曲率驱动的基于亥姆霍兹涡量方程的图像修复模型

图像修复模型根据已知区域信息自动修复目标区域,且保证修复后的图像满足人眼视觉系统的要求,目标区域轮廓自然.理论分析证明,流体力学中无粘亥姆霍兹涡量方程可以实现图像修复.根据曲线和曲面运动方程,使用曲率驱动亥姆霍兹修复模型中的等照度线传输方向.曲率是图像几何特征作用的结果,所以新模型可以很好地保持图像中的线性特征.图像中涡量为平滑程度的度量,二维图像域中涡量具有扩散性,对涡量方程的结果做各向异性扩散,使等照度线间的图像信息交互.亥姆霍兹涡量方程修复模型中的各个参数及扩散过程是涡量扩散性和耗散性以及图像几何特征作用的结果,扩散后的修复模型稳定且不存在错误的传输方向.理论和实验证明了曲率驱动的亥姆霍兹涡量方程模型在图像修复中的有效性.     

求解 Burgers 方程的决定方程的两种方法研究

论文研究了两种求解偏微分方程的决定方程的方法,一种是运用向量场及其延拓方法,另一种是通过符号计算软件 maple 自动求解软件包。论文以 Burgers 方程为例,证明两种方法得到的结果相同。但运用 maple 自动求解软件包能够避免复杂的代数计算,提高计算的速度与准确率,适用于求解复杂的高阶非线性偏微分方程。     

基于结构张量的图像修复方法

针对传统各向异性扩散方程在修复图像时仅考虑梯度模的大小,且在修复彩色图像时易产生虚假边缘等缺陷,提出基于结构张量的图像修复方法。将结构张量作为各向异性扩散方程的扩散系数,实现在不同区域有不同的扩散方式。实验结果显示:该方法与整体变分(TV)和BSCB方法相比,提高了图像修复效果,有效地完成对于彩色图像的修复。     

A Multigrid Platform for Real-Time Motion Computation with Discontinuity-Preserving Variational Methods

Variational methods are among the most accurate techniques for estimating the optic flow. They yield dense flow fields and can be designed such that they preserve discontinuities, estimate large displacements correctly and perform well under noise and varying illumination. However, such adaptations render the minimisation of the underlying energy functional very expensive in terms of computational costs: Typically one or more large linear or nonlinear equation systems have to be solved in order to obtain the desired solution. Consequently, variational methods are considered to be too slow for real-time performance. In our paper we address this problem in two ways: (i) We present a numerical framework based on bidirectional multigrid methods for accelerating a broad class of variational optic flow methods with different constancy and smoothness assumptions. Thereby, our work focuses particularly on regularisation strategies that preserve discontinuities. (ii) We show by the examples of five classical and two recent variational techniques that real-time performance is possible in all cases—even for very complex optic flow models that offer high accuracy. Experiments show that frame rates up to 63 dense flow fields per second for image sequences of size 160 × 120 can be achieved on a standard PC. Compared to classical iterative methods this constitutes a speedup of two to four orders of magnitude.     

A method for the numerical solution of some elliptic boundary value problems for a strip

A boundary integral equation for the numerical solution of a class of elliptic boundary value problems for a strip is derived. The equation should be particularly useful for the solution of an important class of problems governed by Laplace's equation and also for the solution of relevant problems in anisotropic thermostatics and elastostatics     

Finite element methods for the time-dependent Ginzburg-Landau model of superconductivity

The initial-boundary value problem for the time-dependent Ginzburg-Landau equations that model the macroscopic behavior of superconductors is considered. The convergence of finite-dimensional, semidiscrete Galerkin approximations is studied as is a fully-discrete scheme. The results of some computational experiments are presented.     

Solving large systems of linear ordinary differential equations on a vector computer

This paper is concerned with the development, analysis and implementation on a computer consisting of two vector processors of the arithmetic mean method for solving numerically large sparse sets of linear ordinary differential equations. This method has second-order accuracy in time and is stable.

The special class of differential equations that arise in solving the diffusion problem by the method of lines is considered. In this case, the proposed method has been tested on the CRAY X-MP/48 utilizing two CPUs. The numerical results are largely in keeping with the theory; a speedup factor of nearly two is obtained.     

复小波域维纳滤波与偏微分扩散相结合的图像去噪方法

针对图像噪声的去除,提出了一种基于复数小波域上的多方向窗维纳滤波与偏微分方程保持边缘细节相结合的方法。针对小波域维纳滤波的方向性差,去噪后图像容易产生哑铃效应,该方法首先进行双树复数小波变换,集中6个方向上的图像信号能量,之后,再在该6个方向上进行方向维纳滤波,对图像进行初步去噪,再以此引导偏微分方程中的扩散函数,实现各项异性进行扩散,最大限度地在保持图像细节的同时,去除噪声。实验结果表明,该方法的峰值信噪比,以及视觉质量都较复小波去噪或各项异性非线性扩散去噪方法有明显的改善。     

Properties of Higher Order Nonlinear Diffusion Filtering

This paper provides a mathematical analysis of higher order variational methods and nonlinear diffusion filtering for image denoising. Besides the average grey value, it is shown that higher order diffusion filters preserve higher moments of the initial data. While a maximum-minimum principle in general does not hold for higher order filters, we derive stability in the 2-norm in the continuous and discrete setting. Considering the filters in terms of forward and backward diffusion, one can explain how not only the preservation, but also the enhancement of certain features in the given data is possible. Numerical results show the improved denoising capabilities of higher order filtering compared to the classical methods.

A Self-Tuning Evolutionary Algorithm Applied to an Inverse Partial Differential Equation

Evolutionary algorithms (EAs) are becoming increasingly popular tools for solving complex search problems. Their popularity in various problem domains has led to the introduction and development of numerous variants of two standard EA operators—crossover and mutation. Unfortunately, there are few if any effective guidelines for choosing which operators will be most effective in a given problem. In this paper, a self-tuning EA is introduced that employs several crossover and mutation operators simultaneously. The probability of using a given operator changes during the course of an evolutionary run whereby the most effective operators are selected based on which part of the search space is currently being explored. The self-tuning EA is used to solve an inverse partial differential equation—considered to be one of the more difficult problems in the realm of engineering mathematics. Results indicate that for the particular inverse partial differential equation considered, the self-tuning EA provides an effective solution methodology.     

偏微分方程约束的基于模块的图像修复模型

为了对大规模目标区域的图像进行修复,提出了一种既能保持线性特征和纹理信息,又能应用于大规模目标区域的基于模块的图像修复模型。该模型首先使用像素切向等照度线强度约束修复优先级,由于切向等照度线方程在图像边缘宽度约束下扩散,因此具有很好的线性特征保持性能。这种扩散具备形态学不变性,可修复自然场景图像;然后采用Euclidean距离计算模块相似度,并加入偏微分方程约束,使得线性特征位置的像素点在匹配中影响较大;使用散度约束下总体变分插值法对修复结果进行无接缝效应处理,使得最终修复结果平滑;最后扩展模型相似度函数,使目标区域可被修复为指定的纹理特征。理论和实验结果证明该模型在图像修复中是有效的。     

偏微分方程(PDEs)模型在图像处理中的若干应用

介绍了偏微分方程(PDEs)模型在图像处理与分析中的应用,基本思想,发展历史和解决问题的基本框架。主要阐述了变分方法和形变模型(曲线演化)在图像恢复和图像分割中的应用。理论和实验结果表明,应用偏微分方程模型进行图像处理是一种有效的工具。最后,分析了这种方法的优点和面临的挑战。     

一种基于非线性扩散方程的指纹预处理方法

目前用于实时比对的活休指标,由于受指纹受集设备的限制,图像中常混有噪声,必须经过预处理才能进行可靠的比对。本文根据指纹纹理密集的特点,采用改进的基于梯度算子的解非线性扩散方程的算法,克服了退化扩散方程对大曲率边缘点的模糊效应,兼顾了降噪和保边界这一平滑降噪中矛盾的两个方面。对比几种常用的平滑降噪的预处理算法,实验结果证明,本文的改进算法对于带噪指纹图像的预处理,优于其他几种对比算法。     

Quaternion Diffusion for Color Image Filtering

How to combine color and multiscale information is a fundamental question for computer vision, and quite a few color diffusion techniques have been presented. Most of these proposed techniques do not consider the direct interactions between color channel pairs. In this paper, a new method of color diffusion considering these effects is presented, which is based on quaternion diffusion （QD） equation. In addition to showing the solution to linear QD and its analysis, one form of nonlinear QD is discussed. Compared with other color diffusion techniques, considering the interactions between channel pairs, QD has the following advantages： 1） staircasing effect is avoided; 2） as diffusion tensor, the image derivative is regularized without requiring additional convolution; 3） less time is needed. Experimental results demonstrate the effectiveness of linear and nonlinear QD applied to natural color images for denoising by both visual and quantitative evaluations.     

Variational Methods for Multimodal Image Matching

Matching images of different modalities can be achieved by the maximization of suitable statistical similarity measures within a given class of geometric transformations. Handling complex, nonrigid deformations in this context turns out to be particularly difficult and has attracted much attention in the last few years. The thrust of this paper is that many of the existing methods for nonrigid monomodal registration that use simple criteria for comparing the intensities (e.g. SSD) can be extended to the multimodal case where more complex intensity similarity measures are necessary. To this end, we perform a formal computation of the variational gradient of a hierarchy of statistical similarity measures, and use the results to generalize a recently proposed and very effective optical flow algorithm (L. Alvarez, J. Weickert, and J. Sánchez, 2000, Technical Report, and IJCV 39(1):41–56) to the case of multimodal image registrationOur method readily extends to the case of locally computed similarity measures, thus providing the flexibility to cope with spatial non-stationarities in the way the intensities in the two images are related. The well posedness of the resulting equations is proved in a complementary work (O.D. Faugeras and G. Hermosillo, 2001, Technical Report 4235, INRIA) using well established techniques in functional analysis. We briefly describe our numerical implementation of these equations and show results on real and synthetic data.     

A Powerful Numerical Technique Solving Zakai Equation for Nonlinear Filtering

In this paper we have developed a simple but powerful numerical method forthe approximation of the unnormalized conditional (probability) density offiltered diffusion process which satisfies Zakai equation and solves thenonlinear filtering problem. Using Galerkin technique the solution of Zakaiequation is approximated by means of a sequence of nonstandard basisfunctions given by a parameterized family of Gaussian densities. The methodis then illustrated by some examples.     

Alternative Method of Solution of the Regulator Equation: L2‐Space Approach

An alternative method for the proof of solvability of the differential equation that is a part of the regulator equation which arises from the solution of the output regulation problem. The proof uses the L²‐space based theory of solutions of partial differential equations for the case of the linear output regulation problem. In the nonlinear case, a sequence of linear equations is defined so that their solutions converge to the solution of the nonlinear problem. This is proved using the Banach Contraction Theorem. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

一种非线性扩散线形纹理图像增强的方法

对于运用非线性扩散滤波器对线形纹理图像的增强,提出了一种根据图像纹线方向对传统扩散滤波器的扩散系数的修正,使修正后的“半各向异性”扩散滤波器适合于线形纹理图像的滤波。文中直接给出了滤波器的离散形式,使滤波过程易于实现,并缩短了滤波时间。     

Zakai equation of nonlinear filtering with unbounded coefficients. The case of dependent noises

In this paper, we prove that the unnormalized filter associated with nonlinear filtering problems with dependent noises and a one-dimensional observations process the coefficients of which are unbounded solves a parabolic stochastic partial differential equation, the Zakai equation. The robust form of the Zakai equation is also computed.     

基于小波变换的非线性尺度扩散方法

小波变换用于图像处理之所以具有特别的优势，因为它能够聚焦到图像的细微变化。与小波变换不同，用偏微分方程来进行图像处理则需要对图像进行反复的迭代，直到得到一个稳定解，其整个过程是对图像进行整体处理，而且在去噪的同时，可以很好地保持边缘形状不变。若将两者有机结合，则不仅会克服各自的缺点，而且会得到更好的效果。为了在进行图像处理时，既能很好地抑制噪声，又能尽可能多地保持图像细节，提出了一种小波变换与非线性尺度扩散相结合的图像处理方法。该方法是利用小波变换的时频局部性和非线性尺度扩散的边缘增强特性来对图像进行处理，实验结果表明，其不仅能很好地抑制噪声，而且可保留尽可能多的图像细节，可见该方法是有效的。