Block circulant matrices and applications in free vibration analysis of cyclically repetitive structures

In this paper, block circulant matrices and their properties are investigated. Basic concepts and the necessary theorems are presented and then their applications are discussed. It is shown that a circulant matrix can be considered as the sum of Kronecker products in which the first components have the commutativity property with respect to multiplication. The important fact is that the method for block diagonalization of these matrices is much simpler than the previously developed methods, and one does not need to find an additional matrix for orthogonalization. As it will be shown not only the matrices corresponding to domes in the form of Cartesian product, strong Cartesian product and direct product are circulant, but for other structures such as diamatic domes, pyramid domes, flat double layer grids, and some family of transmission towers these matrices are also block circulant.     

Analysis of structures transformable to circulant form using U-transformation and Kronecker products

In this paper, eigenvalues and eigenvectors of the specific types of structural matrices are studied, and a simple method is presented for calculating their eigenvalues. First, the required formulation to diagonalize circulant and block circulant matrices is presented by using U-matrix transformation. Then utilizing the method of this paper, matrices with non-circulant forms are converted into their circulant counterpart matrices. In order to demonstrate the efficiency of the method, some examples are provided using numerical methods such as finite differences, finite element and finite stripe methods.     

鳞状因子循环矩阵方程解的条件与求解的快速算法

利用多项式快速算法,给出了鳞状因子循环矩阵方程AX=b可解的条件与求解的快速算法.当鳞状因子循环矩阵非奇异时,该快速算法求出线性方程组的唯一解;当鳞状因子循环矩阵奇异时,该快速算法求出线性方程组的特解与通解.该快速算法仅用到鳞状因子循环矩阵的第一行元素及对角矩阵中的对角上的常数进行计算.在计算机上实现时只有舍入误差.特别地,在有理数域上用计算机求得的结果是精确的.     

The use of cyclic symmetry in EFG analysis for heat transfer problems

Rotationally periodic (or cyclic) symmetry is exploited in the element‐free Galerkin (EFG) analysis for heat transfer problems of two‐dimensional systems. It is proved that the coefficient matrices of the global EFG equations are block‐circulant. Furthermore, a partitioning algorithm is presented, and the computational convenience and efficiency are demonstrated. A technique dealing with asymmetric boundary conditions is developed to extend the application of the proposed approach. Numerical examples are given to illustrate the advantages of such exploitation of symmetry in the context. Copyright © 2004 John Wiley & Sons, Ltd.     

Circulant preconditioners for ill-conditioned boundary integral equations from potential equations

In this paper, we consider solving potential equations by the boundary integral equation approach. The equations so derived are Fredholm integral equations of the first kind and are known to be ill-conditioned. Their discretized matrices are dense and have condition numbers growing like O(n) where n is the matrix size. We propose to solve the equations by the preconditioned conjugate gradient method with circulant integral operators as preconditioners. These are convolution operators with periodic kernels and hence can be inverted efficiently by using fast Fourier transforms. We prove that the preconditioned systems are well conditioned, and hence the convergence rate of the method is linear. Numerical results for two types of regions are given to illustrate the fast convergence. © 1998 John Wiley & Sons, Ltd.     

Fast Algorithms for Structured Least Squares and Total Least Squares Problems

We consider the problem of solving least squares problems involving a matrix M of small displacement rank with respect to two matrices Z₁ and Z₂. We develop formulas for the generators of the matrix M ^HM in terms of the generators of M and show that the Cholesky factorization of the matrix M ^HM can be computed quickly if Z₁ is close to unitary and Z₂ is triangular and nilpotent. These conditions are satisfied for several classes of matrices, including Toeplitz, block Toeplitz, Hankel, and block Hankel, and for matrices whose blocks have such structure. Fast Cholesky factorization enables fast solution of least squares problems, total least squares problems, and regularized total least squares problems involving these classes of matrices.     

Efficient MIMO channel estimation using complementary sequences

Low complexity channel estimation for single-carrier block transmission systems over multiple-input multiple-output time varying frequency-selective channels is investigated. A time slot structure that uses Golay complementary sequences with perfect periodic autocorrelations as two-sided pilot blocks is presented. Employing this structure, optimal least square estimate in the minimum mean square error (MMSE) sense is achieved. Furthermore, a computationally efficient algorithm which is named as fast periodic Golay correlation is proposed based on the specific generator and the properties related to circulant matrices. Finally, the simulation results show the MMSE performance of the proposed scheme and algorithm.     

High-resolution restoration of dynamic image sequences

We investigated an approach to reconstructing high-resolution images from dynamic image sequences using local spectral analysis. High-resolution reconstruction from linearly shifted multiple static image frames has previously been studied, in which the aliasing relationship between the spectrum of the original signal and the DFTs of shifted and sampled signals is exploited. In the high-resolution reconstruction of dynamic image sequences, difficulties occur as a result of nonuniform shifts in the frames. We use loca spectral analysis to achieve high-resolution reconstruction by overlapped block decomposition and motion compensation. For each block image in a reference frame in the sequence, motion estimation and subpixel registrations are performed against adjacent frames. High resolution reconstruction is performed on such motion-compensated block-image sequences. For some blocks containing motion boundaries, high resolution reconstruction is difficult; a new scene emerges or disappears producing inconsistent information. An interpolation technique is used in such blocks to generate the number of pixels consistent with other high-resolution blocks. The flower-garden image sequence is used for the computer simulations. The quality of the restored images are very encouraging; the aliasing effects in the original frames are significantly reduced and sharper edges are produced. The overall procedure to generate such higher-resolution images from a dynamic image sequence is described in detail. The result can be applied to various image sequence restoration applications including up-conversion of conventional video signals.©1994 John Wiley & Sons Inc     

一种改进的视频序列超分辨率重建算法及应用

在凸集优化基础上,充分利用最大后验概率和凸集投影技术,提出了一种高效强鲁棒性视频序列分辨率提升算法。首先,在空域设计一个简单的预处理共轭梯度估计器,预测原始高分辨率图像;然后,在小波域分别创建帧间和帧内两个不同的凸集,并实施不同的投影运算,提取出隐含在相邻低分辨率图像中的细节信息;最后,利用空域估计器中相邻因子间的关系约束凸集投影解的可行域,保证快速获得图像重建的唯一最优解。仿真实验和实际交通监测系统应用结果均表明,该方法较其他方法不仅可获得更高的峰值信噪比和更好的可视化效果,而且收敛更快,鲁棒性更强。     

基于敏感性分析的协方差随机子空间方法参数优化

在工作模态分析中,结构模态的准确识别在包括结构健康监测在内的许多应用中至关重要.该文基于敏感性分析,研究了模型系统阶数N和Toeplitz矩阵行块数i在协方差驱动随机子空间法(covariance-driven stochastic subspace identification,SSI-Cov)中对模态识别结果的影响...     

求置换因子循环矩阵的极小多项式和逆的算法

本文引入了任意域上置换因子循环矩阵,利用多项式环的理想的Gr(?)bner基的算法给出了任意域上置换因子循环矩阵的极小多项式和公共极小多项式的算法,同时给出了这类矩阵逆矩阵的两种算法最后,利用Schur补给出了任意域上具有置换因子循环矩阵块的分块矩阵逆的一个算法,在有理数域或模素数剩余类域上,这一算法可由代数系统软件CoCoA4．0实现。     

Galerkin formulation and singularity subtraction for spectral solutions of boundary integral equations

A new spectral Galerkin formulation is presented for the solution of boundary integral equations. The formulation is carried out with an exact singularity subtraction procedure based on analytical integrations, which provides a fast and precise way to evaluate the coefficient matrices. The new Galerkin formulation is based on the exact geometry of the problem boundaries and leads to a non-element method that is completely free of mesh generation. The numerical behaviour of the method is very similar to the collocation method; for Dirichlet problems, however, it leads to a symmetric coefficient matrix and therefore requires half the solution time of the collocation method. © 1998 John Wiley & Sons, Ltd.     

Use of Chebychev Polynomials in Thin Film Computations

From Herpin’s expression for the mth power of a multilayer matrix, very simple closed formulas are derived for the matrices and optical constants of any multilayer with a periodic structure.According to Epstein’s theorem, any symmetrical multilayer is equivalent to a fictitious monolayer. A simple expression for the equivalent index and thickness of this monolayer is deduced for the case of a periodic and symmetrical sequence of equally thick films.As compared to any other method of numerical computation, the suggested formulation provides a considerable saving of time and work. In a numerical example, this saving amounts to about 80 percent.     

A fast algorithm for multifield representation and multiscale simulation of high‐quality 3D stochastic aggregate microstructures by concurrent coupling of stationary Gaussian and fractional Brownian random fields

This study presents a multiscale computational framework for the representation and generation of concrete aggregate microstructures on the basis of the multifield theory, which couples the stationary Gaussian random field with the fractional Brownian random field. Specifically, the stationary Gaussian field is utilized to simulate the morphological shape of an aggregate on the coarse scale, whereas the surface topography of the aggregate on the fine scale is represented by the fractional Brownian field. To bridge the 2 scales, a concurrent coupling formula is proposed. This coupled technique allows for smooth transition between the coarse and fine scales and permits the rapid generation of highly realistic concrete aggregates that can be tailored to the desired quality and requirements, making the algorithm computationally appealing. In the generation of the random fields on the 2 scales, the Fourier representation of block circulant covariance matrices with circulant blocks is exploited, which yields substantial efficiency advantages over the conventional Cholesky decomposition approach in factorizing covariance matrices as well as simulating random fields. Meanwhile, a microsurface postprocessing and reconstruction procedure is also developed to convert the generated random fields into realistic 3D shapes. The numerical methodology proposed in this study offers tremendous potential for a plethora of applications in cement‐based materials.     

A Matrix Decomposition MFS Algorithm for Biharmonic Problems in Annular Domains

The Method of Fundamental Solutions (MFS) is a boundary-type method for the solution of certain elliptic boundary value problems. In this work, we develop an efficient matrix decomposition MFS algorithm for the solution of biharmonic problems in annular domains. The circulant structure of the matrices involved in the MFS discretization is exploited by using Fast Fourier Transforms. The algorithm is tested numerically on several examples.     

一种机械振动信号的结构化随机测量矩阵构造方法

针对压缩采集在机械振动信号采集的过程中,现有随机测量矩阵不易硬件实现、确定性测量矩阵重构误差较大的问题。将高斯序列的优点和循环原理的优点相结合,提出一种高斯分布循环测量矩阵,其是一种结构化随机测量矩阵。高斯分布循环测量矩阵的第一行元素由服从高斯分布的序列生成,通过循环移位生成剩余的所有行向量;随机取出除第一行的其他所有行的部分元素,每个元素再乘不同的随机数或者同一个随机数,并放回原位置;基于高斯分布循环测量矩阵得到的机械振动信号压缩测量值采用正交匹配追踪算法对原始振动信号进行重构。高斯分布循环测量矩阵的所有元素的随机性可以满足测量矩阵对随机性的要求,循环原理的内在确定性又可满足测量矩阵硬件实现的要求。仿真表明:高斯分布循环测量矩阵的感知性能略优于与高斯矩阵的性能,整体上基本相当。     

Simple image intensity compensation (SIMIC) method prior to application of distortion correction algorithms in brain diffusion Tensor Magnetic Resonance Imaging: Validation test for two cost functions of distortion correction algorithms

The purpose of this study is to design a simple image intensity compensation (SIMIC) method prior to the application of a variety of cost functions for distortion correction in diffusion tensor imaging (DTI). The synthetic dataset consists of each direction of diffusion weighted imaging (DWI) made by multiplication of nondiffusion weighted image (b = 0 image) and tensor matrices. We added the effects of patient motion and eddy current distortion using translation, rotation, scaling and shearing matrices. We calculated the b = 0 image of each direction from original DTI, inversely. A co‐registration method was applied to the extracted b = 0 images of each direction based on the original b = 0 image and then, the transformation matrices were generated and the original DTI were transformed using this transformation matrix. For the DTI distortion correction, two kinds of cost functions, normalized mutual information (NMI) and normalized cross‐correlation (NCC), were used. Visual assessments and quantitative measurements were used to evaluate the results. When using the NMI as a cost function, the quantitative results showed no significant differences between NMI and NMI with SIMIC method. However, there are significant differences compared with using the NCC as a cost function. Our study showed cost function for image distortion correction with SIMIC method improved the results both quantitatively and in terms of qualitative accuracy. This method may be helpful for DTI analysis and helpful for increasing accuracy. © 2015 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 25, 328–33, 2015     

A chaotic image cryptosystem based on Toeplitz and Hankel matrices

Abstract

Because of properties in chaos system such as the sensitive dependence on initial conditions, system parameters, pseudorandom property and ergodicity, chaotic image encryption algorithm can suggest a new and efficient way of encryption scheme, which has been studied more and more in recent years. A novel chaotic image encryption algorithm based on Toeplitz matrix and Hankel matrix is proposed in this paper. We shuffle totally the positions of image pixels to confuse the relationship between the plain image and cipher image combined with Toeplitz matrix, Hankel matrix and logistic chaotic system. Another hyper-chaos system of Chen's chaotic system is taken to change the grey values of image pixels to enhance the security further. Experimental results in Sections 3 and 4 demonstrate that the key space is large enough and the key is sensitive to initial conditions to resist the brute force attack in the proposed algorithm. Additionally, the distribution of grey values in encrypted image has a random-like behaviour to resist statistical analysis.     

二维线性实对称可解反降晰的一种简化算法

本文讨论了用传统Toeplitz方法进行图像恢复的特点和不足之处,针对传统Toeplitz反降晰方法的缺点, 提出了一种应用于图像恢复的线性实对称反降晰简化方法及相关算法.该方法通过合理的对应映射和截断处理, 近似得到反降晰有限尺寸的逆核.围绕映射矩阵的形式和逆核截断尺寸的选择这两个关键问题,文中作了详细的 分析和探讨.对提出的算法进行仿真检验,并得到的效果明显优于传统Toeplitz方法,说明了该算法的优点.     

Matrix decomposition RBF algorithm for solving 3D elliptic problems

In this study, we propose an efficient algorithm for the evaluation of the particular solutions of three-dimensional inhomogeneous elliptic partial differential equations using radial basis functions. The collocation points are placed on concentric spheres and thus the resulting global matrix possesses a block circulant structure. This structure is exploited to develop an efficient matrix decomposition algorithm for the solution of the resulting system. Further savings in the matrix decomposition algorithm are obtained by the use of fast Fourier transforms. The proposed algorithm is used, in conjunction with the method of fundamental solutions for the solution of three-dimensional inhomogeneous elliptic boundary value problems.