Reconstruction of B-spline surface by interpolating boundary curves and approximating inner points

In this paper, we present an algorithm for reconstruction of B-spline surface such that it interpolates the four given bound- ary curves and simultaneously approximates some given inner points. The main idea of our method is： first, we construct an initial surface which interpolates the four given boundary curves; then, while keeping the boundary control points of the initial surface un- changed, we reposition the inner control points of the surface with energy optimization method. Examples show that our algorithm is practicable and effective.

Using Surface Splines for GIS- ready Imagery

High resolution data have become an important source of data, before they can be integrating into a GIS database. It requires processing for ortho-rectification to generate image map with high accuracy and low cost. Using surface splines interpolation for rectification is quite different with traditional grid method in photogrammetric. To introduce surface splines is important. In fact the actual name is mechanical surface splines because it must add mechanical conditions to form the formula. The main advantages of the surface splines are that the coordinate of the known points are not located in a rectangular array and the function may be differentiated in find slopes. Surface splines are a mathematical tool to interpolate a function of two variables. It bases upon small deflection equation of an infinite plate, it originally developed for interpolation wing deflection of aircraft 1972 by Harderaed and Desmarais contributed. An example map size is beyond the custom. Only five control points for rectification （60 cm × 200 cm） if the number of control points is increasing also the map size increasing large and operating easy. The other UAV in Civilian world gives low cost. Obvious aerial imaging using surface splines in cartographic is a tool for supporting geospatial decisions.

Regional magnetic anomaly fields:3D Taylor polynomial and surface spline models

We used data from 1960.0,1970.0,1980.0,1990.0,and 2000.0 to study the geomagnetic anomaly field over the Chinese mainland by using the three-dimensional Taylor polynomial（3DTP） and the surface spline（SS） models.To obtain the pure anomaly field,the main field and the induced field of the ionospheric and magnetospheric fields were removed from measured data.We also compared the SS model anomalies and the data obtained with Kriging interpolation（KI）.The geomagnetic anomaly distribution over the mainland was analyzed based on the SS and 3DTP models by transferring all points from 1960.0-1990.0 to 2000.0.The results suggest that the total intensity F anomalies estimated based on the SS and KI for each year are basically consistent in distribution and intensity.The anomalous distributions in the X-,Y-,and Z-direction and F are mainly negative.The 3DTP model anomalies suggest that the intensity in the X-direction increases from-100 nT to 0 nT with longitude,whereas the intensity in the Y-direction decreases from 400 nT to 20 nT with longitude and over the eastern mainland is almost negative.The intensity in the Z-direction and F are very similar and in most areas it is about-50 nT and higher in western Tibet.The SS model anomalies overall reflect the actual distribution of the magnetic field anomalies;however,because of the uneven distribution of measurements,it yields several big anomalies.Owing to the added altitude term,the 3DTP model offers higher precision and is consistent with KI.

Theoretical Modeling and Surface Roughness Prediction of Microtextured Surfaces in Ultrasonic Vibration-Assisted Milling

Textured surfaces with certain micro/nano structures have been proven to possess some advanced functions,such as reducing friction,improving wear and increasing wettability.Accurate prediction of micro/nano surface textures is of great significance for the design,fabrication and application of functional textured surfaces.In this paper,based on the kinematic analysis of cutter teeth,the discretization of ultrasonic machining process,transformation method of coordinate systems and the cubic spline data interpolation,an integrated theoretical model was established to characterize the distribution and geometric features of micro textures on the surfaces machined by different types of ultrasonic vibration-assisted milling(UVAM).Based on the theoretical model,the effect of key process parameters(vibration directions,vibration dimensions,cutting parameters and vibration parameters)on tool trajectories and microtextured surface morphology in UVAM is investigated.Besides,the effect of phase difference on the elliptical shape in 2D/3D ultrasonic elliptical vibration-assisted milling(UEVAM)was analyzed.Compared to conventional numerical models,the method of the cubic spline data interpolation is applied to the simulation of microtextured surface morphology in UVAM,which is more suitable for characterizing the morphological features of microtextured surfaces than traditional methods due to the presence of numerous micro textures.The prediction of surface roughness indicates that the magnitude of ultrasonic amplitude in z-direction should be strictly limited in 1D rotary UVAM,2D and 3D UEVAM due to the unfavorable effect of axial ultrasonic vibration on the surface quality.This study can provide theoretical guidance for the design and fabrication of microtextured surfaces in UVAM.

Modeling of CHAMP satellite data according to the 3D surface spline model of geomagnetic fi elds

Surface observations and CHAMP measurement data are employed to develop a three-dimensional surface spline(3DSS)model of China's Mainland.The magnetic field distribution at the satellite level is then demonstrated using the model obtained.The results of this model are compared and verifi ed by deriving the corresponding two(2DTY)and threedimensional(3DTY)Taylor polynomial models.Issues such as the removal of disruptive geomagnetic fi elds,the data gap between the surface and satellite levels,and boundary eff ects are carefully considered during modeling.We then focus on evaluating the modeling eff ect of the satellite data.Ten satellite points not involved in the modeling procedure are selected,and the residuals,absolute change rates,and RMSEs of these points are calculated.Results show that the distribution of the magnetic fi eld determined by the 3DSS model is highly consistent with that obtained from the IGRF12 model.Expect for component Y,the absolute change rates of other components are less than 0.5%.Specifi cally,the RMSE of Y of 3DSS is nearly 60%lower than those of 3DTY and 2DTY;the RMSE of other components of the former are also over 90%lower than those of the latter.This fi nding implies that the 3DSS model has good performance for modeling satellite data and its results are reliable.Moreover,the modeling eff ect of 3DTY is better than that of 2DTY.

一类在Box Splines空间中封闭曲面的重构算法

利用Box Splines空间中的拟插值算子对封闭曲面进行重构,大大改进了张量积型样条的曲面重构的特性和效率。此方法以S_2~1(Δ_(mn)^(2))样条空间的基为基础,然后利用拟插值算子W_(mn)(f)对定义在封闭曲面S上的三维数据进行拟合,并且用边界宽度重合的方法处理曲面封闭的条件。通过示例试验以及分析比较,该方法能够实现在Box Splines空间中曲面的封闭,而且对于封闭曲面的重构有很好的效果。

On Complete Bicubic Fractal Splines

Fractal geometry provides a new insight to the approximation and modelling of experimental data. We give the construction of complete cubic fractal splines from a suitable basis and their error bounds with the original function. These univariate properties are then used to investigate complete bicubic fractal splines over a rectangle Bicubic fractal splines are invariant in all scales and they generalize classical bicubic splines. Finally, for an original function , upper bounds of the error for the complete bicubic fractal splines and derivatives are deduced. The effect of equal and non-equal scaling vectors on complete bicubic fractal splines were illustrated with suitably chosen examples.

C^2-(C^3-) Continuous Interpolation Spline Curve and Surface

Free-formed or sculptured surfaces in engineering products are frequently constructed from a set of measured 3D data points. C2- (C3-) continuity approach is important in this field. This paper presents a method of rectangular interpolation of given 3D data array which is regularly arranged. The interpolation surface which is constructed by tensor product has desirable properties (second-order or third-order continuity locality) and is implemented and adjusted easily. Higher order continuity methods are also briefly discussed.

C^2 quartic spline surface interpolation

This paper discusses the problem of constructing C2 quartic spline surface interpolation. Decreasing the continuity of the quartic spline to C2 offers additional freedom degrees that can be used to adjust the precision and the shape of the interpolation surface. An approach to determining the freedom degrees is given, the continuity equations for constructing C2 quartic spline curve are discussed, and a new method for constructing C2 quartic spline surface is presented. The advantages of the new method are that the equations that the surface has to satisfy are strictly row diagonally dominant, and the discontinuous points of the surface are at the given data points. The constructed surface has the precision of quartic polynomial. The comparison of the interpolation precision of the new method with cubic and quartic spline methods is included.

基于纹理畸变与B-Spline插值的计算机表面重建方法

提出一种基于网格纹理畸变的方法,获取物体2D断层轮廊序列,用B-Spline内插形成新的断层切片,实现物体表面重建.该方法不仅具有断层扫描法的优点,还避免了使用特殊设备采集断层图像如CT、MRI,适用于一般物体的重建.

局部可调的四次Catmull-Rom样条曲面及其形状优化

当控制顶点保持不变时,无法对三次Catmull-Rom样条曲面的形状进行局部调整。为满足实际工程对插值曲面形状设计的要求,给出了一种含局部参数的四次Catmull-Rom样条曲面,并讨论了2种对四次Catmull-Rom样条曲面形状进行优化的方案。结果表明,四次Catmull-Rom样条曲面不仅继承了三次Catmull-Rom样条曲面的插值性与连续性,而且容易利用所含的参数对曲面形状进行局部或整体调整,通过计算所含参数的最优值获得尽可能光顺和尽可能保形的四次Catmull-Rom样条曲面,为插值曲面的构造提供了方法。

基于“D”型深孔测斜曲线的滑坡滑动面位置确定方法研究

在实际的深孔位移监测中,测斜曲线的突变特征是滑动面辨识的关键依据,前人通过大量的研究总结将滑动面迹象显著的测斜曲线类型分为了“B”型、“D”型、“r”型等几种。其中,对于“D”型测斜曲线通常是将曲线的鼓包凸起点作为滑动面的位置,但这种方法容易受到测点布置间隔和横纵坐标观测尺度的影响,存在滑面位置定义不清晰、数值不确定的问题。为了能够有效地克服这些缺点,提升测斜曲线滑动面辨识的准确度,基于“D”型测斜曲线变化特征,将滑坡抽象为由“滑动体”“滑动区间”以及“不动体”三者组成的概化模型,根据三者抗弯刚度的差异建立外界荷载作用下的杆件力学模型,深入分析滑坡运动过程中不同深度处土体的变形特点。研究表明,由于“D”型曲线滑动面并未完全贯通,使得土体沿深度方向变形连续无突变,力学模型中杆件正负弯矩的分界点是变形曲线水平位移最大处,能够真实地反映滑坡变形特点以及滑动面的位置。将土体累计位移转化为相对位移,则“D”型深孔测斜曲线变为了“S”型相对位移-深度曲线,且“S”型曲线的拐点与滑动面的位置相近;通过提取监测期内不同深度处土体的平均相对位移,运用三次样条插值法计算“S”型区段内拐点的深度值,能够更加精准地确定滑动面位置,更好地提升深孔位移监测的可靠度和准确度,具有较大的实用价值。

基于B型深孔测斜曲线的滑坡滑动面位置确定方法

在实际的深孔位移监测中,测斜曲线的突变特征是滑动面辨识的关键依据,对于“B”型测斜曲线通常是将曲线的鼓包凸起点作为滑动面的位置,但这种方法容易受到测点布置间隔和横纵坐标观测尺度的影响,存在滑面位置定义不清晰、数值不确定的问题。为了能够有效地克服这些缺点,提升测斜曲线滑动面辨识的准确度,基于“B”型测斜曲线变化特征,研究将滑坡抽象为由“滑动体”、“滑动区间”以及“不动体”三者组成的概化模型,并对滑坡运动过程中不同深度处土体的位移速率和加速度变化特点进行了深入分析。结果表明:土体位移速率沿深度方向会形成多道曲线簇,且不同曲线簇分别对应各滑面所在的滑动区间及非滑动区间。此外,将位移速率转化为加速度,则土体加速度曲线具有更加显著的分簇特征,能够为准确区分滑动体和不动体的区间范围提供重要依据。进一步地,提取监测期内加速度曲线分簇特征显著的当日土体加速度并绘制散点图,能够快速准确地确定坡体不同区间的深度范围。基于当日各滑面所在滑动区间内的加速度数据,运用三次样条插值法计算该区间内的位移加速度最大值,由于滑面处的土体加速度最大,则该值对应的土体深度可作为滑面位置的计算深度。研究方法能够更加精准地确定滑动面位置,并使滑面位置的确定可计算化,有效提升了深孔位移监测数据的利用程度和可靠度,具有较大的实用价值。

DEM地面形态重构方法的精度差异特征研究

依据DEM地形建模过程,阐述了DEM地面形态重构概念,并以黄土丘陵1∶5万DEM数据为例,采用高程数值误差场和局地坡面形态误差场相结合的分析技术,实验分析了双线性、三次卷积、局部二次多项式、规则样条函数等常用插值方法在基于加密格网的DEM地面形态重构过程中的精度差异,以及DEM地面形态可重构的基本条件等问题。研究表明:在高程数值误差的极值大小、离散程度、空间分布等方面,规则样条函数插值法具有最好的地面形态重构精度,其次为局部二次多项式、三次卷积和双线性插值法;对于已确定综合尺度的地形,存在满足高保真地面形态描述要求的最佳DEM格网分辨率阈值,只有当实际DEM格网分辨率等于或高于该阈值时,才有可能无歧义、高精度地重构出DEM所描述的综合地形的地面形态。

光滑曲面上的G^1插值曲线

在计算机图形学和计算机辅助几何设计中 ,限制在光滑曲面上保持几何连续的曲线插值技术显现出越来越重要的作用 文中用直纹面投影的思想研究了这一问题 ,给出了一种在光滑曲面上保持G1连续的样条曲线插值技术 首先构造一条插值曲面上已知点列的空间 3次B啨ｚｉｅｒ样条曲线 ,然后通过一张直纹面将这条空间插值曲线投影到已知曲面上 ,即可得到限制在已知光滑曲面上的G1插值曲线 理论推导和实例显示表明 。

图像匹配在海底地图匹配中的应用

应用海底地图匹配技术提高水下载体(AUV)定位精度。通过双三次B-样条曲面插值重建参考地图(RDM),选取RDM中部分地图附加正态白噪声模拟当地水深地图(LDM)。将两块地图同时转化为8位256级灰度图像,应用图像相关法进行匹配。仿真显示,该方法具有较高的匹配精度和一定的抗差特性,具有适应大面积海底地形,包括平坦区域和复杂区域的特点。

基于B样条函数的散乱数据曲面拟合和数据压缩

针对大规模散乱数据,本文提出一种基于B样条函数的曲面拟合和数据压缩方法.在此方法中,构建最小平面矩形区域覆盖所有的散乱数据点,根据二维情况下的B样条函数的线性插值公式,利用最小二乘法拟合规则网格控制点的数值,存储有限规则网格控制点的数值,并通过插值可以得到任意点的数值,从而达到曲面拟合和数据压缩的目的.通过算例验证了此方法的可行性以及使用三次B样条函数曲面拟合的优越性.由于此方法具有局部误差小、计算高效的特征,可以适用于数据梯度较小的二维系统数据的曲面拟合和数据压缩,如地形高程数据和射线追踪走时表.

起伏地表声波方程的数值模拟

在实际地震勘探中,经常会遇到形状复杂的起伏地表,而实际采集的有关地表形状和介质的数据在进行数值模拟时都可视为对某一区域进行矩形剖分后所得网格点上的值,也就是说所获得的地表是一个阶梯状的地表,与实际的地表存在很大差异。利用阶梯状地表数据对地震记录进行波动方程模拟时,在阶梯的角点处会产生严重的干扰,进而影响数值模拟结果。本文利用声波方程,对阶梯状地表数据进行样条插值处理,得到了与起伏地表比较吻合的四边形网格剖分。经理论模型和实际数据的计算结果对比表明,用样条插值的方法逼近地表后进行计算的结果可以消除由于角点的奇性而产生的误差,减少了干扰,提高了计算精度。

用参数样条插值挖补方法进行大规模散乱数据曲面造型

利用矩形域中带连续边界条件的多元散乱数据最优插值方法,结合张量积型参数样条插值,从挖补的思想得到启发,提出一种适合大规模散乱数据曲面造型的参数样条插值挖补方法.用该方法构造的参数曲面内部Cm,n连续,挖补的矩形边界分别为Cm-1,0和C0,n-1连续.最后就常见的m=n=2时的双三次样条给出一些数值例子,说明该算法简单易行,效果良好.

B样条曲面在严格约束状态下的光顺拟合

提出一种新的Ｂ样条曲面造型方法：光顺地拟合空间型值点且同时严格地通过其中部分点．运用Ｌａｇｒａｎｇｅ乘子的条件极值法并引入光顺加权项，求出位置偏离和形状弯曲的最小二乘解，可以得到被称为Ｂ样条光顺准拟合的良好造型曲面．这一方法在包含门窗的交通工具外壳曲面设计、机械产品装配联接件制造中具有明显的实用价值．