High-resolution image reconstruction with multisensors

This article considers the problem of reconstructing a high-resolution image from multiple undersampled, shifted, degraded frames with subpixel displacement errors. This leads to a formulation involving a periodically shift-variant system model. The maximum a posteriori (MAP) estimation scheme is used subject to the assumption that the original high-resolution image is modeled by a stationary Markov-Gaussian random field. The resulting MAP formulation is expressed as a complex linear matrix equation, where the characterizing matrix involves the periodic block Toeplitz with Toeplitz block (BTTB) blur matrix and banded-BTTB inverse covariance matrix associated with the original image. By approximating the periodic-BTTB and the banded-BTTB matrices with, respectively, the periodic block circulant with circulant block (BCCB) and the banded-BCCB matrices, it is shown that the computation-intensive MAP formulation can be decomposed into a set of smaller matrix equations by using the two-dimensional discrete Fourier transform. Exact solutions are also considered through the use of the preconditioned conjugate gradient algorithm. Computer simulations are given to illustrate the procedure. © 1998 John Wiley & Sons, Inc. Int J Imaging Syst Technol, 9, 294–304, 1998     

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.     

Factorization for efficient solution of eigenproblems of adjacency and Laplacian matrices for graph products

Many structural models can be generated as the graph products of two or three subgraphs known as their generators. The main types of graph products consist of Cartesian, strong Cartesian, direct, and lexicographic products. In this paper, a general method is presented for the factorization of these graph products, such that the eigenvalues of the entire graph are obtained as the union of the eigenvalues of the weighted subgraphs defined here. The adjacency and Laplacian matrices for each graph product are studied separately. For graphs with missing elements (cut‐outs), the eigenvalues are calculated with the additional use of the Rayleigh quotient approach. The main idea stems from the rules recently developed by the authors for block diagonalization of matrices. These products have many applications in computational mechanics, such as ordering, graph partitioning, dynamic analysis, and stability analysis of structures. Some of these applications are addressed in this paper. Copyright © 2007 John Wiley & Sons, Ltd.     

Block diagonalization of adjacency and Laplacian matrices for graph product; applications in structural mechanics

Eigenvalues and eigenvectors of graphs have many applications in structural mechanics and combinatorial optimization. For a regular space structure, the visualization of its graph model as the product of two simple graphs results in a substantial simplification in the solution of the corresponding eigenproblems. In this paper, the adjacency and Laplacian matrices of four graph products, namely, Cartesian, strong Cartesian, direct and lexicographic products are diagonalized and efficient methods are obtained for calculating their eigenvalues and eigenvectors. An exceptionally efficient method is developed for the eigensolution of the Laplacian matrices of strong Cartesian and direct products. Special attention is paid to the lexicographic product, which is not studied in the past as extensively as the other three graph products. Examples are provided to illustrate some applications of the methods in structural mechanics. Copyright © 2006 John Wiley & Sons, Ltd.     

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。     

On the decomposition of generalized eigenproblems for the free vibration analysis of cyclically symmetric finite element models

Free vibration analysis is a major part of any dynamic analysis. Natural frequencies and related mode shapes may be obtained from free vibration analysis as the solutions of generalized eigenproblems. Although the eigensolutions of large‐scale structures require large computational efforts, these solutions may be achieved simply for symmetric structures. We present an efficient method for the decomposition of generalized eigenproblems related to finite element models with cyclic symmetry (having nodes at the axis of symmetry) into eigensubproblems with significantly smaller dimensions. This decomposition is obtained by block diagonalization of a matrix with a special pattern known as a block circulant, using the concept of the Kronecker product and similarity transformations. The proposed method is applied to three finite element models discretized by triangular and four‐node quadrilateral plate and shell elements, and its efficiency, accuracy and simplicity are evaluated. Copyright © 2009 John Wiley & Sons, Ltd.     

Geometrically nonlinear analysis of circulant structures using an efficient eigensolution method

In this paper, an efficient method is presented for the geometrically nonlinear analysis of circulant structures. A structure is called regular if its model can be formed as the product of two subgraphs, and if one of the subgraphs is a cycle then it is termed circulant. In the present method, we deal with the eigensolution of circulant structures by simply finding the eigenvalues and eigenvectors of the blocks of the stiffness matrix rather than those of the entire structural matrix, leading to a considerable reduction in the computational time. The developed method utilizes concepts from product graphs and linear algebra for the nonlinear analysis of structures. Graph products are used to perform the eigensolution of circulant structures through those of its constituting subgraphs. Here, using the existing numerical methods, nonlinear static and dynamic analyses can be performed. In the presented method instead of solving the characteristic equation in each iteration of each step, the eigensolution of the structure is obtained using the eigensolution of the primary structure (the structure of the first iteration of the first step) together with some simple mathematical operations. In the present method instead of heavy matrix operations such as inverting a matrix and solution of the corresponding equations, one needs only to perform simple matrix multiplications and additions. The advantage of the presented method becomes more apparent when it is applied to the nonlinear analysis of a structure, where analysis should be performed many times.     

Programming Univariate and Multivariate Analysis of Variance

A formulation of analysis of variance based on a model for the subclass means is presented. The deficiency of rank in the model matrix is handled, not by restricting the parameters, but by factoring the matrix as a product of two matrices, one providing a column basis for the model and the other representing linear functions of the parameters. In terms of the column basis and a diagonal matrix of subclass or incidence numbers, a compact matrix solution is derived which provides for testing a hierarchy of hypotheses in the non-orthogonal case. Two theorems are given showing that a column basis for crossed and/or nested designs can be constructed from Kronecker products of equi-angular vectors, contrast matrices, and identity matrices. This construction can be controlled in machine computation by a symbolic representation of each degree of freedom for hypothesis in the analysis. Provision for a multivariate analysis of variance procedure for multiple response data is described. Analysis of covariance, both in the univariate and multivariate case, is shown to be most convenient computationally as part of the multivariate procedure.     

Finite block method for transient heat conduction analysis in functionally graded media

Based on the one‐dimensional differential matrix derived from the Lagrange series interpolation, the finite block method is proposed first time to solve both stationary and transient heat conduction problems of anisotropic and functionally graded materials. The main idea is to establish the first order one‐dimensional differential matrix constructed by using Lagrange series with uniformly distributed nodes. Then the higher order of derivative matrix for one‐dimensional problem is obtained. By introducing the mapping technique, a block of quadratic type is transformed from Cartesian coordinate (xyz) to normalised coordinate (ξη?) with 8 seeds or 20 seeds for two or three dimensions. Then the differential matrices in physical domain are determined from that in normalised transformed coordinate system. In addition, the time dependent partial differential equations are analysed in the Laplace transformed domain, and the Durbin inversion method is used to determine the values in time domain. Illustrative two‐dimensional and three‐dimensional numerical examples are given, and comparisons have been made with analytical solutions. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

Compound matrix block diagonalization for efficient solution of eigenproblems in structural mechanics

Summary Recently, methods were developed for the decomposition of special matrices, leading to canonical forms I-IV. In this paper, the conditions required for the decomposability of such matrices are studied. It is shown that these canonical forms are obtainable by special block diagonalization of matrices, having certain properties. Here, tri- to penta-block diagonal matrices are studied and methods are developed for their decomposition. These methods are incorporated in the efficient solution of numerous eigenproblems of structural mechanics.     

Residual correction applied to a sequence of block tridiagonal matrices

The decomposition of a block tridiagonal matrix into the product of block lowe and upper matrices is described. The cost of solving a block tridiagonal system of equations is given and compared to profile gaussian elimination. The desirability of a less expensive method is coupled to physical intuition about a common problem of solving a slowly varying sequence of such systems to motivate an iterative method based on residual correction. The method is described and convergence criteria are derived. An expression of the cost is developed and is shown to compare favourably with decomposition in many cases. Problems and advantages in computer implementation of the method are discussed and results of tests of a particular implementation on a well-known problem are given.     

Strict doubly ergodic infinite transformations

We give examples of rank-one transformations that are (weak) doubly ergodic and rigid (so all their Cartesian products are conservative), but with non-ergodic two-fold Cartesian product. We give conditions for rank-one infinite measure-preserving transformations to be (weak) doubly ergodic and for their k-fold Cartesian product to be conservative. We also show that a (weak) doubly ergodic nonsingular group action is ergodic with isometric coefficients, and that the latter strictly implies W-measurable sensitivity.     

Efficient dynamic finite element method for 3D objects with uniform cross‐section

An efficient implicit dynamic finite element method (FEM) for elastic 3D objects with uniform cross‐sections was developed. In this method, the finite element mesh is generated in such a way that the object to be analysed is at first sliced into layers with the same thickness along its generatrix and then each layer is discretized into finite elements of the same pattern. This way of discretization makes the mass, viscosity, and stiffness matrices into the repetitive block tridiagonal matrices. The repetitive block tridiagonal matrix has the characteristic, that the sequence of matrices which appears in the Gaussian elimination for the repetitive block tridiagonal matrix is a rapid convergent sequence. The process of the Gaussian elimination can be terminated when the sequence converges. The rest of the sequence is not necessary to be stored. The present method can save the computational time and memory by utilising this characteristic of the repetitive block tridiagonal matrix. A few examples of analyses including whole Hopkinson‐bar analysis were performed to demonstrate the effectiveness of the present method. The present method is applicable not only to the elasto‐dynamics but also to many other problems, such as thermal problems, electrical problems, and plastic problems without geometric non‐linearity. Copyright © 2005 John Wiley & Sons, Ltd.     

(广义)块对角占优矩阵的奇异与非奇异性

本文得到了块对角占优矩阵奇异与非奇异的几个充分必要条件,并由此得到了广义块对角占优矩阵奇异与非奇异的一些充分必要的判定条件.     

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.     

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.     

Higher-order inverse mass matrices for the explicit transient analysis of heterogeneous solids

New methods are presented for the direct computation of higher-order inverse mass matrices (also called reciprocal mass matrices) that are used for explicit transient finite element analysis. The motivation of this work lies in the need of having appropriate sparse inverse mass matrices, which present the same structure as the consistent mass matrix, preserve the total mass, predict suitable frequency spectrum and dictate sufficiently large critical time step sizes. For an efficient evaluation of the reciprocal mass matrix, the projection matrix should be diagonal. This condition can be satisfied by adopting dual shape functions for the momentum field, generated from the same shape functions used for the displacement field. A theoretically consistent derivation of the inverse mass matrix is based on the three-field Hamilton principle and requires the projection matrix to be evaluated from the integral of these shape functions. Unfortunately, for higher-order FE shape functions and serendipity FE elements, the projection matrix is not positive definitive and can not be employed. Therefore, we study several lumping procedures for higher order reciprocal mass matrices considering their effect on total-mass preserving, frequency spectra and accuracy in explicit transient simulations. The article closes with several numerical examples showing suitability of the direct inverse mass matrix in dynamics.     

余弦变换矩阵及其应用

本文介绍笛卡尔坐标与正交曲线坐标的＂余弦变换矩阵＂，证明这个变换矩阵是正交矩阵。利用这个变换矩阵可以方便地将笛卡尔坐标的张量表达式、微分算子及有关公式变换成正交曲线坐标的相应公式。