图像放大的偏微分方程方法

在分析一些常见的图像放大方法的基础上,根据图像像素值特点及近期偏微分方程在图像处理中的应用,将图像的像素值看作是平面物体的温度;利用偏微分方程理论中的热传导数学模型,提出了基于一种新颖的热传导方程初边值问题的图像放大法;并根据其物理意义,设计相应的差分算法．实验证明,这是一种有效的图像放大方法．     

Image Sequence Analysis via Partial Differential Equations

This article deals with the problem of restoring and motion segmenting noisy image sequences with a static background. Usually, motion segmentation and image restoration are considered separately in image sequence restoration. Moreover, motion segmentation is often noise sensitive. In this article, the motion segmentation and the image restoration parts are performed in a coupled way, allowing the motion segmentation part to positively influence the restoration part and vice-versa. This is the key of our approach that allows to deal simultaneously with the problem of restoration and motion segmentation. To this end, we propose a theoretically justified optimization problem that permits to take into account both requirements. The model is theoretically justified. Existence and unicity are proved in the space of bounded variations. A suitable numerical scheme based on half quadratic minimization is then proposed and its convergence and stability demonstrated. Experimental results obtained on noisy synthetic data and real images will illustrate the capabilities of this original and promising approach.     

Solving nonlinear Partial Differential Equations on hypercubes

This paper presents an efficient parallel algorithm for solving nonlinear Partial Differential Equations (PDEs), occurring in heat transfer and fluid flow simulation, on hypercube machines. To evaluate its performance, an expression for the efficiency of the algorithm is derived. The results show that the hypercubes are well suited for solving nonlinear PDEs.     

对称四阶偏微分方程去噪算法

通过分析整体变分模型的去噪原理和效果,提出一个新的四阶偏微分方程去噪模型,用于克服二阶偏微分方程去噪后使图像分块的缺点,同时保持去噪后图像的高保真性,并发展一个基于四邻域系统的对称离散化算法用于求解新模型,应用中值滤波去除四阶偏微分方程去噪所引起的亮点。实验结果表明,与传统方法相比,以该算法去噪后的图像具有更好的质量和视觉效果。     

Factoring and Solving Linear Partial Differential Equations

The problem of factoring a linear partial differential operator is studied. An algorithm is designed which allows one to factor an operator when its symbol is separable, and if in addition the operator has enough right factors then it is completely reducible. Since finding the space of solutions of a completely reducible operator reduces to the same for its right factors, we apply this approach and execute a complete analysis of factoring and solving a second-order operator in two independent variables. Some results on factoring third-order operators are exhibited.AMS Subject Classifications: 35A25, 35C05, 35G05.     

Vanishing Moment Method and Moment Solutions for Fully Nonlinear Second Order Partial Differential Equations

This paper concerns with numerical approximations of solutions of fully nonlinear second order partial differential equations (PDEs). A new notion of weak solutions, called moment solutions, is introduced for fully nonlinear second order PDEs. Unlike viscosity solutions, moment solutions are defined by a constructive method, called the vanishing moment method, and hence, they can be readily computed by existing numerical methods such as finite difference, finite element, spectral Galerkin, and discontinuous Galerkin methods. The main idea of the proposed vanishing moment method is to approximate a fully nonlinear second order PDE by a higher order, in particular, a quasilinear fourth order PDE. We show by various numerical experiments the viability of the proposed vanishing moment method. All our numerical experiments show the convergence of the vanishing moment method, and they also show that moment solutions coincide with viscosity solutions whenever the latter exist. This work was partially supported by the NSF grants DMS-0410266 and DMS-0710831.     

Multivariate Median Filters and Partial Differential Equations

Multivariate median filters have been proposed as generalizations of the well-established median filter for gray-value images to multichannel images. As multivariate median, most of the recent approaches use the \(L^1\) median, i.e., the minimizer of an objective function that is the sum of distances to all input points. Many properties of univariate median filters generalize to such a filter. However, the famous result by Guichard and Morel about approximation of the mean curvature motion PDE by median filtering does not have a comparably simple counterpart for \(L^1\) multivariate median filtering. We discuss the affine equivariant Oja median and the affine equivariant transformation–retransformation \(L^1\) median as alternatives to \(L^1\) median filtering. We analyze multivariate median filters in a space-continuous setting, including the formulation of a space-continuous version of the transformation–retransformation \(L^1\) median, and derive PDEs approximated by these filters in the cases of bivariate planar images, three-channel volume images, and three-channel planar images. The PDEs for the affine equivariant filters can be interpreted geometrically as combinations of a diffusion and a principal-component-wise curvature motion contribution with a cross-effect term based on torsions of principal components. Numerical experiments are presented, which demonstrate the validity of the approximation results.     

Partial Differential Equations and Bivariate Orthogonal Polynomials

In 1929, S. Bochner identified the families of polynomials which are eigenfunctions of a second-order linear differential operator. What is the appropriate generalization of this result to bivariate polynomials? One approach, due to Krall and Sheffer in 1967 and pursued by others, is to determine which linear partial differential operators have orthogonal polynomial solutions with all the polynomials in the family of the same degree sharing the same eigenvalue. In fact, such an operator only determines a multi-dimensional eigenspace associated with each eigenvalue; it does not determine the individual polynomials, even up to a multiplicative constant. In contrast, our approach is to seek pairs of linear differential operators which have joint eigenfunctions that comprise a family of bivariate orthogonal polynomials. This approach entails the addition of some “normalizing" or “regularity" conditions which allow determination of a unique family of orthogonal polynomials. In this article we formulate and solve such a problem and show with the help of Mathematica that the only solutions are disk polynomials. Applications are given to product formulas and hypergroup measure algebras.     

Exact Solutions of Iterative Functional Differential Equations

In general, it is difficult to compute explicit solutions for nonlinear differential equations. In this note, we show how power function solutions can be computed for a class of nonlinear functional equations involving derivatives and iterates of the unknown functions.     

Expansions of Differential Operators and Nonsmooth Solutions of Differential Equations

Elements of F-space theory are presented, which made it possible to develop a promising procedure of extending differential operators.     

基于偏微分方程的隐式曲面光顺方法

提出隐式曲面的光顺问题．针对该问题,给出刻画隐式曲面光顺程度的能量模型,并将能量解释为关于隐函数的泛函．基于变分原理,构造出隐函数关于时间的偏微分方程。通过求解该方程得到隐函数序列,使得光顺能量逐渐变小,从而达到光顺隐式曲面的目的．另外．针对光顺问题提出的其它约束条件,如尽可能保持面积不变,保持原有的形状特征等,对模型进行修正．最后,给出方程的实用解法及实验结果。并作简单讨论．实验结果表明该方法通用、灵活、有效,而且程序易于实现．     

Using Chebyshev Polynomials to Approximate Partial Differential Equations

This paper suggests a simple method based on a Chebyshev approximation at Chebyshev nodes to approximate partial differential equations (PDEs). It consists in determining the value function by using a set of nodes and basis functions. We provide two examples: pricing a European option and determining the best policy for shutting down a machine. The suggested method is flexible, easy to programme and efficient. It is also applicable in other fields, providing efficient solutions to complex systems of PDEs.     

偏微分方程的算子自定义小波解耦算法研究

利用小波算法求解偏微分方程最困难的问题是随着尺度的升高,系统方程的耦合度越来越高,极大降低了计算效率和精度.针对此问题提出了采用算子自定义小波的多尺度解耦算法,首先建立有限元多分辨空间和小波细化关系,提出偏微分方程的多尺度计算理论方法.在优化方案的基础上,提出算子自定义小波的构造方法及解耦条件.改进方法的突出优点在于根据工程问题的实际需要灵活构造具有期望特性的小波基.提出偏微分方程的多尺度算子自定义小波算法,充分利用算子自定义小波的嵌套逼近和尺度解耦特性,实现问题的高效求解.仿真结果表明,改进的算子自定义小波解耦算法具有计算效率高、精度高等特点.     

Neural Network Method for Solving Partial Differential Equations

A method is presented to solve partial differential equations (pde's) and its boundary and/or initial conditions by using neural networks. It uses the fact that multiple input, single output, single hidden layer feedforward networks with a linear output layer with no bias are capable of arbitrarily well approximating arbitrary functions and its derivatives, which is proven by a number of authors and well known in literature. Knowledge about the pde and its boundary and/or initial conditions is incorporated into the structures and the training sets of several neural networks. In this way we obtain networks of which some are specifically structured. To find the solution of the pde and its boundary and/or initial conditions we have to train all obtained networks simultaneously. Therefore we use an evolutionary algorithm to train the networks. We demonstrate the working of our method by applying it to two problems.     

偏微分方程在图像放大中的应用

本文提出基于偏微分方程的图像放大算法,利用扩散模型来消除放大图像的边缘锯齿化和细节模糊化。实验表明,该方法具有较好的实用性。     

Genetic Programming Approaches for Solving Elliptic Partial Differential Equations

In this paper, we propose a technique based on genetic programming (GP) for meshfree solution of elliptic partial differential equations. We employ the least-squares collocation principle to define an appropriate objective function, which is optimized using GP. Two approaches are presented for the repair of the symbolic expression for the field variables evolved by the GP algorithm to ensure that the governing equations as well as the boundary conditions are satisfied. In the case of problems defined on geometrically simple domains, we augment the solution evolved by GP with additional terms, such that the boundary conditions are satisfied by construction. To satisfy the boundary conditions for geometrically irregular domains, we combine the GP model with a radial basis function network. We improve the computational efficiency and accuracy of both techniques with gradient boosting, a technique originally developed by the machine learning community. Numerical studies are presented for operator problems on regular and irregular boundaries to illustrate the performance of the proposed algorithms.      

基于偏微分方程的图像平滑处理技术

传统的图像平滑方法在去除噪声的同时往往会破坏边缘、线务、纹理等图像特征,而基于偏微分方程(PDE)的各向异性扩散算法在抑制噪声的同时能够保持这些特征。研究基于PDE的图像去噪平滑算法并对PDE算法的不足作出改进,将其用于图像去噪处理效果非常理想,并在精密光学元件表面疵病识别的实验中取得较好的效果。     

一种基于偏微分方程的信号增强模型

在峰检测中,由于重叠峰和弱峰的存在,峰的漏检和错检率较高。为此本文在峰检测环节之前增加一个峰增强环节,使重叠峰的分离度以及弱峰幅度均增大。增强环节中采用的方法是将经典非线性扩散与导数谱相结合,即将导数谱增强后的信号作为经典非线性扩散的初始信号,经过一定时间的扩散得到增强后的信号。作为效果检验,首先对比了信号经过所提模型增强前后的效果,之后对比了其他信号增强模型的效果,结果表明本文所提模型有效。最后将本文模型应用于MALDI质谱峰增强。     

An Estimation Problem for Quasilinear Stochastic Partial Differential Equations

A problem of estimating a functional parameter (x) and functionals () based on observation of a solution u (t, x) of the stochastic partial differential equation is considered. The asymptotic problem setting, as the noise intensity 0, is investigated.