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.

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.

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.

scholarly journals A Method of Constructing Measurement Matrix for Compressed Sensing by Chebyshev Chaotic Sequence

Entropy ◽  
2020 ◽  
Vol 22 (10) ◽  
pp. 1085
Author(s):  
Renjie Yi ◽  
Chen Cui ◽  
Yingjie Miao ◽  
Biao Wu
Keyword(s):  
Compressed Sensing ◽  
High Probability ◽  
Hardware Implementation ◽  
Electric Circuit ◽  
Chaotic Sequence ◽  
Measurement Matrix ◽  
Chaotic Sequences ◽  
Reconstruction Performance ◽  
Simulation Results ◽  
Sample Distance

In this paper, the problem of constructing the measurement matrix in compressed sensing is addressed. In compressed sensing, constructing a measurement matrix of good performance and easy hardware implementation is of interest. It has been recently shown that the measurement matrices constructed by Logistic or Tent chaotic sequences satisfy the restricted isometric property (RIP) with a certain probability and are easy to be implemented in the physical electric circuit. However, a large sample distance that means large resources consumption is required to obtain uncorrelated samples from these sequences in the construction. To solve this problem, we propose a method of constructing the measurement matrix by the Chebyshev chaotic sequence. The method effectively reduces the sample distance and the proposed measurement matrix is proved to satisfy the RIP with high probability on the assumption that the sampled elements are statistically independent. Simulation results show that the proposed measurement matrix has comparable reconstruction performance to that of the existing chaotic matrices for compressed sensing.

scholarly journals A Novel Measurement Matrix Optimization Approach for Hyperspectral Unmixing

2017 ◽  
Vol 2017 ◽  
pp. 1-13
Author(s):  
Su Xu ◽  
Xiping He
Keyword(s):  
Computational Cost ◽  
Descent Method ◽  
Qr Factorization ◽  
Eigenvalue Decomposition ◽  
Optimization Approach ◽  
Gradient Descent Method ◽  
Measurement Matrix ◽  
Hyperspectral Unmixing ◽  
Matrix Optimization ◽  
Reconstruction Performance

Each pixel in the hyperspectral unmixing process is modeled as a linear combination of endmembers, which can be expressed in the form of linear combinations of a number of pure spectral signatures that are known in advance. However, the limitation of Gaussian random variables on its computational complexity or sparsity affects the efficiency and accuracy. This paper proposes a novel approach for the optimization of measurement matrix in compressive sensing (CS) theory for hyperspectral unmixing. Firstly, a new Toeplitz-structured chaotic measurement matrix (TSCMM) is formed by pseudo-random chaotic elements, which can be implemented by a simple hardware; secondly, rank revealing QR factorization with eigenvalue decomposition is presented to speed up the measurement time; finally, orthogonal gradient descent method for measurement matrix optimization is used to achieve optimal incoherence. Experimental results demonstrate that the proposed approach can lead to better CS reconstruction performance with low extra computational cost in hyperspectral unmixing.

A New Method for Dictionary Matrix Optimization in ECG Compressed Sensing

2020 ◽  
Author(s):  
Enrico Picariello ◽  
Eulalia Balestrieri ◽  
Francesco Picariello ◽  
Sergio Rapuano ◽  
Ioan Tudosa ◽  
...  
Keyword(s):  
Compressed Sensing ◽  
New Method ◽  
Matrix Optimization

A Randomly Permuted Fourier Measurement Matrix for UWB-PPM Signals

2014 ◽  
Vol 556-562 ◽  
pp. 2646-2649 ◽  
Author(s):  
Hai Bo Yin ◽  
Jun An Yang ◽  
Wei Dong Wang
Keyword(s):  
Compressed Sensing ◽  
Structural Features ◽  
Measurement Matrix ◽  
Reconstruction Accuracy ◽  
High Sampling ◽  
Reconstruction Performance ◽  
Simulation Results ◽  
Hardware Costs ◽  
Fourier Matrix

Compressed Sensing is likely to provide an effective way for lowering the extremely high sampling speed of UWB signal while the design of CS measurement matrix is of great significance for reducing the number of observations and hardware costs as long as improving the reconstruction accuracy. In this paper, with the combination of the structural features of the Fourier matrix and the idea of entry permutation of determined matrices, we propose a new measurement matrix of which the Fourier transformed entries are randomly permuted. Simulation results show that the same algorithm has a better reconstruction performance with the proposed measurement matrix rather than Gaussian/ Bernoulli matrix.

scholarly journals Image Encryption Scheme with Compressed Sensing Based on a New Six-Dimensional Non-Degenerate Discrete Hyperchaotic System and Plaintext-Related Scrambling

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 291
Author(s):  
Chunyang Sun ◽  
Erfu Wang ◽  
Bing Zhao
Keyword(s):  
Compressed Sensing ◽  
Image Compression ◽  
Image Encryption ◽  
Chaotic System ◽  
Chaotic Systems ◽  
Chaotic Map ◽  
Hyperchaotic System ◽  
Encryption Scheme ◽  
Measurement Matrix ◽  
Reconstruction Performance

Digital images can be large in size and contain sensitive information that needs protection. Compression using compressed sensing performs well, but the measurement matrix directly affects the signal compression and reconstruction performance. The good cryptographic characteristics of chaotic systems mean that using one to construct the measurement matrix has obvious advantages. However, existing low-dimensional chaotic systems have low complexity and generate sequences with poor randomness. Hence, a new six-dimensional non-degenerate discrete hyperchaotic system with six positive Lyapunov exponents is proposed in this paper. Using this chaotic system to design the measurement matrix can improve the performance of image compression and reconstruction. Because image encryption using compressed sensing cannot resist known- and chosen-plaintext attacks, the chaotic system proposed in this paper is introduced into the compressed sensing encryption framework. A scrambling algorithm and two-way diffusion algorithm for the plaintext are used to encrypt the measured value matrix. The security of the encryption system is further improved by generating the SHA-256 value of the original image to calculate the initial conditions of the chaotic map. A simulation and performance analysis shows that the proposed image compression-encryption scheme has high compression and reconstruction performance and the ability to resist known- and chosen-plaintext attacks.

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