图像去噪的混合正则化LSQR算法

本文研究了图像去模糊去噪问题.利用正则化技术结合Krylov子空间方法,提出了混合正则化LSQR算法.实验结果表明该算法有效改善了问题的不适定性,获得了逼真度较高的复原图像.     

基于分数阶微积分正则化的图像处理

全变分正则化方法已被广泛地应用于图像处理,利用此方法可以较好地去除噪声,并保持图像的边缘特征,但得到的优化解会产生"阶梯"效应.为了克服这一缺点,本文通过分数阶微积分正则化方法,建立了一个新的图像处理模型.为了克服此模型中非光滑项对求解带来的困难,本文研究了基于不动点方程的迫近梯度算法.最后,本文利用提出的模型与算法进行了图像去噪、图像去模糊与图像超分辨率实验,实验结果表明分数阶微积分正则化方法能较好的保留图像纹理等细节信息.     

求解l1极小化问题的Bregman迭代算法

在Tikhonov正则化方法的基础上将其转化为一类l1极小化问题进行求解,并基于Bregman迭代正则化构建了Bregman迭代算法,实现了l1极小化问题的快速求解.数值实验结果表明,Bregman迭代算法在快速求解算子方程的同时,有着比最小二乘法和Tikhonov正则化方法更高的求解精度.     

变分正则化图像复原模型与算法综述

图像复原就是对图像退化模型进行处理以恢复图像的原始信息.由于引起图像退化的因素和性质各不相同,所以图像复原是一个复杂的数学过程,本质上是求解不适定的反问题.本文综述了变分正则化图像复原模型与算法.首先,系统阐述了图像复原中的去噪(分解)模型、去模糊模型、修复模型.其次,构建了一个统一的变分正则化图像复原模型并总结了各种典型的数值求解方法.最后,指出在今后进一步研究中值得关注的8个问题.     

求解岭回归问题的加速增广投影算法

正1引言考虑求解岭回归或者Tikhonov正则化最小二乘回归问题■这里X是一个m×n的复矩阵,β是一个n维未知向量,y是一个m维的复向量,λ是正则化参数,‖·‖2表示向量的欧拉范数.岭回归问题对病态数据的拟合效果要强于最小二乘法.目前,岭回归问题已广泛应用于数据分析、机器学习、电网等领域.近年来,一系列随机算法被用来求解大规模线性系统.Strohmer和Vershynin [1]提出     

广义全变差正则化图像去噪模型研究

本文研究了全变差正则化模型在图像去噪过程中易产生阶梯效应的问题,依据图像的局部结构特利用联合高斯滤波器和边缘检测算子的方法,构建了广义全变差正则化图像去噪模型,获得了在消除噪声的同时能够保留图像边缘细节和纹理信息的结果.实验结果表明,广义全变差正则化模型在平滑噪声的同时能够保留图像的边缘轮廓等细节信息,得到的复原图像在峰值信噪比、平均结构相似度和主观视觉效果方面均有所提高.     

牛顿拉夫逊算法在ERT反演问题中的应用研究

本文研究了电阻率反演成像(ERT)中的牛顿拉夫逊基础算法及改进问题.利用最小二乘法和Tikhonov正则化等方法将反演算法予以优化,获得了与实验样本结构吻合的碳纤维复合层的电阻率分布图像,推广了牛顿拉夫逊算法的数理反演模型.     

基于对偶的不精确交替方向乘子法求解核范数正则化最小二乘问题

数据时代的所有事物都可以用数据描述记录.在数据分析中,对部分缺失数据补充,即矩阵补全问题.此类问题已有一定的研究,如通过求解核范数正则化最小二乘问题来达到所需效果.该文从对偶问题出发,使用交替方向乘子法(ADMM)来求解.在一定假设条件下,讨论了不精确对偶交替方向乘子法(dADMM)的全局收敛性.数值试验中,通过与原问题交替方向乘子法(pADMM)进行比较,验证了该算法的优越性.     

非精确信赖域类型算法及收敛性分析

在求解大规模数据的优化问题时,由于数据规模和维数较大,传统的算法效率较低.本文通过采用非精确梯度和非精确Hessian矩阵来降低计算成本,提出了非精确信赖域算法和非精确自适应三次正则化算法.在一定条件下,证明了算法有限步停止,并估计了算法迭代的复杂度.特别地,我们分析了采用随机抽样时算法在给定概率下的复杂度.最后,通过二分类问题的数值求解,比较了本文提出的随机信赖域算法,随机自适应三次正则化算法和已有算法收敛效率.数值结果表明在相同精度下,本文提出的算法效率更高,并且随机自适应三次正则化算法的效率优于随机信赖域算法.     

稀疏优化模型及其正则化方法

稀疏优化模型是目前最优化领域中非常热门的研究前沿课题,在压缩感知、图像处理、机器学习和统计建模等领域都获得了成功的应用.本文以光谱分析技术、数字信号处理和推荐系统等多个应用问题为例,阐述稀疏优化模型的建模过程与核心思想.稀疏优化模型属于组合优化模型,非常难以求解(NP-难).正则化方法是稀疏优化模型的一类常用的求解方法.我们将介绍正则化方法的原理与几类常见的正则化模型,并阐述正则化模型的稳定性理论与多种先进算法.数值实验表明,这些算法都具有快速、高效、稳健等显著优点.稀疏正则化模型将在大数据时代中发挥更显著的计算优势与应用价值.     

Linear least squares problems with data over incomplete grids

The paper addresses bivariate surface fitting problems, where data points lie on the vertices of a rectangular grid. Efficient and stable algorithms can be found in the literature to solve such problems. If data values are missing at some grid points, there exists a computational method for finding a least squares spline by fixing appropriate values for the missing data. We extended this technique to arbitrary least squares problems as well as to linear least squares problems with linear equality constraints. Numerical examples are given to show the effectiveness of the technique presented. AMS subject classification (2000)  65D05, 65D07, 65D10, 65F05, 65F20     

数据拟合函数的最小二乘积分法

数据拟合的方法很多,每种方法各有特点.本文探讨了积分准则下的数据拟合函数的方法,称为最小二乘积分法,并给出了两个常用拟合函数具体形式.     

数据拟合函数的加权最小二乘积分法

数据拟合的方法很多,每种方法各有特点.探讨了积分准则下的数据加权拟合函数的方法,称为加权最小二乘积分法,并给出了三个常用拟合函数具体形式.     

线性数据拟合方法的误差分析及其改进应用

利用最小二乘法进行线性数据拟合在一定条件下存在着误差较大的缺陷,为使线性数据拟合方法在科学实验和工程实践中能够更加准确地求解量与量之间的关系表达式,本文通过对常用线性数据拟合方法———最小二乘法进行了误差分析,并在此基础上提出了最小距离平方和法以对最小二乘法作改进处理.最后,通过举例分析对两种线性数据拟合方法的优劣加以讨论并分别给出其较为合理的应用控制条件.     

A dynamic meshless method for the least squares problem with some noisy subdomains

The interpolation method by radial basis functions is used widely for solving scattered data approximation. However, sometimes it makes more sense to approximate the solution by least squares fit. This is especially true when the data are contaminated with noise. A meshfree method namely, meshless dynamic weighted least squares (MDWLS) method, is presented in this paper to solve least squares problems with noise. The MDWLS method by Gaussian radial basis function is proposed to fit scattered data with some noisy areas in the problem’s domain. Existence and uniqueness of a solution is proved. This method has one parameter which can adjusts the accuracy according to the size of noises. Another advantage of the developed method is that it can be applied to problems with nonregular geometrical domains. The new approach is applied for some problems in two dimensions and the obtained results confirm the accuracy and efficiency of the proposed method. The numerical experiments illustrate that our MDWLS method has better performance than the traditional least squares method in case of noisy data.     

The variable projection algorithm in time-resolved spectroscopy,microscopy and mass spectrometry applications

Nonlinear least squares optimization problems in which the parameters can be partitioned into two sets such that optimal estimates of parameters in one set are easy to solve for given fixed values of the parameters in the other set are common in practice. Particularly ubiquitous are data fitting problems in which the model function is a linear combination of nonlinear functions, which may be addressed with the variable projection algorithm due to Golub and Pereyra. In this paper we review variable projection, with special emphasis on its application to matrix data. The generalization of the algorithm to separable problems in which the linear coefficients of the nonlinear functions are subject to constraints is also discussed. Variable projection has been instrumental for model-based data analysis in multi-way spectroscopy, time-resolved microscopy and gas or liquid chromatography mass spectrometry, and we give an overview of applications in these domains, illustrated by brief case studies.     

Compression and denoising using <Emphasis Type="Italic">l</Emphasis><Subscript>0</Subscript>-norm

In this paper, we deal with l ₀-norm data fitting and total variation regularization for image compression and denoising. The l ₀-norm data fitting is used for measuring the number of non-zero wavelet coefficients to be employed to represent an image. The regularization term given by the total variation is to recover image edges. Due to intensive numerical computation of using l ₀-norm, it is usually approximated by other functions such as the l ₁-norm in many image processing applications. The main goal of this paper is to develop a fast and effective algorithm to solve the l ₀-norm data fitting and total variation minimization problem. Our idea is to apply an alternating minimization technique to solve this problem, and employ a graph-cuts algorithm to solve the subproblem related to the total variation minimization. Numerical examples in image compression and denoising are given to demonstrate the effectiveness of the proposed algorithm.     

Application of gradient descent method to the sedimentary grain-size distribution fitting

Existence of a least squares solution for a sum of several weighted normal functions is proved. The gradient descent (GD) method is used to fit the measured data (i.e. the laser grain-size distribution of the sediments) with a sum of three weighted lognormal functions. The numerical results indicate that the GD method is not only easy to operate but also could effectively optimize the parameters of the fitting function with the error decreasing steadily. Meanwhile the overall fitting results are satisfactory. As a new way of data fitting, the GD method could also be used to solve other optimization problems.     

A class of quasi-variational inequalities for adaptive image denoising and decomposition

We introduce a class of adaptive non-smooth convex variational problems for image denoising in terms of a common data fitting term and a support functional as regularizer. Adaptivity is modeled by a set-valued mapping with closed, compact and convex values, that defines and steers the regularizer depending on the variational solution. This extension gives rise to a class of quasi-variational inequalities. We provide sufficient conditions for the existence of fixed points as solutions, and an algorithm based on solving a sequence of variational problems. Denoising experiments with spatial and spatio-temporal image data and an adaptive total variation regularizer illustrate our approach.     

HERMITE SCATTERED DATA FITTING BY THE PENALIZED LEAST SQUARES METHOD

Given a set of scattered data with derivative values. If the data is noisy or there is an extremely large number of data, we use an extension of the penalized least squares method of von Golitschek and Schumaker [Serdica, 18 （2002）, pp.1001-1020] to fit the data. We show that the extension of the penalized least squares method produces a unique spline to fit the data. Also we give the error bound for the extension method. Some numerical examples are presented to demonstrate the effectiveness of the proposed method.