基于C~∞基函数的自然邻点插值(NNI)方法在科学计算可视化上的应用

插值方法及插值基函数的选择是可视化技术的一个关键问题。该文首先根据平面域上分布的数据点集,将平面域剖分为 Ｖｏｒｏｎｏｉ ｃｅｌｌｓ进而得到相应的 Ｄｅｌａｕｎａｒ三角化网格。然后基于 Ｖｏｒｏｎｏｉ ｃｅｌｌｓ的几何性质,应用自然邻点插值（ＮＮＩ）方法,介绍了一种具有无穷次连续可微的 Ｃ∞插值基函数及其数学性质。将基于 Ｃ”基函数的 ＮＮＩ方法用于把复杂数据场重构成一个具有规则网格的经验模型。最后再根据规则网格,生成三维立体图、等值线图、矢量分布图等。文章通过实例表明了基于Ｃ∞基函数的ＮＮＩ方法能够很好地应用于处理极不规则分布数据场的可视化。     

基于Voronoi Cells的自然邻点插值法在地形可视化中的应用

本文基于Voronoi Cells的数据结构，实现了二维平面中大规模散乱数据点的非结构网格剖分的相应算法，并在此基础上运用自然邻点插值法构造出了具有良好数学性质的插值基函数，然后将该剖分方法和插值算法应用到GIS地形可视化的实际数据中，并利用OpenGL三维显示技术显示结果，收到了良好的效果。     

基于样条模型插值的方法在科学可视化上的应用

插值方法和插值基函数的选择是可视化技术的一个关键问题。在规则数据场中，本文提出了一个基于样条模型插值来重建三维结构的方法。该模型以样条基函数为背景，其函数系数则由周围采样点数据重复迭代产生。实验表明该方法的逼近效果明显优于传统的线性插值拟合得出的图像质量。     

一类改进的插值基函数

为了构造具有良好性质的插值基函数用来构造插值曲线与曲面,引入一类具有精确的局部支撑和无穷次可微的函数;将其与sinc函数结合并优化,构造一类相似于插值细分基函数的新基函数,这类新基函数保持了以往基函数的良好性质,并具有以往基函数所不具有的精确局部支撑性的优点.实例结果表明,文中构造的新基函数有很好的效果;与传统的Akima方法相比,所构造的曲线总体上具有较好的光顺性.     

三次PH曲线偶的C1 Hermite插值

对计算机图形中一类特殊的多项式曲线——Pythagorean hodograph(PH)曲线的C1Hermite插值问题进行研究.PH曲线具有诸如有精确的有理Offset、弧长函数可由多项式函数表示以及几何解释优美等一系列优良性质.基于复分析方法,避免了实分析讨论中出现的复杂表示及繁琐计算,构造了满足给定C1 Hermite插值条件且以C1拼接连续的三次PH曲线偶.该曲线偶可灵活处理拐点,从而克服了一般三次PH曲线因恒凸而无法处理拐点的缺陷.相应的两条Bézier曲线表示及其控制顶点的计算简单方便.所得4条插值曲线中,通常有1条曲线具有很好的几何形状特征.结果可直接应用于各工业产品设计及加工领域中.     

基于C1自然邻近插值的曲面拟合

以C0连续non-Sibsonian 插值作为三次单纯形Bernstein-Bézier多项式的基坐标,构造C1连续自然邻近插值函数。介绍了高阶连续函数的构建原理和性质。将C1连续自然邻近插值函数应用于曲面拟合场合,由于Voronoi图能够自动调整数据点分布不规则和密度不均匀在空间上的差异,即使对于散乱数据点,也能获得较好的拟合结果。     

紧支径向基函数插值实现多维数据可视化

通过分析某城市空气质量数值预报数据的时空组织结构,构建出了多维空间数据的整体框架。论述了几种插值方法的优缺点,在比较的基础上,将新的紧支径向基函数局部径向点插值方法引入到多维数据处理中,在空间、时间维度上对数据进行局部插值,从而实现数据的重构。以新的基于封装回调函数的多线程方法实现了大规模空气质量预报数据的三维动态可视化。实验结果表明,以上方法应用于大规模数据可视化时,其质量和运算速度都能满足实际需要。     

基于径向基函数的单级插值隐式曲面重构

研究基于径向基函数单级插值隐式曲面重构问题．探讨基于标准紧支撑径向基函数和变形径向基函数插值的参数求解过程。实验结果表明,该方法能有效地构造隐式曲面。并且插值过程相当快。     

二元有理双四次插值曲面的点控制问题

构造了一种含有两个参数的仅基于函数值的二元有理双四次插值样条函数,插值函数具有简洁的显示表示,既便于应用,又便于理论研究。给出了插值曲面在插值区域上C1光滑的一个充分条件,讨论了插值基函数的性质和插值曲面形状的点控制问题.在插值条件不变的情况下,插值区域内任一点插值函数的值可以根据设计需要通过对参数的选取进行插值曲面的局部修改。     

一种径向基函数插值的ENO格式

将ENO格式和径向基函数插值相结合,提出了求解双曲型偏微分方程的径向基函数插值的ENO方法。该方法依据ENO思想建立自适应模板,在选定的模板上利用径向基函数进行逼近,能够很好地处理具有间断解的问题,消除间断点处数值振荡现象。以一维双曲型偏微分方程为例,对该方法进行了验证,并通过与多项式ENO格式比较,表明该方法更具有优势。     

大幅面地图的三维地形重建

研究完成了一个实用的可用于大幅面像素地图的三维地形重建系统.提出一个实用的 等高线内插算法,算法克服了网格绘制等值线方法和三角网绘制等值线方法因只考虑了点的位 置属性、未考虑等高线的线属性而使插出的等高线常常会与母线相交的弱点;同时提出了一个 实用的等高线高程识别算法和一个可用于大幅面地图的三维规则数据场建立算法,算法充分利 用了等高线的先验知识,不仅建立的数据场的质量很高,而且算法的时间复杂性与数据点数 (m)和网格点数(n)成线性关系O(m+n),因此计算速度很快.最后利用三维规则数据场生成真 实感图形.     

A simple algorithm for structured-grid generation with application to efficient Navier-Stokes computation

A grid generation procedure using a simple algorithm is presented. The present method is essentially based on the electro-static theory. By distributing electric charges at the boundary points, an electro-static vector field is produced inside the region between the inner and outer boundaries. The resulting electric force lines, which all emanate from grid points on the body, can be utilized as one family of grid lines. The other family of grid lines is easily constructed by plotting points at regular intervals along the electric force lines. The electric charge distribution is uniquely determined by an appropriate boundary condition, as long as the total amount of charges on each closed boundary is constant. The actual Navier-Stokes simulation using the block-structured grid generated by the present method was also carried out for a complex space vehicle configuration.     

Measuring homogeneity of planar point-patterns by using kurtosis

Kurtosis is generally associated with measurements of peakedness of a distribution. In this paper, we suggest a method where kurtosis can be used as a measure of homogeneity of any quantifiable property on a planar surface. A 2-dimensional, continuous and uniform distribution has kurtosis equal to 5.6. This value is also the limiting value for a discrete uniform distribution defined on a regular, rectangular grid when the number of grid points tend to infinity. Measurements of a planar surface, taken at regular grid points, are considered as realizations of random fields. These are associated with 2-dimensional random variables from which the value of kurtosis can be computed and used as a measure of the homogeneity of the field. A deviation from 5.6 indicates that the stochastic variable is not uniformly distributed and that the corresponding random field is not homogeneous. The model is applied on the spatial variation of the roughness on the surface of newsprint, an application where homogeneity is very important.     

实时生成三维地形中关于DEM数据插值的探讨

使用DEM来生成地形表面的可视化技术已经应用于军事、测绘、林业和娱乐等多个领域.讨论了将等高线离散采样点数据内插为规则格网数据的方法,采取了实用有效的移动拟合和加权平均结合的内插方法,并且根据规则格网的专有特点,确定了待插范围.在插值的过程中,每个数据块中格网点的高程值通过与其周围邻近的八块地形数据块共同插值而获得,因此优化了相邻数据块之间高程值差异过大而造成的地形过于突兀的问题.     

A FORTRAN IV program for interpolating irregularly spaced data using the difference equations for minimum curvature

The interpolation of potential field data is a common problem in geophysics. It is accomplished each time a contour map or profile is drawn. These operations can be difficult and subjective if the data points are irregularly spaced. The method described here produces a “numerical surface”, or grid of values, that approaches the smoothest surface passing through the control points. Such a grid can be contoured automatically by standard routines, producing acceptable results. Alternatively, sections can be drawn through the surface on any azimuth—a more satisfactory procedure for constructing profiles than projecting the data points onto straight lines.     

Diffusion tensor visualization with glyph packing

A common goal of multivariate visualization is to enable data inspection at discrete points, while also illustrating larger-scale continuous structures. In diffusion tensor visualization, glyphs are typically used to meet the first goal, and methods such as texture synthesis or fiber tractography can address the second. We adapt particle systems originally developed for surface modeling and anisotropic mesh generation to enhance the utility of glyph-based tensor visualizations. By carefully distributing glyphs throughout the field (either on a slice, or in the volume) into a dense packing, using potential energy profiles shaped by the local tensor value, we remove undue visual emphasis of the regular sampling grid of the data, and the underlying continuous features become more apparent. The method is demonstrated on a DT-MRI scan of a patient with a brain tumor.     

规则格网DTM快速构建算法研究*

规则格网(Grid)是数字地形模型(DTM)最常用的表示方法,有着广泛的应用。研究了基于离散点、三角网两种空间数据的规则格网快速构建算法。实验表明,分块索引方法及扫描线填充方法的引入实现了规则格网的快速生成,在大数据量条件下,效果也非常明显。     

A finite volume local defect correction method for solving the transport equation

The local defect correction (LDC) method is applied in combination with standard finite volume discretizations to solve the advection-diffusion equation for a passive tracer. The solution is computed on a composite grid, i.e. a union of a global coarse grid and local fine grids. For the test a dipole colliding with a no-slip wall is used to provide an actively changing velocity field. The LDC method is tested for the problem of localized patch of tracer material that is transported by the provided velocity field. The LDC algorithm can be formulated to conserve the total amount of tracer material. However, if the local fine grids are moved to adaptively follow the behaviour of the solution, a loss or gain in the total amount of tracer material is produced. This deficit in tracer material is created when the solution is interpolated to obtain data for the moved fine grid. The data obtained by the interpolation scheme in the new refined region can be adapted in such a way that the surplus or deficit is spread over the new grid points and conservation of tracer material is satisfied. Finally, the results of the conservative finite volume LDC method are compared and validated with results from a spectral method.     

结构光视觉三维点云逐层三角网格化算法

针对结构光视觉恢复的大规模三维点云的可投影特点,提出一种基于投影网格的底边驱动逐层网格化曲面重建算法。该算法首先将点云投影到一个二维平面上;然后基于点云投影区域建立规则投影网格,并将投影点映射到规则二维投影网格上,建立二维网格点与三维点云间的映射关系;接着对投影网格进行底边驱动的逐层网格化,建立二维三角网格;最后根据二维投影点与三维点的对应关系及二维三角网格拓扑关系获得最终的三维网格曲面。实验结果表明,算法曲面重建速度快,可较好地保持曲面细节特征。