Robust recovery of complex exponential signals from random Gaussian projections via low rank Hankel matrix reconstruction

Cai, Jian-Feng; Qu, Xiaobo; Xu, Weiyu; Ye, Gui–Bo

doi:10.1016/j.acha.2016.02.003

Cited by 63 publications

(81 citation statements)

References 30 publications

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“…The term "non-stationary" indicates that the point spread functions are potentially different and comes from the field of non-stationary deconvolution [8]; the term "blind" indicates that the point spread functions are unknown. Non-stationary blind super-resolution problems also appear in applications involving radar imaging [9], astronomy [10], photography [11], 3D single-molecule microscopy [12], seismology [8] and nuclear magnetic resonance (NMR) spectroscopy [13,14,15]. Exchanging the roles of time and frequency, the conventional line spectral estimation problem is modified as follows: each complex exponential is modulated (pointwise multiplied) by an unknown waveform, and this waveform can vary from one complex exponential to the next.…”

Section: Introductionmentioning

confidence: 99%

Atomic Norm Denoising for Complex Exponentials With Unknown Waveform Modulations

Wakin

Tang

2020

IEEE Trans. Inform. Theory

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Non-stationary blind super-resolution is an extension of the traditional super-resolution problem, which deals with the problem of recovering fine details from coarse measurements. The non-stationary blind super-resolution problem appears in many applications including radar imaging, 3D single-molecule microscopy, computational photography, etc. There is a growing interest in solving non-stationary blind super-resolution task with convex methods due to their robustness to noise and strong theoretical guarantees. Motivated by the recent work on atomic norm minimization in blind inverse problems, we focus here on the signal denoising problem in non-stationary blind super-resolution. In particular, we use an atomic norm regularized least-squares problem to denoise a sum of complex exponentials with unknown waveform modulations. We quantify how the mean square error depends on the noise variance and the true signal parameters. Numerical experiments are also implemented to illustrate the theoretical result.Index terms-Atomic norm denoising, non-stationary blind deconvolution, line spectrum estimation, demodulation, mean square error.

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Section: Introductionmentioning

confidence: 99%

Atomic Norm Denoising for Complex Exponentials With Unknown Waveform Modulations

Wakin

Tang

2020

IEEE Trans. Inform. Theory

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“…The second term in Equation is the sum of nuclear norms for the pixel‐wise Hankel matrices

{\sum_{i = 1}^{N_{P}} ‖ h ({boldR}_{i} Q x) ‖}_{*}

, where

N_{P}

is the number of pixels,

{‖ \cdot ‖}_{*}

is the nuclear norm,

h (\cdot)

is the Hankel matrix formation ,

{boldR}_{i}

is an operator that selects the recovery time course for a pixel i from the data

Q x

, and

Q = ({boldD}_{1} - {boldD}_{N_{T} + 1}) | | ({boldD}_{2} - {boldD}_{N_{T} + 1}) | | \dots | | ({boldD}_{N_{T}} - {boldD}_{N_{T} + 1})

is an linear operator that performs the subtraction of MP image

{boldD}_{t} x

with the reference image

{boldD}_{N_{T} + 1} x

for every RT

t = 1 \dots N_{T}

and every pixel (as shown in Fig. c).…”

Section: Methodsmentioning

confidence: 99%

“…The second term in Equation [4] is the sum of nuclear norms for the pixel-wise Hankel matrices P NP i¼1 jjhðR i QxÞjj Ã , where N P is the number of pixels, jj Á jj Ã is the nuclear norm, hðÁÞ is the Hankel matrix formation (25), R i is an operator that selects the recovery time course for a pixel i from the data Q x, and Q ¼ ðD 1 ÀD NT þ1 Þ jj ðD 2 À D NT þ1 Þ jj . .…”

Section: Pixel-wise Hankel Matrix Completion In Combination With Compmentioning

confidence: 99%

Shuffled magnetization‐prepared multicontrast rapid gradient‐echo imaging

Cao

Zhu

Tang

et al. 2017

Magnetic Resonance in Med

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Purpose: To develop a novel acquisition and reconstruction method for magnetization-prepared 3-dimensional multicontrast rapid gradient-echo imaging, using Hankel matrix completion in combination with compressed sensing and parallel imaging. Methods: A random k-space shuffling strategy was implemented in simulation and in vivo human experiments at 7 T for 3-dimensional inversion recovery, T 2 /diffusion preparation, and magnetization transfer imaging. We combined compressed sensing, based on total variation and spatial-temporal low-rank regularizations, and parallel imaging with pixel-wise Hankel matrix completion, allowing the reconstruction of tens of multicontrast 3-dimensional images from 3-or 6-min scans. Results: The simulation result showed that the proposed method can reconstruct signal-recovery curves in each voxel and was robust for typical in vivo signal-to-noise ratio with 16-times acceleration. In vivo studies achieved 4 to 24 times accelerations for inversion recovery, T 2 /diffusion preparation, and magnetization transfer imaging. Furthermore, the contrast was improved by resolving pixel-wise signal-recovery curves after magnetization preparation. Conclusions: The proposed method can improve acquisition efficiencies for magnetization-prepared MRI and tens of multicontrast 3-dimensional images could be recovered from a single scan. Furthermore, it was robust against noise, applicable for recovering multi-exponential signals, and did not require any previous knowledge of model parameters. Magn Reson Med 79:62-70,

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“…The nuclear norm minimization for the block‐Hankel matrix completion can be written as

arg min_{boldX} | | H (boldX) | |_{*} s . t . M (X) = Y

where

M

is a k‐space undersampling operator,

H

a 3D sliding window and concatenation operator describing the conversion into a block‐Hankel matrix, X the reconstructed MRSI data set, and Y the undersampled MRSI data set. This nuclear norm minimization can be solved by a singular value thresholding method , as shown in Figure .…”

Section: Methodsmentioning

confidence: 99%

“…Several reconstruction methods have also been used successfully in previous CS MRSI studies, eg, L 1 ‐minimization , total variation minimization , and maximum entropy reconstruction , and group sparsity–based reconstruction . Recently, Hankel or block‐Hankel matrix completion has been used for recovering undersampled spectral data and calibrationless parallel imaging reconstruction . The low‐rank property of dynamic MRI data has also been demonstrated and utilized in accelerated reconstructions .…”

Section: Introductionmentioning

confidence: 99%

Accelerated high-bandwidth MR spectroscopic imaging using compressed sensing

et al. 2016

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Purpose To develop a compressed sensing (CS) acceleration method with a high spectral bandwidth exploiting the spatial-spectral sparsity of MR spectroscopic imaging (MRSI). Methods Accelerations were achieved using blip gradients during the readout to perform non-overlapped and stochastically delayed random walks in kx-ky-t space, combined with block-Hankel matrix completion for efficient reconstruction. Both retrospective and prospective CS accelerations were applied to 13C MRSI experiments, including in vivo rodent brain and liver studies with administrations of hyperpolarized [1-13C] pyruvate at 7T and [2-13C] dihydroxyacetone at 3T, respectively. Results In retrospective undersampling experiments using in vivo 7T data, the proposed method preserved spectral, spatial and dynamic fidelities with R2 ≥ 0.96 and ≥ 0.87 for pyruvate and lactate signals, respectively, 750-Hz spectral separation and up to 6.6-fold accelerations. In prospective in vivo experiments, with 3.8-fold acceleration, the proposed method exhibited excellent spatial localization of metabolites and peak recovery for pyruvate and lactate at 7T as well as for dihydroxyacetone and its metabolic products with a 4.5-kHz spectral span (140 ppm at 3T). Conclusion This study demonstrated the feasibility of a new CS approach to accelerate high spectral bandwidth MRSI experiments.

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Robust recovery of complex exponential signals from random Gaussian projections via low rank Hankel matrix reconstruction

Cited by 63 publications

References 30 publications

Atomic Norm Denoising for Complex Exponentials With Unknown Waveform Modulations

Atomic Norm Denoising for Complex Exponentials With Unknown Waveform Modulations

Shuffled magnetization‐prepared multicontrast rapid gradient‐echo imaging

Accelerated high-bandwidth MR spectroscopic imaging using compressed sensing

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