionmentioning

confidence: 99%

A Fast Algorithm for Convolutional Structured Low-Rank Matrix Recovery

Ongie

Jacob

2017

IEEE Trans. Comput. Imaging

Self Cite

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Fourier domain structured low-rank matrix priors are emerging as powerful alternatives to traditional image recovery methods such as total variation (TV) and wavelet regularization. These priors specify that a convolutional structured matrix, i.e., Toeplitz, Hankel, or their multi-level generalizations, built from Fourier data of the image should be low-rank. The main challenge in applying these schemes to large-scale problems is the computational complexity and memory demand resulting from a lifting the image data to a large scale matrix. We introduce a fast and memory efficient approach called the Generic Iterative Reweighted Annihilation Filter (GIRAF) algorithm that exploits the convolutional structure of the lifted matrix to work in the original un-lifted domain, thus considerably reducing the complexity. Our experiments on the recovery of images from undersampled Fourier measurements show that the resulting algorithm is considerably faster than previously proposed algorithms, and can accommodate much larger problem sizes than previously studied.

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“…Vetterli et al (2002) proposed, for example, a sampling scheme permitting the exact recovery of a stream of Dirac from a few Fourier series coefficients. The framework has since been applied successfully in other fields, and extended to 2D signals as well as noisy measurements (Maravić & Vetterli 2005;Shukla & Dragotti 2007;Pan et al 2014;Ongie & Jacob 2016). Having been originally designed to work only with equally spaced Fourier samples as input, Pan et al (2017b) extended the FRI framework to cases with non-uniform samples (as is the case in radio interferometry).…”

Section: Introductionmentioning

confidence: 99%

LEAP: Looking beyond pixels with continuous-space EstimAtion of Point sources

Pan

Simeoni

Hurley

et al. 2017

A&A

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Context. Two main classes of imaging algorithms have emerged in radio interferometry: the CLEAN algorithm and its multiple variants, and compressed-sensing inspired methods. They are both discrete in nature, and estimate source locations and intensities on a regular grid. For the traditional CLEAN-based imaging pipeline, the resolution power of the tool is limited by the width of the synthesized beam, which is inversely proportional to the largest baseline. The finite rate of innovation (FRI) framework is a robust method to find the locations of point-sources in a continuum without grid imposition. The continuous formulation makes the FRI recovery performance only dependent on the number of measurements and the number of sources in the sky. FRI can theoretically find sources below the perceived tool resolution. To date, FRI had never been tested in the extreme conditions inherent to radio astronomy: weak signal / high noise, huge data sets, large numbers of sources. Aims. The aims were (i) to adapt FRI to radio astronomy, (ii) verify it can recover sources in radio astronomy conditions with more accurate positioning than CLEAN, and possibly resolve some sources that would otherwise be missed, (iii) show that sources can be found using less data than would otherwise be required to find them, and (iv) show that FRI does not lead to an augmented rate of false positives. Methods. We implemented a continuous domain sparse reconstruction algorithm in Python. The angular resolution performance of the new algorithm was assessed under simulation, and with visibility measurements from the LOFAR telescope. Existing catalogs were used to confirm the existence of sources. Results. We adapted the FRI framework to radio interferometry, and showed that it is possible to determine accurate off-grid pointsource locations and their corresponding intensities. In addition, FRI-based sparse reconstruction required less integration time and smaller baselines to reach a comparable reconstruction quality compared to a conventional method. The achieved angular resolution is higher than the perceived instrument resolution, and very close sources can be reliably distinguished. The proposed approach has cubic complexity in the total number (typically around a few thousand) of uniform Fourier data of the sky image estimated from the reconstruction. It is also demonstrated that the method is robust to the presence of extended-sources, and that false-positives can be addressed by choosing an adequate model order to match the noise level.

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Off-the-Grid Recovery of Piecewise Constant Images from Few Fourier Samples

Cited by 114 publications

References 69 publications

A General Framework for Compressed Sensing and Parallel MRI Using Annihilating Filter Based Low-Rank Hankel Matrix

A General Framework for Compressed Sensing and Parallel MRI Using Annihilating Filter Based Low-Rank Hankel Matrix

A Fast Algorithm for Convolutional Structured Low-Rank Matrix Recovery

LEAP: Looking beyond pixels with continuous-space EstimAtion of Point sources

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