Optimal twist vectors as a tool for interpolating a network of curves with a minimum energy surface

This paper provides a new answer to the old problem of specifying the mixed partial derivatives (MPDs) or ‘twist vectors’ at the grid points for an interpolating surface over a rectangular network of curves. An algorithm is presented for finding the MPDs that minimizes a generalized energy integral over the entire surface. The integrand may be any quadratic form in the second partial derivatives of the surface. This results in a surface design technique for interpolating over a network of curves by automatically selecting the optimal twist vectors at the grid points.     

共形几何代数与运动和形状的刻画

共形几何代数在基于运动和形状刻画的视觉和图形学若干问题中的应用,反映了它能够提供统一和有效的表示和算法,这些应用主要集中在采纳几何体的Grassmann分级表示以及刚体运动的旋量和扭量表示.着重介绍了Grassmann分级表示如何被应用于单眼视觉问题并带来解决方法的简化;通过对刚体运动不同表示的分析,介绍旋量和扭量表示如何克服刚体运动蹬矩阵表示中参数空间具有过多非线性约束的缺点,从而为姿态估计、形状逼近和曲线拼接等问题的解决提供简化方案.     

Behavior of the beta-splines with values of the parameters beta2 negative

In this paper we study the behavior of the beta-spline functions in the case the parameter β₂(i) is negative. We prove that a negative value exists so that if , the beta-spline functionsN_i(u) are positive. Moreover, if the control vertices are such that x₀ x_m−1, we have proved that the design curve keeps the properties already proved in the case β₂(i) 0.     

On the convexity of parametric Bézier triangular surfaces

In this paper, we discuss the convexity of parametric Bézier triangular patches, give some sufficient conditions of it to be convex, which only depend on the edge vectors and twist vectors. All the conditions we obtained can be served as the extension of the convexity preserving conditions of functional Bézier triangular patches.     

Ensuring compatibility of G-continuous surface patches around a nodepoint

The problem of ensuring compatibility of mixed partial derivative vectors of surface patches joining G²-continuously around a common nodepoint is essential in modelling G²-continuous n-sided surfaces. Although the compatibility constraints can be removed by using C² Gregory patches, these patches have singularities at their corner points. This paper presents conditions for ensuring the compatibility of the mixed partial derivative vectors of surface patches joining G²-continuously around a common nodepoint. After investigating the solvability of these compatibility conditions, a new solution method exploiting G³-continuity of surface patches at a common nodepoint is given. Example surfaces based on this solution method are also provided.     

带参数Coons插值曲面的图像插值方法

提出一种根据图像边缘特征调整插值曲面形状的图像插值方法．首先把数字图像构造为分片双三次Coons曲面;然后引入曲面形状控制参数,调整曲面的形状以符合图像的边缘特征;最后对插值曲面进行重采样．实验结果表明,该方法可以大大改善传统图像插值方法的平滑作用,保持插值图像边缘清晰、光滑。     

Constrained multibody system dynamics an automated approach

The governing equations for constrained multibody systems are formulated in a manner suitable for their automated, numerical development and solution. Specifically, the “closed loop” problem of multibody chain systems is addressed.

The governing equations are developed by modifying dynamical equations obtained from Lagrange's form of d'Alembert's principle. This modification, which is based upon a solution of the constraint equations obtained through a “zero eigenvalues theorem,” is, in effect, a contraction of the dynamical equations.

It is observed that, for a system with n generalized coordinates and m constraint equations, the coefficients in the constraint equations may be viewed as “constraint vectors” in n-dimensional space. Then, in this setting the system itself is free to move in the n−m directions which are “orthogonal” to the constraint vectors.     

基于几何思想的快速支持向量机算法

为了快速地进行分类,根据几何思想来训练支持向量机,提出了一种快速而简单的支持向量机训练算法——几何快速算法。由于支持向量机的最优分类面只由支持向量决定,因此只要找出两类样本中所有支持向量,那么最优分类面就可以完全确定。该新的算法根据两类样本的几何分布,先从两类样本的最近点开始;然后通过不断地寻找违反KKT条件的样本点来找出支持向量;最后确定最优分类面。为了验证新算法的有效性,分别利用两个公共数据库,对新算法与SMO算法及DIRECTSVM算法进行了实验对比,实验结果显示,新算法的分类精度虽与其他两个方法相当,但新算法的运算速度明显比其他两个算法快。     

Non-self-overlapping Hermite interpolation mapping: a practical solution for structured quadrilateral meshing

This paper addresses the problem of constructing a structured quadrilateral grid inside a given four-sided 2D region by a particular boundary-conforming mapping scheme—Hermite Interpolation Mapping (HIM). When the four given boundary curves are concave and convoluted, all boundary-conform mapping methods suffer from potential self-overlapping problem. Under HIM, the geometry of the grid depends on both the four boundary curves and the tangent vector functions associated with the curves. While the four boundary curves are fixed, the tangent functions in HIM can be varied to suit the need of controlling the characteristics of the mesh inside the given region to prevent self-overlapping. Besides tangent functions, the four twist vectors at the corners of the region can also be adjusted to influence the distribution of the inner grid elements. In our approach, a constrained functional optimization scheme is adopted to adjust the tangent functions and the twist vectors, adaptive to the geometry of the boundary curves, so that the resulting HIM will be free of self-overlapping. The optimization is carried out on the shape control energy that measures the overall mesh quality of the underlying HIM while the self-overlapping is strongly prevented in the form of constraints to the optimization. Experimental results show the promise of the proposed method as a practical and effective solution for structured grid generation.     

A non-parametric clustering scheme for landsat

A 4-dimensional histogram is computed to reduce the large LANDSAT pixel data (up to 7.6 million pixels to a frame) to the much smaller number (6,000) of distinct vectors and their frequency of occurrence in the scene. The vectors are clustered by a recent non-parametric clustering algorithm⁽³⁾ using the histogram count as a probability density estimate. The resultant clusters are unimodal m the 4-dimensional histogram and can possess arbitrary shapes. The algorithm is non-iterative and does not require specification of the number of clusters a priori.

Hashing is used to generate the histogram and also subsequent table look-up classification of the individual pixels in the image after the histogram vectors are clustered. The resultant clustering scheme is very efficient and a 512 × 512 LANDSAT scene can be clustered in less than 2 min of CPU time on a PDP-10 computer. Results of the application of the clustering scheme on representative LANDSAT scenes are included.     

Computation of generalised inverse of a special class of matrices

The computation of the generalised Inverse A⁺ of matrix A depends critically of the rank of A and involves several matrix multiplications. It is shown here that if A is of the form where A_i are now vectors, then A⁺ can be computed efficiently and accurately by a simple algebraic method.     

THz-TDS signal analysis and substance identification via the conformal split

A terahertz time-domain spectroscopy(THz-TDS)imaging system can obtain high-dimensional signals with substance fingerprint information.By introducing geometric algebra,a novel signal analysis approach to THz-TDS signals is developed based on an optical physical mechanism.Using this approach,signals are represented with vectors in the high-dimensional real vector space.Geometric distribution properties and algebraic relationships of THz-TDS signals are deduced.It is proved that every complex refractive index of substances relates to a unique 2-blade,the vectors corresponding to the samples of the same substance are collinear and belong to the intrinsic 2-blade of the substance.When decomposed through the conformal split with respect to a 2-blade,THz-TDS signals of high dimensionality can be related to vectors in a 2-dimensional subspace.Based on the conformal split properties we deduced,two criteria for substance identification on the basis of THz-TDS signals are proposed.Accordingly,a novel substance identification method via the conformal split is presented.In the method,the 2-blade related to each "known" substance is calculated with two vectors corresponding to THz-TDS signals measured from samples of the substance but with dierent thicknesses.Using the conformal split with respect to those 2-blades,an identified vector corresponding to a THz-TDS signal is linearly related to the vector in a 2-dimensional subspace.The substance of a sample can be identified using criteria on the projected vectors in the subspaces.This method can contribute to accurate classification and identification.Finally,two experiments are presented that show the feasibility and accuracy of this method.     

Kinestatic Analysis of Nonsingular Lower Mobility Manipulators

This paper presents a method of determining exact solutions to kinestatic analyses of lower mobility manipulators by means of the reciprocal Jacobian. The existence and uniqueness of the reciprocal Jacobian are shown, which enables the complete kinestatic analyses of both serial and fully parallel manipulators in a unified way, as well as the derivation of the constraints on the end-effector twist and wrench.      

Segmentation of ultrasound images by using a hybrid neural network

A hybrid neural network is presented for the segmentation of ultrasound images.

Feature vectors are formed by the discrete cosine transform of pixel intensities in region of interest (ROI). The elements and the dimension of the feature vectors are determined by considering only two parameters: The amount of ignored coefficients, and the dimension of the ROI.

First-layer-nodes of the proposed hybrid network represent hyperspheres (HSs) in the feature space. Feature space is partitioned by intersecting these HSs to represent the distribution of classes. The locations and radii of the HSs are found by the genetic algorithms.

Restricted Coulomb energy (RCE) network, modified RCE network, multi-layer perceptron and the proposed hybrid neural network are examined comparatively for the segmentation of ultrasound images.     

Simulations of Using Vectors in Natural Sciences Education

The implementation of ICT （information and communication technologies） into the educational process is becoming a reality in the 21st century. Today＇s students grow up with technology. To keep their attention, scientific problems should be solved through visualization, which is made possible using ICT in the educational process. In the modem educational process, students still have difficulties in learning science concepts. Also, it is a very common problem that students cannot apply mathematical language and concepts into other science areas such as physics, engineering, etc. For example, students start learning about vectors in mathematics in secondary school. Vectors are very important because they have a wide area of applications especially in physics, engineering and navigation to represent forces, tension, velocity, etc.. Using the free mathematical software GeoGebra, a simulation of using vectors in these areas is made. It will be shown that such simulations increase students＇ interest, keep their attention, and make this knowledge more real and more understandable and connected to the physical world and thus more applicable to their other studies.     

形状插值的G1 Hermite曲线

提出了在给定2个端点及其切矢方向的条件下生成一条形状较好的三次Hermite曲线的方法.把未知的形状最好曲线的端点切矢模长看作端点条件的函数;然后建立该函数应当满足的条件,并根据工程制图人员在一些特殊的端点条件下的绘图得到一些经验数据;最后把该函数近似用三角函数的二次以下谐波分解表示,根据已有的经验数据和建立的条件得到谐波分量的大小.目标曲线的计算简单,在经验数据的情况下,目标曲线端点切矢模长范围为(0.5,2.9).与已有方法相比,曲线形状较好.     

Shape preserving interpolation by curvature continuous parametric curves

An interpolation scheme for planar curves is described, obtained by patching together parametric cubic segments and straight lines. The scheme has, in general, geometric continuity of order 2 (G² continuity) and is similar in approach to that of [Goodman & Unsworth ′86], but whereas this earlier scheme, when applied to cubics, produces curves with zero curvature at the interpolation points, the corresponding curvature values in this scheme are in general non-zero. The choice of a tangent vector at each interpolation point guarantees that the interpolating curve is local convexity preserving, and in the case of functional data it is single-valued and local monotonicity preserving. The algorithm for generating the cubic curve segments usually requires the solution of two non-linear equations in two unknowns, and lower bounds are obtained on the magnitude of the curvature at the relevant interpolation points in order that this system of equations has a unique solution. Particular attention is given to cubic segments which are adjacent to straight line segments. Two methods for calculating these segments are described, one which preserves G² continuity, and one which only gives G¹ continuity. A number of examples of the application of the scheme are presented.