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...     

薄板样条函数在空间数据插值中的应用

薄板样条函数是空间数据插值中一种重要的方法,介绍了该方法的基本原理,并以珠江河道地形数据为例,借助地理信息系统的二次开发功能,将薄板样条函数应用于空间插值,通过与测试样本点以及克里金插值在最大值、标准误等方面的比较分析,证明薄板样条函数是一种有效的空间数据插值方法。     

A convex version of multivariate adaptive regression splines

Multivariate adaptive regression splines (MARS) provide a flexible statistical modeling method that employs forward and backward search algorithms to identify the combination of basis functions that best fits the data and simultaneously conduct variable selection. In optimization, MARS has been used successfully to estimate the unknown functions in stochastic dynamic programming (SDP), stochastic programming, and a Markov decision process, and MARS could be potentially useful in many real world optimization problems where objective (or other) functions need to be estimated from data, such as in surrogate optimization. Many optimization methods depend on convexity, but a non-convex MARS approximation is inherently possible because interaction terms are products of univariate terms. In this paper a convex MARS modeling algorithm is described. In order to ensure MARS convexity, two major modifications are made: (1) coefficients are constrained, such that pairs of basis functions are guaranteed to jointly form convex functions and (2) the form of interaction terms is altered to eliminate the inherent non-convexity. Finally, MARS convexity can be achieved by the fact that the sum of convex functions is convex. Convex-MARS is applied to inventory forecasting SDP problems with four and nine dimensions and to an air quality ground-level ozone problem.     

Fair, G- and C-continuous circle splines for the interpolation of sparse data points

This paper presents a carefully chosen curve blending scheme between circles, which is based on angles, rather than point positions. The result is a class of circle splines that robustly produce fair-looking G²-continuous curves without any cusps or kinks, even through rather challenging, sparse sets of interpolation points. With a simple reparameterization the curves can also be made C²-continuous. The same method is usable in the plane, on the sphere, and in 3D space.     

Upsilon-quaternion splines for the smooth interpolation of orientations

We present a new method for smoothly interpolating orientation matrices. It is based upon quaternions and a particular construction of /spl nu/-spline curves. The new method has tension parameters and variable knot (time) spacing which both prove to be effective in designing and controlling key frame animations.     

Spherical splines and orientation interpolation

This paper presents a method for designing spherical curves by two weighted spatial rotations. This approach is for the design of interpolating spherical curves and orientation interpolation. The same approach can be used for smoothing orientations or corners on a sphere. The designed curves have the following features: C¹ continuity, local control, and invariance under orthogonal transformations of coordinate systems.     

Quasi interpolation with Voronoi splines

We present a quasi interpolation framework that attains the optimal approximation-order of Voronoi splines for reconstruction of volumetric data sampled on general lattices. The quasi interpolation framework of Voronoi splines provides an unbiased reconstruction method across various lattices. Therefore this framework allows us to analyze and contrast the sampling-theoretic performance of general lattices, using signal reconstruction, in an unbiased manner. Our quasi interpolation methodology is implemented as an efficient FIR filter that can be applied online or as a preprocessing step. We present visual and numerical experiments that demonstrate the improved accuracy of reconstruction across lattices, using the quasi interpolation framework.     

Spatial interpolation of large climate data sets using bivariate thin plate smoothing splines

Thin plate smoothing splines are widely used to spatially interpolate surface climate, however, their application to large data sets is limited by computational efficiency. Standard analytic calculation of thin plate smoothing splines requires O(n³) operations, where n is the number of data points, making routine computation infeasible for data sets with more than around 2000 data points. An O(N) iterative procedure for calculating finite element approximations to bivariate minimum generalised cross validation (GCV) thin plate smoothing splines operations was developed, where N is the number of grid points. The key contribution of the method lies in the incorporation of an automatic procedure for optimising smoothness to minimise GCV. The minimum GCV criterion is commonly used to optimise thin plate smoothing spline fits to climate data. The method discretises the bivariate thin plate smoothing spline equations using hierarchical biquadratic B-splines, and uses a nested grid multigrid procedure to solve the system. To optimise smoothness, a double iteration is incorporated, whereby the estimate of the spline solution and the estimate of the optimal smoothing parameter are updated simultaneously. When the method was tested on temperature data from the African and Australian continents, accurate approximations to analytic solutions were obtained.     

Shape-preserving interpolation of irregular data by bivariate curvature-based cubic splines in spherical coordinates

We investigate C¹-smooth bivariate curvature-based cubic L₁ interpolating splines in spherical coordinates. The coefficients of these splines are calculated by minimizing an integral involving the L₁ norm of univariate curvature in four directions at each point on the unit sphere. We compare these curvature-based cubic L₁ splines with analogous cubic L₂ interpolating splines calculated by minimizing an integral involving the square of the L₂ norm of univariate curvature in the same four directions at each point. For two sets of irregular data on an equilateral tetrahedron with protuberances on the faces, we compare these two types of curvature-based splines with each other and with cubic L₁ and L₂ splines calculated by minimizing the L₁ norm and the square of the L₂ norm, respectively, of second derivatives. Curvature-based cubic L₁ splines preserve the shape of irregular data well, better than curvature-based cubic L₂ splines and than second-derivative-based cubic L₁ and L₂ splines. Second-derivative-based cubic L₂ splines preserve shape poorly. Variants of curvature-based L₁ and L₂ splines in spherical and general curvilinear coordinate systems are outlined.     

Positive interpolation with rational quadratic splines

A necessary and sufficient criterion is presented under which the property of positivity carry over from the data set to rational quadratic spline interpolants. The criterion can always be satisfied if the occuring parameters are properly chosen.     

Convexity preserving interpolation with exponential splines

Sufficient and necessary conditions are derived under which interpolating splines are convex if the data set is in convex position. In order to select one of the interpolants, by means of a well-known objective function a quadratic optimization problem is stated which can be solved effectively by passing to a dual program.     

Uniform convergence of interpolation by cubic splines

For a sequence of meshes on [0, 1] sufficient conditions are given to obtain uniform convergence of cubic spline interpolants for continous functions respectively for the third derivatives of cubic spline interpolants for functions fromC ³ [0, 1].     

Free knot splines for biochemical data

The use of spline functions in the analysis of empirical two-dimensional (2-D) data (y(i), x(i)) is described. Spline functions are excellent empirical functions, which can be used with advantage instead of other ones, such as polynomials or exponentials. The knot location seen as variable value corresponds to classical parameter used to describe oxidation curves. An application on characterization of LDL oxidability shows free knot splines in a regression context.     

An explicit method ofC 2 interpolation using splines

The interpolation of a discrete set of data, on the interval [a, b], representing the functionf is obtained using explicit splines. Estimations of interpolation accuracy are obtained.     

A new family of life distributions for dependent data: Estimation

One of the biggest problems in reliability analysis is determining an appropriate distribution of life data. Therefore, this paper develops the estimation aspect of a family of life distributions obtained from spherical distributions. Additionally, a new family of life distributions is proposed for dependent life data, together with an optimization algorithm based on the simulated annealing method. This algorithm is very efficient for optimization purposes and does not require any manipulation of the log-likelihood functions for the distributions proposed in this study.     

A new family of life distributions for dependent data: Estimation

One of the biggest problems in reliability analysis is determining an appropriate distribution of life data. Therefore, this paper develops the estimation aspect of a family of life distributions obtained from spherical distributions. Additionally, a new family of life distributions is proposed for dependent life data, together with an optimization algorithm based on the simulated annealing method. This algorithm is very efficient for optimization purposes and does not require any manipulation of the log-likelihood functions for the distributions proposed in this study.     

具有局部性质的球面插值样条曲线的构造

高维球面样条曲线拟合技术在计算机动画和惯性导航等领域都受到广泛地关注.实际中常需球面曲线插值给定的数据点,并要求曲线具有一定的连续性和良好的局部性质.此前的方法存在一定的局限性.为此,基于球面Bézier曲线,提出了一种仅利用插值点位置信息便可在任意维空间中构造C 2球面插值样条曲线的新方法.首先,通过映射拟合出了插值...     

Monotone and convex cubic spline interpolation

Given a set of monotone and convex data, we present a necessary and sufficient condition for the existence of cubic differentiable interpolating splines which are monotone and convex. Further, we discuss their approximation properties when applied to the interpolation of functions having preassigned degree of smoothness.