HERMITE—BIRKHOFF INTERPOLATION OF SCATTERED DATA BY RADIAL BASIS FUNCTIONS

For Hermite-Birkhoff interpolation of scattered multidumensional data by radial basis function (?),existence and characterization theorems and a variational principle are proved. Examples include (?)(r)=r^b,Duchon's thin-plate splines,Hardy's multiquadrics,and inverse multiquadrics.

Surface Reconstruction of 3D Scattered Data with Radial Basis Functions

We use Radial Basis Functions （RBFs） to reconstruct smooth surfaces from 3D scattered data. An object＇s surface is defined implicitly as the zero set of an RBF fitted to the given surface data. We propose improvements on the methods of surface reconstruction with radial basis functions. A sparse approximation set of scattered data is constructed by reducing the number of interpolating points on the surface. We present an adaptive method for finding the off-surface normal points. The order of the equation decreases greatly as the number of the off-surface constraints reduces gradually. Experimental results are provided to illustrate that the proposed method is robust and may draw beautiful graphics.

Large Scattered Data Fitting Based on Radial Basis Functions

Solving large radial basis function （RBF） interpolation problem with non-customized methods is computationally expensive and the matrices that occur are typically badly conditioned. In order to avoid these difficulties, we present a fitting based on radial basis functions satisfying side conditions by least squares, although compared with interpolation the method loses some accuracy, it reduces the computational cost largely. Since the fitting accuracy and the non-singularity of coefficient matrix in normal equation are relevant to the uniformity of chosen centers of the fitted RBE we present a choice method of uniform centers. Numerical results confirm the fitting efficiency.

Scattered Data Interpolation Using Cubic Trigonometric Bézier

This paper discusses scattered data interpolation using cubic trigonometric Bézier triangular patches with C1 continuity everywhere.We derive the C1 condition on each adjacent triangle.On each triangular patch,we employ convex combination method between three local schemes.The final interpolant with the rational corrected scheme is suitable for regular and irregular scattered data sets.We tested the proposed scheme with 36,65,and 100 data points for some well-known test functions.The scheme is also applied to interpolate the data for the electric potential.We compared the performance between our proposed method and existing scattered data interpolation schemes such as Powell–Sabin(PS)and Clough–Tocher(CT)by measuring the maximum error,root mean square error(RMSE)and coefficient of determination(R^(2)).From the results obtained,our proposed method is competent with cubic Bézier,cubic Ball,PS and CT triangles splitting schemes to interpolate scattered data surface.This is very significant since PS and CT requires that each triangle be splitting into several micro triangles.

Trivariate Polynomial Natural Spline for 3D Scattered Data Hermit Interpolation

Consider a kind of Hermit interpolation for scattered data of 3D by trivariate polynomial natural spline, such that the objective energy functional （with natural boundary conditions） is minimal. By the spline function methods in Hilbert space and variational theory of splines, the characters of the interpolation solution and how to construct it are studied. One can easily find that the interpolation solution is a trivariate polynomial natural spline. Its expression is simple and the coefficients can be decided by a linear system. Some numerical examples are presented to demonstrate our methods.

Rational Quasi-Interpolation Approximation of Scattered Data in R^(3)

This paper is concerned with a piecewise smooth rational quasi-interpolation with algebraic accuracy of degree(n+1)to approximate the scattered data in R 3.We firstly use the modified Taylor expansion to expand the mean value coordinates interpolation with algebraic accuracy of degree one to one with algebraic accuracy of degree(n+1).Then,based on the triangulation of the scattered nodes in R^(2),on each triangle a rational quasi-interpolation function is constructed.The constructed rational quasi-interpolation is a linear combination of three different expanded mean value coordinates interpolations and it has algebraic accuracy of degree(n+1).By comparing accuracy,stability,and efficiency with the C^(1)-Tri-interpolation method of Goodman[16]and the MQ Shepard method,it is observed that our method has some computational advantages.

FITTING C1 SURFACES TO SCATTERED DATA WITH S 1 2 （△ m,n （2））

This paper presents a fast algorithm （BS2 Algorithm） for fitting C 1 surfaces to scat- tered data points. By using energy minimization, the bivariate spline space S 2 1（△ m,n （2） ） is introduced to construct a Cl-continuous piecewise quadratic surface through a set of irregularly 3D points. Moreover, a multilevel method is also presented. Some experimental results show that the accuracy is satisfactory. Furthermore, the BS2 Algorithm is more suitable for fitting surfaces if the given data points have some measurement errors.

Adaptive Hierarchical B-spline Surface Representation of Large-Scale Scattered Data

The representation of large scale scattered data is a difficult problem, especially when various features of the representation, such as C 2-continuity, are required. This paper describes a fast algorithm for large scale scattered data approximation and interpolation. The interpolation algorithm uses a coarse-to-fine hierarchical control lattice to fit the scattered data. The refinement process is only used in the regions where the error between the scattered data and the result in a surface is greater than a specified tolerance. A method to ensure C 2-continuity is introduced to calculate the control lattice under constrained conditions. Experimental results show that this method can quickly represent large scale scattered data set.[

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.

SOME SHAPE-PRESERVING QUASI-INTERPOLANTS TO NON-UNIFORMLY DISTRIBUTED DATA BY MQ-B-SPLINES

Based on the definition of MQ-B-Splines,this article constructs five types of univariate quasi-interpolants to non-uniformly distributed data. The error estimates and the shape-preserving properties are shown in details.And examples are shown to demonstrate the capacity of the quasi-interpolants for curve representation.

Geometric Scaling Analysis of Deep Inelastic Scattering Data Including Heavy Quarks

An analytic massive total cross section of photon proton scattering is derived, which has geometric scaling. A geometric scaling is used to perform a global analysis of the deep inelastic scattering data on inclusive structure function F2 measured in lepton-hadron scattering experiments at small values of Bjorken x. It is shown that the descriptions of the inclusive structure function F2 and longitudinal structure function FL are improved with the massive analytic structure function, which may imply the gluon saturation effect dominating the parton evolution process at HERA. The inclusion of the heavy quarks prevent the divergence of the lepton-hadron cross section, which plays a significant role in the description of the photoproduction region.

Analysis of the Diffractive Deep Inelastic Scattering Data with Running Coupling and Gluon Number Fluctuations

We study the effects of running coupling and gluon number fluctuations in the latest diffractive deep inelastic scattering data. It is found that the description of the data is improved once the running coupling and gluon number fluctuations are included with x2/d.o.f. = 0.867, x2/d.o.f. = 0.923 and x2/d.o.f. = 0.878 for three different groups of experimental data. The values of diffusive coefficient subtracted from the fit are smaller than the ones obtained by considering only the gluon number fluctuations in our previous studies. The smaller values of the diffusive coefficient are in agreement with the theoretical predictions, where the gluon number fluctuations are suppressed by the running coupling which leads to smaller values of the diffusive coefficient.

Numerical Comparison of Shapeless Radial Basis Function Networks in Pattern Recognition

This work focuses on radial basis functions containing no parameters with themain objective being to comparatively explore more of their effectiveness.For this,a total of sixteen forms of shapeless radial basis functions are gathered and investigated under the context of the pattern recognition problem through the structure of radial basis function neural networks,with the use of the Representational Capability(RC)algorithm.Different sizes of datasets are disturbed with noise before being imported into the algorithm as‘training/testing’datasets.Each shapeless radial basis function is monitored carefully with effectiveness criteria including accuracy,condition number(of the interpolation matrix),CPU time,CPU-storage requirement,underfitting and overfitting aspects,and the number of centres being generated.For the sake of comparison,the well-known Multiquadric-radial basis function is included as a representative of shape-contained radial basis functions.The numerical results have revealed that some forms of shapeless radial basis functions show good potential and are even better than Multiquadric itself indicating strongly that the future use of radial basis function may no longer face the pain of choosing a proper shape when shapeless forms may be equally(or even better)effective.

COMPLEX SURFACE RECONSTRUCTION BASED ON OBJECT-ORIENTED DEVELOPING TOOL VBA

Taking AutoCAD2000 as platform, an algorithm for the reconstruction ofsurface from scattered data points based on VBA is presented. With this core technology customerscan be free from traditional AutoCAD as an electronic board and begin to create actual presentationof real-world objects. VBA is not only a very powerful tool of development, but with very simplesyntax. Associating with those solids, objects and commands of AutoCAD 2000, VBA notably simplifiesprevious complex algorithms, graphical presentations and processing, etc. Meanwhile, it can avoidappearance of complex data structure and data format in reverse design with other modeling software.Applying VBA to reverse engineering can greatly improve modeling efficiency and facilitate surfacereconstruction.

A numerical method for one-dimensional nonlinear sine-Gordon equation using multiquadric quasi-interpolation

In this paper, we use a univariate multiquadric quasi-interpolation scheme to solve the one-dimensional nonlinear sine-Gordon equation that is related to many physical phenomena. We obtain a numerical scheme by using the derivative of the quasi-interpolation to approximate the spatial derivative and a difference scheme to approximate the temporal derivative. The advantage of the obtained scheme is that the algorithm is very simple so that it is very easy to implement. The results of numerical experiments are presented and compared with analytical solutions to confirm the good accuracy of the presented scheme.

Inverse Construction Methods of Heterogeneous NURBS Object Based on Additive Manufacturing

According to the requirement of heterogeneous object modeling in additive manufacturing(AM),the Non-Uniform Rational B-Spline(NURBS)method has been applied to the digital representation of heterogeneous object in this paper.By putting forward the NURBS material data structure and establishing heterogeneous NURBS object model,the accurate mathematical unified representation of analytical and free heterogeneous objects have been realized.With the inverse modeling of heterogeneous NURBS objects,the geometry and material distribution can be better designed to meet the actual needs.Radical Basis Function(RBF)method based on global surface reconstruction and the tensor product surface interpolation method are combined to RBF-NURBS inverse construction method.The geometric and/or material information of regular mesh points is obtained by RBF interpolation of scattered data,and the heterogeneous NURBS surface or object model is obtained by tensor product interpolation.The examples have shown that the heterogeneous objects fitting to scattered data points can be generated effectively by the inverse construction methods in this paper and 3D CAD models for additive manufacturing can be provided.

ALGORITHM FOR SPHERICITY ERROR AND THE NUMBER OF MEASURED POINTS

The data processing technique and the method determining the optimal number of measured points are studied aiming at the sphericity error measured on a coordinate measurement machine （CMM）. The consummate criterion for the minimum zone of spherical surface is analyzed first, and then an approximation technique searching for the minimum sphericity error from the form data is studied. In order to obtain the minimum zone of spherical surface, the radial separation is reduced gradually by moving the center of the concentric spheres along certain directions with certain steps. Therefore the algorithm is precise and efficient. After the appropriate mathematical model for the approximation technique is created, a data processing program is developed accordingly. By processing the metrical data with the developed program, the spherical errors are evaluated when different numbers of measured points are taken from the same sample, and then the corresponding scatter diagram and fit curve for the sample are graphically represented. The optimal number of measured points is determined through regression analysis. Experiment shows that both the data processing technique and the method for determining the optimal number of measured points are effective. On average, the obtained sphericity error is 5.78 μm smaller than the least square solution, whose accuracy is increased by 8.63%; The obtained optimal number of measured points is half of the number usually measured.

Triangular Curved Surface Construction of A Series ofArbitrary Disordered Points in Space

Constructing Bernstein-Bezier triangular interpolating curve surface interpolating a series of arbitrary disordered data points is of considerable importance for the design and modeling of surfaces with a variety of continuity information. In this article. a kind of simple and reliable algorithm that can process complex field triangular grid generating is presented, and a group of formulae for determining triangular curved surface with wholly C1 continuity are given. It can process arbitrary non-convex boundary and can be used to construct surfaces inner holes.

Selection of radio astronomical observation sites and its dependence on human generated RFI

We investigate the influence of population density on radio-frequency inter- ference （RFI） affecting radio astronomy. We use a new method to quantify the thresh- old of population density in order to determine the most suitable lower limit for site selection of a radio quiet zone （RQZ）. We found that there is a certain trend in the population density-RFI graph that increases rapidly at lower values and slows down to almost fiat at higher values. We use this trend to identify the thresholds for pop- ulation density that produce RFI. Using this method we found that, for frequencies up to 2.8 GHz, low, medium and high population densities affecting radio astronomy are below 150 ppl km-2, between i50 ppl km-2 and 5125 ppl km-~, and above 5125 ppl km-2 respectively. We also investigate the effect of population density on the environment of RFI in three astronomical windows, namely the deuterium, hydro- gen and hydroxyl lines. We find that a polynomial fitting to the population density produces a similar trend, giving similar thresholds for the effect of population density. We then compare our interference values to the standard threshold levels used by the International Telecommunication Union within these astronomical windows.

EVOLUTION OF THE SCATTERING DATA UNDER THE CLASSICAL DARBOUX TRANSFORM FOR su(2) SOLITON SYSTEMS

The changing of the scattering data for the solutions of su (2) soliton systems which are relatedby a classical Darboux transformation (CDT) is obtained. It is shown that how a CDT creates anderases a soliton.