Some geometric considerations about the Kronecker normal form

In the first part of this paper we develop general algorithms from the Kronecker normal form of any matrix pencil, and give some of their properties. In the second part these results are applied to the so-called restricted matrix pencil associated with a given (C, A, B) system and introduced by Jaffe and Karcanias (1981). This allows us to make a link with well known algorithms of the geometric approach, particularly the (A, B)-invariant subspace algorithm and the almost (A, B)-controllability subspace algorithm. Finally, relationships between Morse's invariants (Morse 1973) and the invariants of the restricted matrix pencil are exhibited.     

Numerical solution of the discrete-time periodic Riccati equation

In this paper we present a method for the computation of the periodic nonnegative definite stabilizing solution of the periodic Riccati equation. This method simultaneously triangularizes by orthogonal equivalences a sequence of matrices associated with a cyclic pencil formulation related to the Euler-Lagrange difference equations. In doing so, it is possible to extract a basis for the stable deflating subspace of the extended pencil, from which the Riccati solution is obtained. This algorithm is an extension of the standard QZ algorithm and retains its attractive features, such as quadratic convergence and small relative backward error. A method to compute the optimal feedback controller gains for linear discrete time periodic systems is dealt with     

一种改进的铅笔画的生成方法

给出了一种改进的根据真实2D图像自动生成相应的非真实感铅笔画的方法。首先将彩色图像进行霓虹处理,再进行反相计算和灰度化,就可以产生铅笔画中的轮廓效果。其次,为了更好地产生铅笔画的光照效果及其局部走势纹理,采用线积分卷积(LIC)的方法来生成类似的效果,并且用适当的图像分割方法来获取进行LIC处理的有意义的区域。实验结果表明,本文的方法与以往的方法相比,能生成具有不同风格的效果,并且生成的速度更快。     

Creating a nilpotent pencil via deadbeat

We consider the problem of finding a square low-rank correction (λC ? B)F to a given square pencil (λE ? A) such that the new pencil λ(E ? CF) ? (A ? BF) has all its generalised eigenvalues at the origin. We give necessary and sufficient conditions for this problem to have a solution and we also provide a constructive algorithm to compute F when such a solution exists. We show that this problem is related to the deadbeat control problem of a discrete-time linear system and that an (almost) equivalent formulation is to find a square embedding that has all its finite generalised eigenvalues at the origin.     

Chained aggregation of singular systems

Extends the notion of aggregation to singular systems (sE-A). Necessary and sufficient conditions for complete aggregability are given which reduce in the state-space case, E=I, to Aoki's conditions. A geometric quotient-space analysis is given which uses a new result on the pencil induced in the quotient space by a regular matrix pencil. A generalized chained aggregation algorithm is used to put the system into a Hessenberg form which provides a reduced-order model for the output dynamics. It is shown that, as in the state-space situation, chained aggregation corresponds to removal of the unobservable subspace. It is also pointed out that the usual state-space chained aggregation algorithm is very efficient and numerically unstable. Specializing the singular system chained aggregation algorithm to the state-space case yields a better and more streamlined approach     

Nano—X图形引擎算法优化

Nano—X是一个基于客户端／服务器（C／S）#系结构的GUI,系统结构采用分层设计,包括设备驱动层、图形引擎层和应用接口层。Nano—X系统稳定且具有良好的可裁剪性和较高的实时性,但图形引擎中未采用任何反走样算法,图形存在较严重的图形阶梯现象。为解决这一问题,基于Wu反走样算法提出了一种新型快速反走样算法,通过降低运算量提高了算法快速性。仿真结果表明该算法提高了绘图效率,明显减少阶梯现象,同时算法具有较高的实时性。     

Equivalence conditions for behaviors and the Kronecker canonical form

In this paper we explore equivalence conditions and invariants for behaviors given in kernel representations. In case the kernel representation is given in terms of a linear matrix pencil, the invariants for strict equivalence are given by the Kronecker canonical form which, in turn, we interpret in geometric control terms. If the behavior is given in a kernel representation by a higher order rectangular polynomial matrix, the natural equivalence concept is behavior equivalence. These notions are closely related to the Morse group that incorporates state space similarity transformations, state feedback, and output injection. A simple canonical form for behavioral equivalence is given that clearly exhibits the reachable and autonomous parts of the behavior. Using polynomial models we also present a unified approach to pencil equivalence that elucidates the close connections between classification problems from linear algebra, geometric control theory, and behavior theory. We also indicate how to derive the invariants under behavior equivalence from the Kronecker invariants.     

The generalized eigendecomposition approach to the blind source separation problem

This paper proposes a novel formulation of the generalized eigendecomposition (GED) approach to blind source separation (BSS) problems. The generalized eigendecomposition algorithms consider the estimation of a pair of correlation matrices (a matrix pencil) using observed sensor signals. Each of various algorithms proposed in the literature uses a different approach to form the pencil. This study proposes a linear algebra formulation which exploits the definition of congruent matrix pencils and shows that the solution and its constraints are independent of the way the matrix pencil is computed. Also an iterative eigendecomposition algorithm, that updates separation parameters on a sample-by-sample basis, is developed. It comprises of: (1) performing standard eigendecompositions based on power and deflation techniques; (2) computing a transformation matrix using spectral factorization. Another issue discussed in this work is the influence of the length of the data segment used to estimate the pencil. The algorithm is applied to artificially mixed audio data and it is shown that the separation performance depends on the eigenvalue spread. The latter varies with the number of samples used to estimate the eigenvalues.     

Solving the infinite-dimensional discrete-time algebraic riccati equation using the extended symplectic pencil

In this paper we present results about the algebraic Riccati equation (ARE) and a weaker version of the ARE, the algebraic Riccati system (ARS), for infinite-dimensional, discrete-time systems. We introduce an operator pencil, associated with these equations, the so-called extended symplectic pencil (ESP). We present a general form for all linear bounded solutions of the ARS in terms of the deflating subspaces of the ESP. This relation is analogous to the results of the Hamiltonian approach for the continuous-time ARE and to the symplectic pencil approach for the finite-dimensional discrete-time ARE. In particular, we show that there is a one-to-one relation between deflating subspaces with a special structure and the solutions of the ARS. Using the relation between the solutions of the ARS and the deflating subspaces of the ESP, we give characterizations of self-adjoint, nonnegative, and stabilizing solutions. In addition we give criteria for the discrete-time, infinite-dimensional ARE to have a maximal self-adjoint solution. Furthermore, we consider under which conditions a solution of the ARS satisfies the ARE as well.     

Total variation and high-order total variation adaptive model for restoring blurred images with Cauchy noise

In this paper, we propose a novel model to restore an image corrupted by blur and Cauchy noise. The model is composed of a data fidelity term and two regularization terms including total variation and high-order total variation. Total variation provides well-preserved edge features, but suffers from staircase effects in smooth regions, whereas high-order total variation can alleviate staircase effects. Moreover, we introduce a strategy for adaptively selecting regularization parameters. We develop an efficient alternating minimization algorithm for solving the proposed model. Numerical examples suggest that the proposed method has the advantages of better preserving edges and reducing staircase effects.     

Adaptive Generalized Eigenvector Estimating Algorithm for Hermitian Matrix Pencil

Generalized eigenvector plays an essential role in the signal processing field. In this paper, we present a novel neural network learning algorithm for estimating the generalized eigenvector of a Hermitian matrix pencil. Differently from some traditional algorithms, which need to select the proper values of learning rates before using, the proposed algorithm does not need a learning rate and is very suitable for real applications. Through analyzing all of the equilibrium points, it is proven that if and only if the weight vector of the neural network is equal to the generalized eigenvector corresponding to the largest generalized eigenvalue of a Hermitian matrix pencil, the proposed algorithm reaches to convergence status. By using the deterministic discrete-time (DDT) method, some convergence conditions, which can be satisfied with probability 1, are also obtained to guarantee its convergence. Simulation results show that the proposed algorithm has a fast convergence speed and good numerical stability. The real application demonstrates its effectiveness in tracking the optimal vector of beamforming.      

A pole-assignment algorithm for linear state feedback

A new algorithm for the pole-assignment problem of a controllable time-invariant linear multivariable system with linear state feedback is presented. The resulting feedback matrix is a least-squares solution and is robust in a loose sense. The method is based on the controllability canonical (staircase) form and amounts to a new proof for the existence of a solution of the pole-assignment problem. Illustrative examples are given.     

A grid generator for 3-D explosion simulations using the staircase boundary approach in Cartesian coordinates based on STL models

In this paper, an automatic grid generator based on STL models is proposed. The staircase boundary treatment is implemented to handle irregular geometries and the computation domain is discretized using a regular Cartesian grid. Using the grid generator, staircase grids that are suitable for fast and accurate finite difference analysis could be generated. Employing the slicing algorithm in RP technologies [1], the STL models are sliced with a set of parallel planes to generate 2D slices after the STL files obtained from a CAD system undergo topology reconstruction. To decrease the staircase error (increase accuracy) and enhance working efficiency, the cross-section at the middle of the layer is taken to represent the cross-section of whole layer. The scan line filling technique of computer graphics [2] is used to achieve grid generation after slicing. Finally, we demonstrate an application of the introduced method to generate staircase grids, which allows successful FDM simulation in the field of explosion. The example shows that the automatic grid generator based on STL models is fast and gives simulation results that are in agreement with practical observations.     

Incorporating object-centered sampling and Delaunay tetrahedrization for visual hull reconstruction

In this paper we present a novel shape from silhouette algorithm. For an object to be modeled, the algorithm first computes a cloud of points located on a pencil of rays and distributed evenly on the visual hull surface, inside and outside the visual hull. Then Delaunay tetrahedrization is applied to the point cloud to partition its convex hull into a set of tetrahedrons. Finally, outlier tetrahedrons are removed by tetrahedron peeling, and a mesh model of the visual hull is extracted. The algorithm is robust, free from discretization artifacts, and produces a mesh model composed of well-shaped triangles.     

广义系统观控性及正则束条件的数值判定

本文提出了一种用于判定广义系统观控性及正则束条件的算法.该算法具有误差小、数 值稳定的特点.     

Network optimization and problems with coupling variables

Network optimization problems are considered. Their statements include numerous variables and equipments. Decomposition methods are used for their solution. In specific situations, intermediate problems solved by the algorithms have the form of the knapsack problem. In the case when the constraints have a staircase structure, an efficient algorithm can be constructed. The procedure for the sequential recalculation of the coefficients of the objective function in this algorithm can be applied to other problems with unimodular matrices, in particular, to various transportation problems.     

应用运动模糊方法仿真铅笔画纹理

通过分析真实铅笔纹理的特点,提出了一种基于运动滤波器的仿真铅笔纹理的新方法.首先对用户输入的图像进行边缘检测,提取轮廓信息,然后对传统白噪声生成法进行改进,将参考图像由RGB色彩空间转换到HSV色彩空间,通过亮度值控制噪声点的产生几率,最后应用运动模糊方法代替传统的线积分卷积(LIC)方法模拟铅笔纹理,得到具有素描风格的结果图像.实验结果表明,该方法生成效果接近于手绘风格,且无需用户交互,适合于动画制作等很多领域的应用.