Multidimensional spline integration of scattered data

We introduce a numerical method for reconstructing a multidimensional surface using the gradient of the surface measured at some values of the coordinates. The method consists of defining a multidimensional spline function and minimizing the deviation between its derivatives and the measured gradient. Unlike a multidimensional integration along some path, the present method results in a continuous, smooth surface, furthermore, it also applies to input data that are non-equidistant and not aligned on a rectangular grid. Function values, first and second derivatives and integrals are easy to calculate. The proper estimation of the statistical and systematical errors is also incorporated in the method.     

Image warping with scattered data interpolation

Discusses a new approach to image warping based on scattered data interpolation methods which provides smooth deformations with easily controllable behavior. A new, efficient deformation algorithm underlies the method     

An algorithm for computing shape-preserving cubic spline interpolation to data

We present an algorithm for the construction of shape-preserving cubic splines interpolating a set of data point. The method is based upon some existence properties recently developed. Graphical examples are given.     

四元数样条插值的人体运动数据重构

为了得到平滑的人体动画,提出一种基于四元数的样条插值算法,利用提取的关键帧实现人体运动序列的有效重构。为减少重构误差、加快收敛速度,将已知关键帧集合作为初始条件,通过迭代算法求出样条曲线的控制点集合。利用样条曲线控制点计算贝塞尔曲线控制点,构造贝塞尔样条曲线段,将各段贝塞尔样条曲线段组合,构造一条基于四元数的样条曲线。根据德卡斯特里奥（de Casteljau）算法插值重构人体运动。实验结果表明,该算法在保证执行效率的同时,可得到光滑的插值结果,实现满足视觉要求的人体运动重构。     

一种基于散乱数据插值的网格图象变形方法

提出一种基于特征点运动分解和散乱灰度数据插值的网格图象变形算法,以改进传统的两步扫描网格变形法在扫描顺序和变形效果上的不足。将原始图象的象素坐标一次性映射至目标图象,再对映射后得到的散乱坐标点的灰度进行散乱数据插值以恢复目标图象的象素信息。为了提高灰度映射的效率,引入一种基于Delaunay三角剖分的三角线性插值的方法来处理大规模散乱数据的插值。最后通过实例证明该算法的变形效果较两步扫描网格变形法有显著提高。     

Powell-Sabin splines in range restricted interpolation of scattered data

Computing - The construction of range restricted bivariateC 1 interpolants to scattered data is considered. In particular, we deal with quadratic spline interpolation on a Powell-Sabin refinement...     

基于空间数据插值算法的空间散乱数据体绘制实现

体绘制技术是一种能够真实地反映空间数据场内部信息的可视化技术。在体绘制研究领域中,非规则的空间散乱数据体绘制目前仍然是一个研究热点。文中采用空间数据插值算法对散乱的原始数据进行网格化插值,然后使用光投影体绘制算法对规则网格数据进行体绘制。最后,通过新方法实现了某油藏区地下流体压力和孔隙度分布结构的体绘制。     

An iterative algorithm for spline interpolation

One of the fundamental results in spline interpolation theory is the famous Schoenberg-Whitney Theorem, which completely characterizes those distributions of interpolation points which admit unique interpolation by splines. However, until now there exists no iterative algorithm for the explicit computation of the interpolating spline function, and the only practicable method to obtain this function is to solve explicitly the corresponding system of linear equations. In this paper we suggest a method which computes iteratively the coefficients of the interpolating function in its B-spline basis representation; the starting values of our one-step iteration scheme are quotients of two low order determinants in general, and sometimes even just of two real numbers. Furthermore, we present a generalization of Newton's interpolation formula for polynomials to the case of spline interpolation, which corresponds to a result of G. Mühlbach for Haar spaces.     

Convexity preserving scattered data interpolation using Powell–Sabin elements

The problem of strict convexity preserving interpolation by a piecewise quadratic smooth function is solved. A constructive procedure is provided.     

Some physical and mathematical properties of inverse distance weighted methods for scattered data interpolation

A class of inverse distance weighted formulas for scattered data interpolation, such as the well-known Shepard one, is reconsidered from a physical viewpoint and in particular with respect to electrostatic or gravitational fields. In this way, simple properties are explained, that permit the construction of either parallel or recursive algorithms to calculate these formulas. The formulas are then interpreted as means in accordance with the definitions of Cauchy, Chisini and others. As a consequence, some significant properties of the means can be used. Finally, a few comments are made on the numerical evaluation of the formulas considered. this work has been supported by the Italian Ministry of Scientific and Technological Research and the National Research Council.     

Computational complexity of spline interpolation

The procedure for composing the spline function of order m that interpolates n data is roughly divided into two stages: (1) constructing the matrix A_n that transforms the fl-spline coefficient vector c into the sample value vector s; and (2) calculating the vector c. Then the effects of boundary conditions and the locations of the sampling points and the knots on the number of computations for spline interpolation are evaluated. There are no boundary condition effects of the construction of A" for equispacing. Non-periodic boundary conditions reduce the number of computations needed to calculate c from O(m²n) to O(m²n/4).

In the construction of A_n, choosing equispaced sampling points reduces the number of computations from O(m²n) to O(m²) in the case of sampling points located at a constant interval between each pair of adjacent knots. In the calculation of c, equispaced sampling points reduce the number of computations from O(m²n) to O(m²n /2) in the case where m is even and the locations of the knots are identical with those of the sampling points.     

基于截面数据的C1插值融合曲面重建

为了避免NURBS曲面重建需要进行节点矢量相容的问题,提出了一种双方向融合插值的[C1]参数曲面重建方法,该方法先后分段插值截面上连续的数据点、截面曲线以构造样条曲线和曲面片,并引入融合算法进行曲线、曲面拼接,从而得到光滑的待建曲面。该方法不会产生由节点插入所带来的大量的数据冗余以及复杂的计算过程,同时采用了融合的思想来处理曲线、曲面的拼接,改良了传统参数曲线、曲面拼接方法需要满足边界条件的缺陷。     

3D scattered data interpolation and approximation with multilevel compactly supported RBFs

In this paper, we propose a hierarchical approach to 3D scattered data interpolation and approximation with compactly supported radial basis functions. Our numerical experiments suggest that the approach integrates the best aspects of scattered data fitting with locally and globally supported basis functions. Employing locally supported functions leads to an efficient computational procedure, while a coarse-to-fine hierarchy makes our method insensitive to the density of scattered data and allows us to restore large parts of missed data. Given a point cloud distributed over a surface, we first use spatial down sampling to construct a coarse-to-fine hierarchy of point sets. Then we interpolate (approximate) the sets starting from the coarsest level. We interpolate (approximate) a point set of the hierarchy, as an offsetting of the interpolating function computed at the previous level. The resulting fitting procedure is fast, memory efficient, and easy to implement.     

A one-pass algorithm for shape-preserving quadratic spline interpolation

Schumaker (1983) and McAllister and Roulier (1981) have proposed algorithms for shape-preserving interpolation using quadratic splines. The former requires the user to provide and perhaps to adjust estimates of the slope at the data points. Here we show that, for a particular slope estimation technique, the two methods are identical, and that in this case the Schumaker algorithm automatically generates shape-preserving interpolants. Furthermore, in case of convex data the slopes are improved iteratively to produce more visually pleasing curves.