Compressed-Sensing MRI Based on Adaptive Tight Frame in Gradient Domain

2018 ◽  
Vol 49 (5) ◽  
pp. 465-477 ◽  
Author(s):  
Xiaoyu Fan ◽  
Qiusheng Lian ◽  
Baoshun Shi
Keyword(s):  
Compressed Sensing ◽  
Tight Frame ◽  
Gradient Domain

scholarly journals Measurement Matrix Optimization for Compressed Sensing System with Constructed Dictionary via Takenaka–Malmquist Functions

Sensors ◽  
2021 ◽  
Vol 21 (4) ◽  
pp. 1229
Author(s):  
Qiangrong Xu ◽  
Zhichao Sheng ◽  
Yong Fang ◽  
Liming Zhang
Keyword(s):  
Compressed Sensing ◽  
Singular Values ◽  
Tight Frame ◽  
Gram Matrix ◽  
Signal Recovery ◽  
Measurement Matrix ◽  
Sensing Matrix ◽  
Matrix Optimization ◽  
Sensing System ◽  
Iterative Minimization

Compressed sensing (CS) has been proposed to improve the efficiency of signal processing by simultaneously sampling and compressing the signal of interest under the assumption that the signal is sparse in a certain domain. This paper aims to improve the CS system performance by constructing a novel sparsifying dictionary and optimizing the measurement matrix. Owing to the adaptability and robustness of the Takenaka–Malmquist (TM) functions in system identification, the use of it as the basis function of a sparsifying dictionary makes the represented signal exhibit a sparser structure than the existing sparsifying dictionaries. To reduce the mutual coherence between the dictionary and the measurement matrix, an equiangular tight frame (ETF) based iterative minimization algorithm is proposed. In our approach, we modify the singular values without changing the properties of the corresponding Gram matrix of the sensing matrix to enhance the independence between the column vectors of the Gram matrix. Simulation results demonstrate the promising performance of the proposed algorithm as well as the superiority of the CS system, designed with the constructed sparsifying dictionary and the optimized measurement matrix, over existing ones in terms of signal recovery accuracy.

Constructing Preconditioner Based on Tight Frame for Bernoulli Random Sensing Matrix in Compressed Sensing

2019 ◽  
Author(s):  
Jiasheng Zhang ◽  
Yue Liu ◽  
Yongxin Wang ◽  
Rong Zhu ◽  
Zixin Liu ◽  
...  
Keyword(s):  
Compressed Sensing ◽  
Tight Frame ◽  
Sensing Matrix

scholarly journals Two-Layer Tight Frame Sparsifying Model for Compressed Sensing Magnetic Resonance Imaging

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Shanshan Wang ◽  
Jianbo Liu ◽  
Xi Peng ◽  
Pei Dong ◽  
Qiegen Liu ◽  
...  
Keyword(s):  
Magnetic Resonance Imaging ◽  
Magnetic Resonance ◽  
Compressed Sensing ◽  
Adaptive Learning ◽  
Tight Frame ◽  
Resonance Imaging ◽  
Acceleration Rate ◽  
Space Data ◽  
Mr Reconstruction

Compressed sensing magnetic resonance imaging (CSMRI) employs image sparsity to reconstruct MR images from incoherently undersampledK-space data. Existing CSMRI approaches have exploited analysis transform, synthesis dictionary, and their variants to trigger image sparsity. Nevertheless, the accuracy, efficiency, or acceleration rate of existing CSMRI methods can still be improved due to either lack of adaptability, high complexity of the training, or insufficient sparsity promotion. To properly balance the three factors, this paper proposes a two-layer tight frame sparsifying (TRIMS) model for CSMRI by sparsifying the image with a product of a fixed tight frame and an adaptively learned tight frame. The two-layer sparsifying and adaptive learning nature of TRIMS has enabled accurate MR reconstruction from highly undersampled data with efficiency. To solve the reconstruction problem, a three-level Bregman numerical algorithm is developed. The proposed approach has been compared to three state-of-the-art methods over scanned physical phantom and in vivo MR datasets and encouraging performances have been achieved.

scholarly journals Improved RIP Conditions for Compressed Sensing with Coherent Tight Frames

2017 ◽  
Vol 2017 ◽  
pp. 1-8
Author(s):  
Yao Wang ◽  
Jianjun Wang
Keyword(s):  
Compressed Sensing ◽  
Sufficient Conditions ◽  
Tight Frame ◽  
Restricted Isometry Property ◽  
Tight Frames ◽  
Analysis Method

This paper establishes new sufficient conditions on the restricted isometry property (RIP) for compressed sensing with coherent tight frames. One of our main results shows that the RIP (adapted to D) condition δk+θk,k<1 guarantees the stable recovery of all signals that are nearly k-sparse in terms of a coherent tight frame D via the l1-analysis method, which improves the existing ones in the literature.

THE RESTRICTED ISOMETRY PROPERTY FOR SIGNAL RECOVERY WITH COHERENT TIGHT FRAMES

2015 ◽  
Vol 92 (3) ◽  
pp. 496-507 ◽  
Author(s):  
FEN-GONG WU ◽  
DONG-HUI LI
Keyword(s):  
Compressed Sensing ◽  
Orthonormal Basis ◽  
Natural Extension ◽  
Tight Frame ◽  
Restricted Isometry Property ◽  
Tight Frames ◽  
Signal Recovery ◽  
Measurement Matrix

In this paper, we consider signal recovery via $l_{1}$-analysis optimisation. The signals we consider are not sparse in an orthonormal basis or incoherent dictionary, but sparse or nearly sparse in terms of some tight frame $D$. The analysis in this paper is based on the restricted isometry property adapted to a tight frame $D$ (abbreviated as $D$-RIP), which is a natural extension of the standard restricted isometry property. Assuming that the measurement matrix $A\in \mathbb{R}^{m\times n}$ satisfies $D$-RIP with constant ${\it\delta}_{tk}$ for integer $k$ and $t>1$, we show that the condition ${\it\delta}_{tk}<\sqrt{(t-1)/t}$ guarantees stable recovery of signals through $l_{1}$-analysis. This condition is sharp in the sense explained in the paper. The results improve those of Li and Lin [‘Compressed sensing with coherent tight frames via $l_{q}$-minimization for $0<q\leq 1$’, Preprint, 2011, arXiv:1105.3299] and Baker [‘A note on sparsification by frames’, Preprint, 2013, arXiv:1308.5249].

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