<Emphasis Type="Italic">X</Emphasis>- and <Emphasis Type="Italic">Y</Emphasis>-invariants of partial differential operators in the plane

In the paper, a classical computer algebra problem—symbolic solution of differential equations—is considered. Particularly, widely used Darboux theorems on transformation of hyperbolic operators in the plane are extended to the space of invariants by means of differential substitutions. X- and Y-invariants for such operators are introduced as solutions of some equations written in terms of the Laplace invariants of operator L with respect to gauge transformations. Explicit formulas of transformations of sets of X- and Y-invariants under the Darboux transformations are obtained.     

图像关于边缘提取算子的微分不变性

图像边缘是一类重要特征,该文研究图像关于边缘提取算子的微分不变性,其目的是考察图像在边缘提取的过程失去了什么信息,利用Ｌｉｅ变换群在节丛上的延拓公式以及由此导出的相关微分方程的不变性条件,文中给出了构造Ｌａｐｌａｃｅ边缘算子微分不变量的详细推导过程,为使该文自封闭,作者发展了所需的数学工具,包括一系列延拓Ｌｉｅ悍数的计算公式和有关微分方程解的不变性条件,最后给出了一个包含尺度、灰度、放置变换的数值     

On approximation of the Laplace–Beltrami operator and the Willmore energy of surfaces

Discrete Laplace–Beltrami operators on polyhedral surfaces play an important role for various applications in geometry processing and related areas like physical simulation or computer graphics. While discretizations of the weak Laplace–Beltrami operator are well‐studied, less is known about the strong form. We present a principle for constructing strongly consistent discrete Laplace–Beltrami operators based on the cotan weights. The consistency order we obtain, improves previous results reported for the mesh Laplacian. Furthermore, we prove consistency of the discrete Willmore energies corresponding to the discrete Laplace–Beltrami operators.     

A unified discrete framework for intrinsic and extrinsic Dirac operators for geometry processing

Spectral mesh analysis and processing methods, namely ones that utilize eigenvalues and eigenfunctions of linear operators on meshes, have been applied to numerous geometric processing applications. The operator used predominantly in these methods is the Laplace‐Beltrami operator, which has the often‐cited property that it is intrinsic, namely invariant to isometric deformation of the underlying geometry, including rigid transformations. Depending on the application, this can be either an advantage or a drawback. Recent work has proposed the alternative of using the Dirac operator on surfaces for spectral processing. The available versions of the Dirac operator either only focus on the extrinsic version, or introduce a range of mixed operators on a spectrum between fully extrinsic Dirac operator and intrinsic Laplace operator. In this work, we introduce a unified discretization scheme that describes both an extrinsic and intrinsic Dirac operator on meshes, based on their continuous counterparts on smooth manifolds. In this discretization, both operators are very closely related, and preserve their key properties from the smooth case. We showcase various applications of our operators, with improved numerics over prior work.     

Secondary Laplace operator and generalized Giaquinta–Hildebrandt operator with applications on surface segmentation and smoothing

Various geometric operators have been playing an important role in surface processing. For example, many shape analysis algorithms have been developed based on eigenfunctions of the ​Laplace–Beltrami operator (LBO), which is defined based on the first fundamental form of the surface. In this paper, we introduce two new geometric operators based on the second fundamental form of the surface, namely the secondary Laplace operator (SLO) and generalized Giaquinta–Hildebrandt operator (GGHO). Surface features such as concave creases/regions and convex ridges can be captured by eigenfunctions of the SLO, which can be used in surface segmentation with concave and convex features detected. Moreover, a new geometric flow method is developed based on the GGHO, providing an effective tool for sharp feature-preserving surface smoothing.     

Differential operators on orbifolds

An algorithm is presented that computes explicit generators for the ring of differential operators on an orbifold, the quotient of a complex vector space by a finite group action. The algorithm also describes the relations among these generators. The algorithm presented in this paper is based on Schwarz’s study of a map carrying invariant operators to operators on the orbifold and on an algorithm to compute rings of invariants using Gröbner bases due to Derksen [Derksen, Harm, 1999. Computation of invariants for reductive groups. Adv. Math. 141 (2), 366–384]. It is also possible to avoid using Derksen’s algorithm, instead relying on the Reynolds operator and the Molien series.     

一种精确的袋装粮图像边缘检测算法

基于图像识别的国家储备粮仓袋装粮食数量自动监管与稽核系统的技术核心是智能识别各种粮仓场景图像中粮袋的数量,杜绝人为的弄虚作假。边缘检测是袋装粮图像识别的首要问题,在分析了经典的以及在其基础上进行各种改进的Laplace算子缺陷的基础上,提出了1种改进的Laplace算子,该算子通过设置合理的模板参数克服了原有算子的不足,提高了图像边缘检测的精度。实验结果证明,该算子检测效果优于其他模板,并且能够精确地检测出各种类型的边缘信息。     

基于改进高斯-拉普拉斯算子的噪声图像边缘检测方法

针对现有梯度算子在图像边缘检测中存在的对噪声比较敏感的问题,提出了一种基于改进高斯-拉普拉斯算子的图像边缘检测方法。噪声图像中的边缘检测是一项关键任务,然而目前常用的几种梯度算子,包括已经提出的高斯-拉普拉斯算子都没能取得理想效果。提出的方法对传统的拉普拉斯边缘检测算子做了改进,并与高斯滤波器相结合。首先,应用高斯滤波器来平滑图像,抑制噪声。然后基于拉普拉斯梯度边缘检测器进行边缘检测。最后在标准图像上进行评估,评估结果显示,提出的边缘检测方法所获得的峰值信噪比(PSNR)和均方误差(MSE)均优于其他几种对比方法。     

A new algorithm for finite spectrum assignment of single-inputsystems with time delay

Finite spectrum assignment for time-delay systems is the elimination of delay operators from the characteristic function of the closed-loop system and the arbitrary assignment of poles. The control consists of polynomials in the delay operator and finite Laplace transforms. An algorithm for computing the control matrix is presented. In particular, the control matrix over rational functions of a delay operator is computed and expanded to partial fractions. Partial fractions are systematically transformed to finite Laplace transforms     

Perfect Laplacians for Polygon Meshes

A discrete Laplace‐Beltrami operator is called perfect if it possesses all the important properties of its smooth counterpart. It is known which triangle meshes admit perfect Laplace operators and how to fix any other mesh by changing the combinatorics. We extend the characterization of meshes that admit perfect Laplacians to general polygon meshes. More importantly, we provide an algorithm that computes a perfect Laplace operator for any polygon mesh without changing the combinatorics, although, possibly changing the embedding. We evaluate this algorithm and demonstrate it at applications.     

Matrix operators for numerically stable representation of stifflinear dynamic systems

A new transformation having features similar to the Laplace transform (but numerically oriented) is developed from the Chebyshev polynomials theory. Signals are represented as vectors of Chebyshev coefficients, and linear subsystems as precomputed matrices. The original problem is preprocessed only once to yield matrix invariants for fast recurrent computations. Theoretical implications of the exact digitizing of a tenth-order transfer function and the reduced-order modeling of a stiff system are discussed     

On laplace and Dini transformations for multidimensional equations with a decomposable principal symbol

Algorithms for solving linear PDEs implemented in modern computer algebra systems are usually limited to equations with two independent variables. In this paper, we propose a generalization of the theory of Laplace transformations to second-order partial differential operators in ?³ (and, generally, ? ⁿ ) with a principal symbol decomposable into the product of two linear (with respect to derivatives) factors. We consider two algorithms of generalized Laplace transformations and describe classes of operators in ?³ to which these algorithms are applicable. We correct a mistake in [8] and show that Dini-type transformations are in fact generalized Laplace transformations for operators with coefficients in a skew (noncommutative) Ore field. Keywords: computer algebra, partial differential equations, algorithms for solution.     

An algorithm for constructing Darboux transformations of type I for third-order hyperbolic operators of two variables

Darboux transformations of type I are invertible Darboux transformations with explicit short formulas for inverse transformations. These transformations are invariant with respect to gauge transformations, and, for gauge transformations acting on third-order hyperbolic operators of two variables, a general-form system of generating differential invariants is known. In the paper, first-order Darboux transformations of type I for this class of operators are considered. The corresponding operator orbits are directed graphs with at most three edges originating from each vertex. In the paper, an algorithm for constructing such orbits is suggested. We have derived criteria for existence of first-order Darboux transformations of type I in terms of the generating invariants, formulas for transforming invariants, and the so-called “triangle rule” property of orbits. The corresponding implementation in the LPDO package is described. The orbits are constructed in two different forms, one of which outputs the graph in the format of the well-known built-in Maple package Graph Theory.     

一种新的非线性边缘检测滤波器

本文提出一种除噪滤波与边缘检测同时进行的方法——非线性检测滤波器法.该方法基于中值滤波器和非线性Laplace算子数学方法.理论与实验表明:该非线性滤波器很少损失边缘信息,而且速度快.     

A bit-level systolic array for digital contour smoothing

Linear operators for digital contour smoothing are described. These operators are defined by circulant Toeplitz matrices and allow to smooth digital contours in the least-squares sense. They minimize the undersampling, digitizing and quantizing error and allow to calculate invariants, such as curvature, which are not possible to calculate without smoothing. A bit-level systolic array which is capable of realizing the proposed operator is described. This array is easy to implement in VLSI, because the array cells involved are very simple. Furthermore, the array is completely pipelined on the bit-level, so that it operates with a high clock frequency achieving very high throughputs.     

Computational experiments in the problem on eigenvalues for the Laplace operator in the polygonal domain

The technique of numerical evaluation of the Laplace operator eigenvalues in a polygon are described. The L-shaped domain is taken as an example. The conformal mapping of the circle is constructed to this area, using the Christoffel-Schwarz integral. In the circle, the problem is solved by the author’s (with K.I. Babenko’s contribution) procedures without saturation developed earlier. The question remains whether this procedure is applicable to piecewise-smooth boundaries (the conformal mapping has special features on the boundary). The performed computations show that it is possible to calculate about five eigenvalues (for the Neumann problem about 100 eigenvalues) of the Laplace operator in this domain with two to five characters after the decimal point.     

Nonlinear invariants for linear loops and eigenpolynomials of linear operators

The concept of an eigenpolynomial of a linear operator is introduced, an algorithm for the construction of eigenpolynomials is formulated, and a relationship between eigenpolynomials and polynomial invariants of linear loops of programs is established. The main result obtained is the construction of sets of L-invariants for loops of operators whose Jordan forms contain nontrivial blocks.     

A Numerical Strategy for Freestream Preservation of the High Order Weighted Essentially Non-oscillatory Schemes on Stationary Curvilinear Grids

The weighted essentially non-oscillatory (WENO) schemes have been extensively employed for the simulation of complex flow fields due to their high order accuracy and good shock-capturing properties. However, the standard finite difference WENO scheme cannot hold freestream automatically in general curvilinear coordinates. Numerical errors from non-preserved freestream can hide small scales such as turbulent flow structures; aero-acoustic waves which can make the results inaccurate or even cause the simulation failure. To address this issue, a new numerical strategy to ensure freestream preservation properties of the WENO schemes on stationary curvilinear grids is proposed in this paper. The essential idea of this approach is to offset the geometrically induced errors by proper discretization of the metric invariants. It includes the following procedures: (1) the metric invariants are retained in the governing equations and the full forms of the transformed equations on the general curvilinear coordinates are solved; (2) the symmetrical, conservative form of the metrics instead of the original ones are used; (3) the WENO schemes which are applied for the inviscid fluxes of the governing equations are employed to compute the outer-level partial derivatives of the metric invariants. In other words, the outer-level derivative operators for the metric invariants are kept the same with those for the corresponding inviscid fluxes. It is verified theoretically in this paper that by using this approach, the WENO schemes hold the freestream preservation properties naturally and thus work well in the generalized coordinate systems. For some well-known WENO schemes, the derivative operators for the metric invariants are explicitly expressed and thus this approach can be straightforwardly employed. The effectiveness of this strategy is validated by several benchmark test cases.     

The induced generalized aggregation operators for intuitionistic fuzzy sets and their application in group decision making

In this paper, we present the induced generalized intuitionistic fuzzy ordered weighted averaging (I-GIFOWA) operator. It is a new aggregation operator that generalized the IFOWA operator, including all the characteristics of both the generalized IFOWA and the induced IFOWA operators. It provides a very general formulation that includes as special cases a wide range of aggregation operators for intuitionistic fuzzy information, including all the particular cases of the I-IFOWA operator, GIFOWA operator and the induced intuitionistic fuzzy ordered geometric (I-IFOWG) operator. We also present the induced generalized interval-valued intuitionistic fuzzy ordered weighted averaging (I-GIIFOWA) operator to accommodate the environment in which the given arguments are interval-valued intuitionistic fuzzy sets. Further, we develop procedures to apply them to solve group multiple attribute decision making problems with intuitionistic fuzzy or interval-valued intuitionistic fuzzy information. Finally, we present their application to show the effectiveness of the developed methods.