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.

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].

scholarly journals A New Method of Measurement Matrix Optimization for Compressed Sensing Based on Alternating Minimization

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 329
Author(s):  
Renjie Yi ◽  
Chen Cui ◽  
Biao Wu ◽  
Yang Gong
Keyword(s):  
Compressed Sensing ◽  
New Method ◽  
Gram Matrix ◽  
Alternating Minimization ◽  
Measurement Matrix ◽  
Mutual Coherence ◽  
Matrix Optimization ◽  
Shrinkage Function ◽  
Reconstruction Performance ◽  
Method Of Measurement

In this paper, a new method of measurement matrix optimization for compressed sensing based on alternating minimization is introduced. The optimal measurement matrix is formulated in terms of minimizing the Frobenius norm of the difference between the Gram matrix of sensing matrix and the target one. The method considers the simultaneous minimization of the mutual coherence indexes including maximum mutual coherence μmax, t-averaged mutual coherence μave and global mutual coherence μall, and solves the problem that minimizing a single index usually results in the deterioration of the others. Firstly, the threshold of the shrinkage function is raised to be higher than the Welch bound and the relaxed Equiangular Tight Frame obtained by applying the new function to the Gram matrix is taken as the initial target Gram matrix, which reduces μave and solves the problem that μmax would be larger caused by the lower threshold in the known shrinkage function. Then a new target Gram matrix is obtained by sequentially applying rank reduction and eigenvalue averaging to the initial one, leading to lower. The analytical solutions of measurement matrix are derived by SVD and an alternating scheme is adopted in the method. Simulation results show that the proposed method simultaneously reduces the above three indexes and outperforms the known algorithms in terms of reconstruction performance.

Research on Compressed Sensing Matrix Based on Data Compressing of Insulator Leakage Current

2013 ◽  
Vol 475-476 ◽  
pp. 451-454
Author(s):  
Xue Ming Zhai ◽  
Xiao Bo You ◽  
Ruo Chen Li ◽  
Yu Jia Zhai ◽  
De Wen Wang
Keyword(s):  
Compressed Sensing ◽  
Data Compression ◽  
Leakage Current ◽  
Sparse Matrix ◽  
Circulant Matrix ◽  
Measurement Matrix ◽  
Sensing Matrix ◽  
Significant Step ◽  
On Line

Insulator fault may lead to the accident of power network,thus the on-line monitoring of insulator is very significant. Low rates wireless network is used for data transmission of leakage current. Making data compression and reconstruction of leakage current with the compressed sensing theory can achieve pretty good results. Determination of measurement matrix is the significant step for realizing the compressed sensing theory. This paper compares multiple measurement matrix of their effect via experiments, putting forward to make data compression and reconstruction of leakage current using Toeplitz matrix, circulant matrix and sparse matrix as measurement matrix, of which the reconstitution effect is almost the same as classical measurement matrix and depletes computational complexity and workload.

scholarly journals Measurement Matrix Optimization via Mutual Coherence Minimization for Compressively Sensed Signals Reconstruction

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Ziran Wei ◽  
Jianlin Zhang ◽  
Zhiyong Xu ◽  
Yong Liu ◽  
Krzysztof Okarma
Keyword(s):  
Optimization Methods ◽  
Superior Performance ◽  
Gram Matrix ◽  
Sparse Signals ◽  
Two Dimensional ◽  
Measurement Matrix ◽  
Mutual Coherence ◽  
One Dimensional ◽  
Two Dimensional Image ◽  
Matrix Optimization

For signals reconstruction based on compressive sensing, to reconstruct signals of higher accuracy with lower compression rates, it is required that there is a smaller mutual coherence between the measurement matrix and the sparsifying matrix. Mutual coherence between the measurement matrix and sparsifying matrix can be expressed indirectly by the property of the Gram matrix. On the basis of the Gram matrix, a new optimization algorithm of acquiring a measurement matrix has been proposed in this paper. Firstly, a new mathematical model is designed and a new method of initializing measurement matrix is adopted to optimize the measurement matrix. Then, the loss function of the new algorithm model is solved by the gradient projection-based method of Gram matrix approximating an identity matrix. Finally, the optimized measurement matrix is generated by minimizing mutual coherence between measurement matrix and sparsifying matrix. Compared with the conventional measurement matrices and the traditional optimization methods, the proposed new algorithm effectively improves the performance of optimized measurement matrices in reconstructing one-dimensional sparse signals and two-dimensional image signals that are not sparse. The superior performance of the proposed method in this paper has been fully tested and verified by a large number of experiments.

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 Compressed Sensing System for a Fingerprint Image Recognition Using Sensing Matrix with Chaotic Model-Based Deterministic Row Indexes

2021 ◽  
Author(s):  
Workneh Wolde Hailemariam ◽  
Pallavi Gupta
Keyword(s):  
Compressed Sensing ◽  
Matching Pursuit ◽  
Signal To Noise Ratio ◽  
Optimization Method ◽  
Security Assessment ◽  
Sensing Matrix ◽  
Sensing System ◽  
Chaotic Model ◽  
Assessment Techniques ◽  
Linear Congruential

Abstract This paper proposes a novel design approach for a secured compressed sensing system for fingerprint sensing and transmission. In the proposed design, the first stage is acquiring the signal followed by sparsely modeling it using Orthogonal Matching Pursuit (OMP) algorithm then compressing. In addition to compressing, we multiply the sparse modeled data by a novel, deterministic, and partially orthogonal Discrete Cosine Transform (DCT) sensing matrix to guarantee its security. Furthermore, the construction of the sensing matrix uses a modified Multiplicative Linear Congruential Generator (MLCG) to select the row index appropriately from chaotically re-arranged rows of DCT pseudo-randomly. On the other hand, the compressed image's simultaneous recovery and decryption accomplished using a convex optimization method—the proposed system tested by employing different image and security assessment techniques. The results show that we have archived a better Peak Signal to Noise Ratio (PSNR) than the recommended value for wireless transmission using samples below 25%.

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