基于正交完备U-系统的图形分类与识别方法

为了探索有效的图形分类与识别的新方法,引进一类正交完备的分段k次多项式系统(简称U-系统).U-系统是一类属于L²[0,1]的正交完备分段k次多项式系统.该系统下的U级数展开式具有良好的平方逼近及一致逼近性质.基于U-系统理论,提出了U描述子的概念,给出了U描述子的性质并在理论上予以证明.为了更好地对图形分类与识别,对U描述子进行了归一化,同时在理论上证明了归一化U描述子具有旋转、平移、尺度大小等不变的性质.实验表明,归一化的U描述子能够高效、准确地对图形进行分类与识别,与Fourier描述子相比,具有更好的识 别率.     

U-系统与 V-系统的理论及应用综述

传统的 Fourier 级数在逼近间断信号时因 Gibbs 现象的干扰,会产生比较大的误差。针对此问 题,国内学者齐东旭教授带领的课题组提出了非连续正交函数系的研究课题,其中 U-系统和 V-系统是两类典 型的非连续完备正交函数系。从数学理论上来说,U-系统和 V-系统分别是对著名的 Walsh 函数和 Haar 函数由 分段常数向分段 k 次多项式进行推广的结果,其最重要的特点是函数系中既有光滑函数又有各个层次的间断函 数。因此,U,V-系统可以处理连续和间断并存的信息,在一定程度上弥补了 Fourier 分析和连续小波的缺憾。 本文从理论与应用 2 个方面对 U,V-系统进行了综述。在理论方面,首先介绍了单变量 U-系统与 V-系统各自 的构造方法,其次介绍三角域上 U,V-系统的构造方法,最后介绍 U,V-系统的主要性质。在应用方面,介绍 了若干具有代表性的应用案例。     

计算机辅助几何设计中的保形插值理论及应用

本文主要研究计算机辅助几何设计中的分段多项式保形插值理论与算法 ,分段参数多项式保形插值方法及GHI问题 ,参数曲线弧长参数化的混合数值算法与近似方法 ,与给定任意切线多边形相切的保形逼近样条曲线 ,Bézier曲线和 NURBS曲线的等距线生成以及一般参数曲线等距线的保形逼近曲线。本文首先系统地研究了分段多项式的保形插值 ,建立了分段多项式的保形插值理论框架 ,导出了分段三次Hermite插值保形的充要条件 ,构造了一个 C1 连续的分段三次多项式保形插值算法 ,导出了 2 k+1次或 2 k次多项式保凸的充要条件 ,给出了插入内结点的区域…     

光学图象几何畸变的快速校正算法

实际光学镜头所成的图象难免会有几何变形，多项式坐标变换法是进行几何修正的有效方法，但是当次数较高的时候，运算量太大，难以应用到实时图象处理系统，为此在分析多项式坐标变换算法的基础上，提出了一次多项式非均匀分片逼近算法。该算法首先将图象非均匀划分成矩形区域，在每个矩形区域内部用一个一次多项式逼近高次多项式。基于对图象畸变的主要因素即径向畸变的分析，该算法的图象划分规则能在保证逼近精度的前提下占用最少的保存模型参数的空间。该算法大大降低了运算量，将运算时间减少了近2／3，同时能很好地保证逼近精度，空间代价也限制在很小的范围内，试验结果表明该算法是图象几何修正的有效方法，具有良好的工程应用价值。     

一类新的正交样条函数——Franklin函数的推广及其应用

为了探索样条曲线曲面的正交表达及其频谱性质,提出了一类新的k次正交样条函数--Franklin函数的推广,简称为k次GF系统.Haar函数及Franklin正交函数恰好分别是GF系统当k=0及k=1时的特殊情形.基于GF系统,给出了用以计算样条曲线曲面频谱的信息转换算法,该算法具有直观、简便、快速的特点.构建的数据处理平台可用于样条曲线曲面的分析与综合;实验表明有限项GF系统能够实现一类几何造型的精确重构,而有限项傅立叶正交甬数则不能精确重构该原图.     

基于三点形状可调的二次三角Bzier曲线

给出了二次三角多项式形式的Bzier曲线,基函数由一组带形状参数的二次三角多项式组成。由三个控制顶点生成的曲线具有与二次Bzier曲线类似的性质,但具有比二次Bzier曲线更好的逼近性。形状参数有明确几何意义:参数越大,曲线越逼近控制多边形。曲线可精确表示椭圆弧,还给出了两段三角多项式曲线的G2和C3连续的拼接条件。     

二维旋转不变U变换及其应用

U-系统是一类L2[0,1]上的正交分段多项式函数系,为了将其推广到二维情形,传统的L2[0,1]2上张量积形式的U变换并不具有旋转不变性.本文提出了一类二维旋转不变U变换（Rotation-invariant U transform,RIUT）. RIUT将U-系统函数与调和函数相结合,使得图像的旋转转化为相位的平移而模保持不变.与经典的正交旋转不变矩（如Zernike矩）相比,RIUT具有诸多特别的性质,从而在图像特征提取中具有良好的潜力.本文将RIUT应用到二值图像检索中的实验结果表明,RIUT具有更高的检索精度.     

一类正交函数系的离散表示及快速变换

V系统是L2[0,1]上一类新的完备正交函数系,它由分段多项式组成,具有多分辨分析特性和全局/局部性,在几何模型的正交表达方面具有明显的优势,但其快速算法难以得到。利用Haar函数和Legendre多项式构造了一类由分段次多项式组成的函数系(文中称为W系),在该函数系上作函数逼近的效果等同于在V系统上的效果,并进一步讨论了一次离散W变换的快速算法,从而部分克服了直接对V系统设计快速算法的困难。     

带有形状参数的Bézier三角曲面片

给出了含有参数的二元(n+1)次多项式基函数,是三角域上二元n次Bernstein基函数的扩展;分析了该组基的性质并定义了带有形状参数的(n+1)次Bézier三角曲面片.该曲面不仅具有n次Bézier三角曲面片的特性,而且具有形状的可调性;其参数有明确的几何意义,参数越大,曲面越逼近控制网格;当参数为0时,曲面可退化为n次Bézier三角曲面片.     

广义Wang-Ball曲线

给出了n次带形状参数λ的Wang-Ball曲线,它具有n次Wang-Ball曲线的类似性质.形状参数λ具有明显的几何意义:λ越大,曲线越逼近控制多边形.当λ=0时,曲线退化为一条线段;当λ=2时,曲线退化为Wang-Ball曲线.给出了曲线的递归求值,升阶和降阶逼近算法,用Bézier形式表达的系数公式及两段曲线G1,C1连续拼接的条件.     

参数曲线近似弧长参数化的插值方法

本文提出了参数曲线近似弧长参数化的一种插值方法。参数曲线的弧长函数的单调增的，近似弧长参数化可以转化为弧长函数的保单调分段有理线性插值。用这种插值得到的近似弧长参数化曲线插值原曲线上的一组点，最后，两个实例表明了近似弧长参数化曲线能很好地逼近原曲线，且没有所不希望的波动。     

Degenerate parametric curves

This paper discusses two degenerate cases of polynomial parametric curves for which the degrees of the defining polynomials can be reduced without altering the curve. The first case is the improperly parametrized curve for which each point on the curve corresponds to several parameter values. The second case, which can only occur for rational polynomial parametric curves, exists when the defining polynomials all have a common factor.

This paper describes how to detect and correct each type of degeneracy. Examples are given which demonstrate that seemingly innocuous Bézier curves may suffer from either of these degeneracies.     

Isometric Piecewise Polynomial Curves

The main preoccupations of research in computer-aided geometric design have been on shape-specification techniques for polynomial curves and surfaces, and on the continuity between segments or patches. When modelling with such techniques, curves and surfaces can be compressed or expanded arbitrarily. There has been relatively little work on interacting with direct spatial properties of curves and surfaces, such as their arc length or surface area. As a first step, we derive families of parametric piecewise polynomial curves that satisfy various positional and tangential constraints together with arc-length constraints. We call these curves isometric curves. A space curve is defined as a sequence of polynomial curve segments, each of which is defined by the familiar Hermite or Bézier constraints for cubic polynomials; as well, each segment is constrained to have a specified arc length. We demonstrate that this class of curves is attractive and stable. We also describe the numerical techniques used that are sufficient for achieving real time interaction with these curves on low-end workstations.     

Surface algorithms using bounds on derivatives

This paper generalizes three very important algorithms for surfaces which previously only worked well with polynomials. The algorithms are calculation of piecewise linear approximations, min-max boxes, and surface/surface intersections. The class of surfaces that can be handled are all parametric C² surfaces. All the algorithms are related by theorems from approximation theory which give information about the maximum deviation an approximation to a surface can have if bounds on partial derivatives are known. We generalize these theorems to work with parametric geometry, and we also show how to obtain the necessary bounds.     

How many different rational parametric cubic curves are there? 3.The catalog

The author is interested in rational parametric cubic curves, particularly in finding out how many essentially different shapes the given equation can make. Types of homogeneous cubic polynomials are considered     

A piecewise parabolic C approximation technique for curved boundaries

The parabola, in parametric form, is discussed as a suitable approximation to curved boundaries. The four degrees of freedom are specified locally by Hermite interpolation. This piecewise curve approximation can then be thought of as a locally defined C′ spline. A geometric norm, i.e. one which is invariant under the axes change is constructed and the approximation technique is analysed in this norm. A selection of graphical studies of piecewise approximations to a variety of curves, using this method, is given.     

Theory of contact for geometric continuity of parametric curves

This paper develops a theory of contact for piecewise parametric curves based on the differential geometry of evolutes, polar curves, and binormal indicatrices. This theory is completely geometric, independent of parametrization and generalizes to any order. Two sets of dimensionless, characteristic numbers describing the local geometry of a curve up tonth order are defined. These characteristic numbers can be used to describe conditions for higher order contacts in an algebraic fashion. The same characteristic numbers can also be used to interpret contact conditions of up tonth order in terms of the geometry of higher evolutes and binormal indicatrices. The resulting geometric contact conditions are used to design piecewise parametric curves for Computer Aided Geometric Design (CAGD) applications.     

Blending curves for landing problems by numerical differential equations II. Numerical methods

A landing curve of airplane is a blending curve that smoothly joins the two given boundary points, which is described by the parametric functions x(s), y(s), and z(s) governed by a system of ordinary differential equations (ODEs) with certain boundary conditions. In Part I, Mathematical Modelling [1], existence and uniqueness of the ODE system are explored to produce the optimal landing curves in minimum energy. In this paper, numerical techniques are provided by the finite element method (FEM) using piecewise cubic Hermite polynomials, to give the optimal solutions. An important issue is how to deal with infinite solutions occurring in the landing problems reported in [1]. Moreover, error analysis is made, and numerical examples are carried to verify the theoretical results made. This paper displays again the effectiveness and flexibility of the ODE approach to complicated blending curves. Besides, the numerical techniques in this paper can be applied directly to other landing and trajectory problems given in [1], as well as other kinds of blending curves and surfaces of airplane, ships, grand building, and astronautic shuttle-station.