scholarly journals Construction of Measurement Matrix Based on Cyclic Direct Product and QR Decomposition for Sensing and Reconstruction of Underwater Echo

2018 ◽  
Vol 8 (12) ◽  
pp. 2510 ◽  
Author(s):  
Tongjing Sun ◽  
Hong Cao ◽  
Philippe Blondel ◽  
Yunfei Guo ◽  
Han Shentu
Keyword(s):  
Direct Product ◽  
Hadamard Matrix ◽  
Triangular Matrix ◽  
Field Measurements ◽  
Qr Decomposition ◽  
Orthogonal Matrix ◽  
Measurement Matrix ◽  
Reconstruction Accuracy ◽  
Weak Signals ◽  
Construction Time

Compressive sensing is a very attractive technique to detect weak signals in a noisy background, and to overcome limitations from traditional Nyquist sampling. A very important part of this approach is the measurement matrix and how it relates to hardware implementation. However, reconstruction accuracy, resistance to noise and construction time are still open challenges. To address these problems, we propose a measurement matrix based on a cyclic direct product and QR decomposition (the product of an orthogonal matrix Q and an upper triangular matrix R). Using the definition and properties of a direct product, a set of high-dimensional orthogonal column vectors is first established by a finite number of cyclic direct product operations on low-dimension orthogonal “seed” vectors, followed by QR decomposition to yield the orthogonal matrix, whose corresponding rows are selected to form the measurement matrix. We demonstrate this approach with simulations and field measurements of a scaled submarine in a freshwater lake, at frequencies of 40 kHz–80 kHz. The results clearly show the advantage of this method in terms of reconstruction accuracy, signal-to-noise ratio (SNR) enhancement, and construction time, by comparison with Gaussian matrix, Bernoulli matrix, partial Hadamard matrix and Toeplitz matrix. In particular, for weak signals with an SNR less than 0 dB, this method still achieves an SNR increase using less data.

scholarly journals Blind Color Image Watermarking Using Fan Beam Transform and QR Decomposition

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 486
Author(s):  
Pranab Kumar Dhar ◽  
Pulak Hazra ◽  
Tetsuya Shimamura
Keyword(s):  
Color Image ◽  
State Of The Art ◽  
Signal To Noise Ratio ◽  
Image Watermarking ◽  
Triangular Matrix ◽  
Structural Similarity ◽  
Qr Decomposition ◽  
Original Image ◽  
Color Image Watermarking ◽  
Multimedia Contents

Digital watermarking has been utilized effectively for copyright protection of multimedia contents. This paper suggests a blind symmetric watermarking algorithm using fan beam transform (FBT) and QR decomposition (QRD) for color images. At first, the original image is transferred from RGB to L*a*b* color model and FBT is applied to b* component. Then the b*component of the original image is split into m × m non-overlapping blocks and QRD is conducted to each block. Watermark data is placed into the selected coefficient of the upper triangular matrix using a new embedding function. Simulation results suggest that the presented algorithm is extremely robust against numerous attacks, and also yields watermarked images with high quality. Furthermore, it represents more excellent performance compared with the recent state-of-the-art algorithms for robustness and imperceptibility. The normalized correlation (NC) of the proposed algorithm varies from 0.8252 to 1, the peak signal-to-noise ratio (PSNR) varies from 54.1854 to 54.1892, and structural similarity (SSIM) varies from 0.9285 to 0.9696, respectively. In contrast, the NC of the recent state-of-the-art algorithms varies from 0.5193 to 1, PSNR varies from 38.5471 to 52.64, and SSIM varies from 0.9311 to 0.9663, respectively.

scholarly journals Unconstrained representation of orthogonal matrices with application to common principal components

2020 ◽  
Author(s):  
Luca Bagnato ◽  
Antonio Punzo
Keyword(s):  
Objective Function ◽  
Normal Distribution ◽  
Triangular Matrix ◽  
Likelihood Estimation ◽  
Optimization Method ◽  
Orthogonal Matrix ◽  
Additional Parameter ◽  
Biometric Data ◽  
Principle Components Analysis ◽  
Common Principle

Abstract Many statistical problems involve the estimation of a $$\left( d\times d\right) $$ d × d orthogonal matrix $$\varvec{Q}$$ Q . Such an estimation is often challenging due to the orthonormality constraints on $$\varvec{Q}$$ Q . To cope with this problem, we use the well-known PLU decomposition, which factorizes any invertible $$\left( d\times d\right) $$ d × d matrix as the product of a $$\left( d\times d\right) $$ d × d permutation matrix $$\varvec{P}$$ P , a $$\left( d\times d\right) $$ d × d unit lower triangular matrix $$\varvec{L}$$ L , and a $$\left( d\times d\right) $$ d × d upper triangular matrix $$\varvec{U}$$ U . Thanks to the QR decomposition, we find the formulation of $$\varvec{U}$$ U when the PLU decomposition is applied to $$\varvec{Q}$$ Q . We call the result as PLR decomposition; it produces a one-to-one correspondence between $$\varvec{Q}$$ Q and the $$d\left( d-1\right) /2$$ d d - 1 / 2 entries below the diagonal of $$\varvec{L}$$ L , which are advantageously unconstrained real values. Thus, once the decomposition is applied, regardless of the objective function under consideration, we can use any classical unconstrained optimization method to find the minimum (or maximum) of the objective function with respect to $$\varvec{L}$$ L . For illustrative purposes, we apply the PLR decomposition in common principle components analysis (CPCA) for the maximum likelihood estimation of the common orthogonal matrix when a multivariate leptokurtic-normal distribution is assumed in each group. Compared to the commonly used normal distribution, the leptokurtic-normal has an additional parameter governing the excess kurtosis; this makes the estimation of $$\varvec{Q}$$ Q in CPCA more robust against mild outliers. The usefulness of the PLR decomposition in leptokurtic-normal CPCA is illustrated by two biometric data analyses.

A novel compressive sensing method based on SVD sparse random measurement matrix in wireless sensor network

2016 ◽  
Vol 33 (8) ◽  
pp. 2448-2462 ◽  
Author(s):  
Zhen Ma ◽  
Degan Zhang ◽  
Si Liu ◽  
Jinjie Song ◽  
Yuexian Hou
Keyword(s):  
Wireless Sensor Network ◽  
Data Collection ◽  
Random Matrix ◽  
Sparse Matrix ◽  
Hadamard Matrix ◽  
Wireless Sensor ◽  
Independent Random Variables ◽  
Measurement Matrix ◽  
Content Type ◽  
The Matrix

Purpose The performance of the measurement matrix directly affects the quality of reconstruction of compressive sensing signal, and it is also the key to solve practical problems. In order to solve data collection problem of wireless sensor network (WSN), the authors design a kind of optimization of sparse matrix. The paper aims to discuss these issues. Design/methodology/approach Based on the sparse random matrix, it optimizes the seed vector, which regards elements in the diagonal matrix of Hadamard matrix after passing singular value decomposition (SVD). Compared with the Toeplitz matrix, it requires less number of independent random variables and the matrix information is more concentrated. Findings The performance of reconstruction is better than that of Gaussian random matrix. The authors also apply this matrix to the data collection scheme in WSN. The result shows that it costs less energy and reduces the collection frequency of nodes compared with general method. Originality/value The authors design a kind of optimization of sparse matrix. Based on the sparse random matrix, it optimizes the seed vector, which regards elements in the diagonal matrix of Hadamard matrix after passing SVD. Compared with the Toeplitz matrix, it requires less number of independent random variables and the matrix information is more concentrated.

scholarly journals Research on Deterministic Measurement Matrix of Power Line Carrier Compressed Sensing

2021 ◽  
Vol 2095 (1) ◽  
pp. 012017
Author(s):  
Jiliang Jin ◽  
Liyun Xing ◽  
Miao Yang ◽  
Jianqiang Shen ◽  
Yuqi Dong
Keyword(s):  
Compressed Sensing ◽  
Power Line ◽  
Measurement Matrix ◽  
Reconstruction Accuracy ◽  
Eighth Order ◽  
Chaotic Mapping ◽  
Power Line Carrier ◽  
Power Line Carrier Communication ◽  
Reconstruction Efficiency ◽  
Carrier Communication

Abstract To verify the advantages of deterministic matrix applied to power line carrier communication (PLCC) based on compressed sensing (CS). This article analyzed the research status of commonly used deterministic measurement matrices, and made simulation comparison. It is found that different types of deterministic measurement matrices generated based on chaotic mapping had higher reconstruction accuracy and higher reconstruction efficiency than Gaussian random matrix. Then, according to simulation results and the characteristics of PLCC signal, the Chebyshev sparse circulant (CSC) measurement matrix was designed by combining eighth-order Chebyshev chaotic and the idea of sparse and circulant. Actual circuit measurement shows that when compression rate was 40% and 60%, the reconstruction loss of CSC is 0.72dB and 0.49dB higher than that of Chebyshev chaotic measurement matrix and Chebyshev circulate measurement matrix, respectively. Obviously, the CSC measurement matrix designed in this paper can effectively improve the reconstruction accuracy.

scholarly journals Improved Generalized Sparsity Adaptive Matching Pursuit Algorithm Based on Compressive Sensing

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Zhao Liquan ◽  
Ma Ke ◽  
Jia Yanfei
Keyword(s):  
Matching Pursuit ◽  
Initial Step ◽  
Convergence Speed ◽  
One Dimension ◽  
Measurement Matrix ◽  
Improved Method ◽  
Reconstruction Accuracy ◽  
Step Size ◽  
Matching Pursuit Algorithm ◽  
Initial Step Size

The modified adaptive orthogonal matching pursuit algorithm has a lower convergence speed. To overcome this problem, an improved method with faster convergence speed is proposed. In respect of atomic selection, the proposed method computes the correlation between the measurement matrix and residual and then selects the atoms most related to residual to construct the candidate atomic set. The number of selected atoms is the integral multiple of initial step size. In respect of sparsity estimation, the proposed method introduces the exponential function to sparsity estimation. It uses a larger step size to estimate sparsity at the beginning of iteration to accelerate the algorithm convergence speed and a smaller step size to improve the reconstruction accuracy. Simulations show that the proposed method has better performance in terms of convergence speed and reconstruction accuracy for one-dimension signal and two-dimension signal.

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.

Vacuum consolidation and its combination with embankment loading

2006 ◽  
Vol 43 (10) ◽  
pp. 985-996 ◽  
Author(s):  
J -C Chai ◽  
J P Carter ◽  
S Hayashi
Keyword(s):  
Penetration Depth ◽  
Lateral Displacement ◽  
Fem Analysis ◽  
Field Measurements ◽  
Numerical Approach ◽  
Vacuum Pressure ◽  
Construction Time ◽  
Vacuum Consolidation ◽  
Prefabricated Vertical Drains ◽  
Prefabricated Vertical Drain

A method is proposed for determining the optimum penetration depth of prefabricated vertical drains (PVDs) in cases where vacuum consolidation is combined with the use of PVDs in a clayey deposit with two-way drainage. The advantages of combining vacuum pressure with embankment loading are discussed in terms of reducing preloading-induced lateral displacement of the subsoil, increasing the effective surcharge loading, and reducing construction time in the case of road construction. A vacuum consolidation project conducted in Saga, Japan, is described, and the results from a fully instrumented test section are presented and analyzed using a two-dimensional finite element approach. The numerical simulations compare well with the field measurements. The validated numerical approach is then used to examine the response of soft subsoil subjected to vacuum consolidation. The results confirm the usefulness of the proposed method for determining the optimum penetration depth of PVDs and the advantages of combining vacuum pressure with embankment loading.Key words: vacuum consolidation, preloading, prefabricated vertical drain, FEM analysis, embankment.

scholarly journals Hadamard matrices and submatrices

1976 ◽  
Vol 22 (4) ◽  
pp. 469-475 ◽  
Author(s):  
K. Vijayan
Keyword(s):  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Orthogonal Matrix

AbstractShrinkande and Bhagwan Das (1970) showed how to extend a (4t – 1, 4t) row-orthogonal matrix with entries ± 1 to a Hadamard matrix of order 4t. Using a slightly different approach we consider extensions of (4t – k, 4t) row-orthogonal matrix to a Hadamard matrix of order 4t.

scholarly journals A parallel algorithm for the sparse QR decomposition of a rectangular upper quasi-triangular matrix with ND-type sparsity

2015 ◽  
pp. 566-577
Author(s):  
С.А. Харченко
Keyword(s):  
Parallel Algorithm ◽  
Triangular Matrix ◽  
Parallel Computations ◽  
Qr Decomposition ◽  
Nested Dissection ◽  
Rectangular Matrix ◽  
Computational Mesh ◽  
The Matrix ◽  
Householder Transformations ◽  
Initial Zero

Рассматривается параллельный алгоритм вычисления разреженного $QR$-разложения специальным образом упорядоченной прямоугольной матрицы на основе разреженных блочных преобразований Хаусхолдера. Для построения необходимого упорядочивания можно использовать столбцевое упорядочивание типа вложенных сечений, построенное по структуре матрицы $A^{T}A$, где $A$ - исходная прямоугольная матрица. Для сеточных задач упорядочивание может быть построено на основе известного объемного разбиения расчетной сетки. В качестве базового алгоритма для организации параллельных вычислений используется $QR$-разложение для наборов строк матрицы с дополнением в виде нулевого начального блока. An algorithm for computing the sparse $QR$ decomposition of a specially ordered rectangular matrix is proposed. This decomposition is based on the block sparse Householder transformations. For ordering computations, the nested dissection ordering is used for the matrix $A^{T}A$, where $A$ is the original rectangular matrix. For mesh based problems, the ordering can be constructed starting from an appropriate volume partitioning of the computational mesh. Parallel computations are based on sparse $QR$ decomposition for sets of rows with an additional initial zero block.

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