The Convergence of Least-Squares Progressive Iterative Approximation for Singular Least-Squares Fitting System

Data fitting is an extensively employed modeling tool in geometric design. With the advent of the big data era, the data sets to be fitted are made larger and larger, leading to more and more least-squares fitting systems with singular coefficient matrices. LSPIA (least-squares progressive iterative approximation) is an efficient iterative method for the least-squares fitting. However, the convergence of LSPIA for the singular least-squares fitting systems remains as an open problem. In this paper, the authors showed that LSPIA for the singular least-squares fitting systems is convergent. Moreover, in a special case, LSPIA converges to the Moore-Penrose (M-P) pseudo-inverse solution to the least- squares fitting result of the data set. This property makes LSPIA, an iterative method with clear geometric meanings, robust in geometric modeling applications. In addition, the authors discussed some implementation detail of LSPIA, and presented an example to validate the convergence of LSPIA for the singular least-squares fitting systems.

LEAST-SQUARES METHOD-BASED FEATURE FITTING AND EXTRACTION IN REVERSE ENGINEERING

The main purpose of reverse engineering is to convert discrete data pointsinto piecewise smooth, continuous surface models. Before carrying out model reconstruction it issignificant to extract geometric features because the quality of modeling greatly depends on therepresentation of features. Some fitting techniques of natural quadric surfaces with least-squaresmethod are described. And these techniques can be directly used to extract quadric surfaces featuresduring the process of segmentation for point cloud.

AN EXPONENTIALLY FITTED DIFFERENCE SCHEME FOR THE HYPERBOLIC-HYPERBOLIC SINGULARLY PERTURBED INITIAL-BOUNDARY VALUE PROBLEM

In this paper we discuss, an initial-boundary value problem of hyperbolic type with first derivative with respect to x. The asymptotic solution is constructed and its uniform validity is proved under weader compatibility conditions. Then we develop an exponentially fitted difference scheme and establish discrete energy inequality. Finally, we prove that the solution of difference problem uniformly converges to the solution of the original problem.

An Asymptotic-Fitted Method for Solving Singularly Perturbed Delay Differential Equations

In this paper, we presented an asymptotic fitted approach to solve singularly perturbed delay differential equations of second order with left and right boundary. In this approach, the singularly perturbed delay differential equations is modified by approximating the term containing negative shift using Taylor series expansion. After approximating the coefficient of the second derivative of the new equation, we introduced a fitting parameter and determined its value using the theory of singular Perturbation;O’Malley [1]. The three term recurrence relation obtained is solved using Thomas algorithm. The applicability of the method is tested by considering five linear problems (two problems on left layer and one problem on right layer) and two nonlinear problems.

基于RMS子线区间拟合法的轴承剩余寿命预测

轴承剩余使用寿命(Residual life,RL)预测模型的预测精度在很大程度上取决于衰退特征的趋势一致性。由于轴承个体差异性,导致同类型不同轴承的相同特征衰退趋势表现不同,从而导致训练集轴承建立的RL预测模型与测试集轴承不匹配。针对上述问题,本文提出一种基于振动信号均方根(RMS)趋势一致性的子线区间拟合法轴承寿命预测方法。依据μ±3σ原则提出一种奇异值替换方法,有效增强RMS衰退曲线的趋势一致性,并基于RMS衰退曲线高度的趋势一致性提出子线区间拟合法,利用XJTU-SY数据集对提出方法进行测试与验证,并与常见的寿命预测方法反向传播神经网络(Back Propagation Neural Network,BPNN)与长短期记忆神经网络(Long Short-term Memory Network,LSTM)预测结果进行对比,可有效提升轴承RL预测精度。

基于再模糊和奇异值分解的局部模糊区域检测

针对局部模糊区域检测与分割,提出一种基于再模糊和奇异值分解的局部模糊检测方法。首先,对待检图像进行高斯模糊和图像连续分块,对分块图像进行奇异值分解;其次,对奇异值向量进行指数函数拟合,得到待测和再模糊图像的奇异值曲线,分别计算拟合曲线的积分并作差,将差值差异性作为描述模糊的特征值;最后,利用Canny边缘检测算法完成图像分割,并利用最大、最小滤波器和相关形态学处理对边缘内部区域进行填充。实验结果表明:该方法可以有效地完成局部模糊区域的检测和分割,同时对纹理平坦区域的识别同样有效。

A unified approach for least-squares surface fitting

This paper presents a novel approach for least-squares fitting of complex surface to measured 3D coordinate points by adjusting its location and/or shape. For a point expressed in the machine reference frame and a deformable smooth surface represented in its own model frame, a signed point-to-surface distance function is defined, and its increment with respect to the differential motion and differential deformation of the surface is derived. On this basis, localization, surface reconstruction and geometric variation characterization are formulated as a unified nonlinear least-squares problem defined on the product space SE(3)×Km. By using Levenberg-Marquardt method, a sequential approximation surface fitting algorithm is developed. It has the advantages of implementational simplicity, computational efficiency and robustness. Applications confirm the validity of the proposed approach.

NUMERICAL SOLUTION OF THE SINGULARLY PERTURBED PROBLEM WITH NONLOCAL BOUNDARY CONDITION

Singularly perturbed boundary value problem with nonlocal conditions is examined. The appopriate solution exhibits boundary layer behavior for small positive values of the perturbative parameter. An exponentially fitted finite difference scheme on a non-equidistant mesh is constructed for solving this problem. The uniform convergence analysis in small parameter is given. Numerical example is provided, too.

Singularity of Some Software Reliability Models and Parameter Estimation Method

According to the principle, “The failure data is the basis of software reliability analysis”, we built a software reliability expert system (SRES) by adopting the artificial intelligence technology. By reasoning out the conclusion from the fitting results of failure data of a software project, the SRES can recommend users “the most suitable model” as a software reliability measurement model. We believe that the SRES can overcome the inconsistency in applications of software reliability models well. We report investigation results of singularity and parameter estimation methods of experimental models in SRES.

DMPFIT: A Tool for Atomic-Scale Metrology via Nonlinear Least-Squares Fitting of Peaks in Atomic-Resolution TEM Images

Despite the wide availability and usage of Gatan’s DigitalMicrograph software in the electron microscopy community for image recording and analysis, nonlinear least-squares fitting in DigitalMicrograph is less straightforward. This work presents a ready-to-use tool, the DMPFIT software package, written in DigitalMicrograph script and C++ language, for nonlinear least-squares fitting of the intensity distribution of atomic columns in atomic-resolution transmission electron microscopy (TEM) images with a general two-dimensional (2D) Gaussian model. Applications of the DMPFIT software are demonstrated both in atomic-resolution conventional coherent TEM (CTEM) images recorded by the negative spherical aberration imaging technique and in high angle annular dark field (HAADF) scanning TEM (STEM) images. The implemented peak-finding algorithm based on the periodicity of 2D lattices enables reliable and convenient atomic-scale metrology as well as intuitive presentation of the resolved atomic structures.

A Numerical Method for Nonlinear Singularly Perturbed Multi-Point Boundary Value Problem

We consider a uniform finite difference method for nonlinear singularly perturbed multi-point boundary value problem on Shishkin mesh. The problem is discretized using integral identities, interpolating quadrature rules, exponential basis functions and remainder terms in integral form. We show that this method is the first order convergent in the discrete maximum norm for original problem (independent of the perturbation parameter ε). To illustrate the theoretical results, we solve test problem and we also give the error distributions in the solution in Table 1 and Figures 1-3.

基于网络导纳矩阵奇异性的配电网谐波谐振分析

为了降低谐振分析的复杂度,减少计算量从而提高计算效率,提出一种配电网谐波谐振改进模态分析法。该方法根据网络导纳矩阵行列式特性及其随频率的变化规律,引入曲线拟合方法来确定配电网谐振频率,并利用模态分析法确定谐振信息。仿真分析及实测数据验证结果表明:应用所提方法对配电网进行谐振分析可显著提高配电网谐波谐振分析的计算速度,降低谐振分析难度。

基于经验小波变换和奇异值分解的冲击波降噪方法

针对爆炸场噪声源多、噪声特点各不相同的问题,提出基于经验小波变换(EWT)和奇异值分解(SVD)的冲击波降噪方法。该方法首先对冲击波信号进行指数拟合,用原始信号减去拟合后对应点的值,对上述信号使用EWT算法进行分解,随后对分解得到的分量采用SVD算法进行降噪重构,实现噪声的滤除。仿真及试验验证表明,基于EWT-SVD联合算法可有效抑制相位偏移,更好地还原出冲击波信号的超压峰值、上升沿时间,可用于冲击波信号的降噪。

空间直线拟合新方法

首先证明了正交距离最小二乘准则下拟合的空间直线必经过数据中心点,再以此为基础提出了一种空间直线拟合新算法。该方法不用迭代,计算简便。证明了正交距离最小二乘准则和整体最小二乘准则在拟合空间直线上的等价性;为增强算法的抗粗差能力,增加了RANSAC策略识别粗差;结合案例,分别采用BFGS、SQP优化算法以及整体最小二乘法对该方法进行验证,从而证明其准确性和有效性。

奇异值分解压制随机噪声的方法及应用

针对随机噪声压制问题,通过对多项式确定的同相轴形状重新选取数据矩阵,再对新选取的数据矩阵进行SVD滤波,从而压制随机噪声。通过对矩阵奇异值分解的研究,揭示了奇异值分解的原理和SVD滤波原理,将一个多道地震记录看作一个图,它可分解为由一系列正交的子图,通过对子图适当选择并结合其它手段,从而达到提高信噪比的目的,为奇异值分解在地球物理中的进一步应用提供了理论依据。实际数据应用表明,该方法去噪手段灵活、保真性好、分辨率高,对一些突然的脉冲干扰、侧反射、以及其他的反向干扰有明显效果。

三轴磁航向传感器误差补偿

用椭球拟合法进行磁航向传感器误差补偿时,约束矩阵奇异导致算法不稳定.对此提出改进的最小二乘椭球拟合算法,讨论了磁航向传感器的误差来源,建立了误差数学模型.结合对约束矩阵的奇异性分析,提出矩阵分块分解方法解决奇异性问题,并用最小二乘法求解椭球系数.该算法消除了传统算法的不稳定性,降低了计算量.计算机仿真和转台实验表明它能有效补偿磁航向传感器的测量误差,补偿后航向最大误差小于0.42°.

基于拟合误差最小化原则的奇异值分解降噪有效秩阶次确定方法

为了最大限度地提高旋转机械设备故障振动信号的信噪比,研究了奇异值分解降噪的原理,提出了一种新的奇异值分解降噪有效秩阶次的确定方法。首先,对振动信号进行相空间重构,对吸引子轨迹矩阵进行奇异值分解;然后,按不同的阶数,将奇异值分成信号组和噪声组,对每次分组的结果,以阶数为自变量、以奇异值为因变量,拟合成信号特征奇异值曲线和噪声特征奇异值曲线,并求拟合误差;最后,将拟合误差最小值对应的奇异值阶数确定为有效秩阶次,并进行奇异值分解降噪。通过数值仿真和实际齿轮故障数据分析,表明该方法可以有效地提高信号的信噪比,为后期的故障特征提取创造有利条件。

椭圆拟合方法在磁罗盘罗差校准中的应用

磁罗盘在使用过程中受周围铁磁物质的影响致使航向精度降低,因此对磁罗盘进行校准是十分必要的。针对传统的磁罗盘罗差校准算法存在系数矩阵易奇异导致算法不稳定的问题,提出了改进的最小二乘椭圆拟合算法。分析了外界磁干扰对磁罗盘输出的影响,建立了罗差的椭圆模型,通过对系数矩阵的奇异性进行分析,采用最小二乘方法实现了对椭圆模型误差系数的辨识。该算法克服了原有算法的不稳定性,对校准过程中的突变干扰有很强的抑制能力,同时将六维特征矢量的求解转化为三维特征矢量求解,在保证校准精度的情况下降低了计算量。多位置转台实验验证了该方法的有效性和实用性,校准后的磁罗盘精度由0.8°提高到0.4°,满足了导航系统的需求。

测速光幕靶用数字信号处理算法

针对光幕靶在使用中易受到蚊虫、冲击波和光电器件随机噪声干扰因而可靠性差的技术难题,提出一种测速光幕靶用数字信号处理算法。利用数据采集仪采集光幕靶输出的弹丸过靶信号,通过莱达因准则剔除偶发性干扰数据。通过阈值比较法判断出弹丸信号的时间范围后,利用4次高斯公式对一定时间范围内的采样数据进行曲线拟合,并准确计算出半峰值触发时刻点,实现低采样率下的高精度测量。研究的信号处理算法利用计算机进行仿真验证,其原理正确,计算效率高。配合XGK-2002型光幕靶进行实弹射击试验,测试数据表明:该算法可准确计算出各类弹丸信号的过幕时刻,极大地提高了光幕靶的稳定性和适用范围,测量相对误差不大于0.1%。研究的算法同样适用于一类动态特征信号的提取与计算。

弱行波信号的奇异点检测方法

当电压过零附近或者高阻接地故障时行波信号较弱,同时受行波上升沿畸变以及噪声干扰等因素的影响,传统小波检测法的检测结果可能会出现较大误差。为可靠检测到故障行波信号的奇异点,文中提出了一种新的检测方法,即根据推导得出的故障行波波头表达式,并结合高速数据采集装置的实测数据,获取故障行波拟合曲线,依此确定故障行波信号奇异点。通过对输电线路中波头上升沿的特征分析以及母线处分布电容对波头上升沿的影响分析,得出了表征行波波头上升沿的表达式;为增强算法可行性,利用麦克劳林(Maclaurin)展开式推导出了行波波头的间接表达形式。利用该表达式拟合出的行波曲线可确定行波信号的奇异点。仿真结果表明,所提出的检测方法可有效地确定弱行波信号的奇异点。