Richard Franke

Smooth Interpolation of Scattered Data by Local Thin Plate Splines

Abstract An algorithm and the corresponding computer program for solution of the scattered data interpolation problem is described. Given points ( x k , y k , f k ), k = 1,…, N a locally defined function F ( x , y ) which has the property F ( x k , y k ) = f k , k = 1,…, N is constructed. The algorithm is based on a weighted sum of locally defined thin plate splines, and yields an interpolation function which is differentiable. The program is available from the author.

Scattered Data Interpolation and Applications: A Tutorial and Survey

The multivariate scattered data interpolation problem is introduced and the reasons for the difficulty of the problem compared to the one dimensional case are discussed. Basic ideas for interpolation (or approximation) of scattered data are introduced. Various types of data sets and some strategies for dealing with some of them are given. Readily available algorithms for the solution of the problem are discussed and suitability for various types of data, along with discussion of situations where they have been useful is given. Some related ideas are briefly mentioned. Throughout there are bountiful references to the existing literature.

Interpolation of scattered data on closed surfaces

Abstract Techniques are presented for the construction and visualization of a function defined over a closed surface domain which depends on a discrete sample of measurements at arbitrary locations on the domain surface. The interpolating function is constructed by first finding a one-to-one correspondence between the closed surface domain and a sphere, and then a corresponding scattered data interpolation problem over the sphere is solved. Visualization techniques include color blended contour regions on the domain surface and transparent surface graphs projected from the domain.

Thin plate splines with tension

The equation of an infinite thin plate under the influence of point loads and mid-plane forces is developed. The properties of the function as the tension goes to zero or becomes large is investigated. This function is then used to interpolate scattered data, giving the user the parameter of tension to give some control over overshoot when the surface has large gradients. Examples illustrating the behavior of the interpolation function are given.

RECENT ADVANCES IN THE APPROXIMATION OF SURFACES FROM SCATTERED DATA

Advances in the mathematical theory behind Hardys multiquadric method, development of methods for surfaces with tension parameters or which satisfy constraints, and methods for least squares approximation and subset selection are discussed.

Computing the separating surface for segmented data

An algorithm for computing a triangulated surface which separates a collection of data points that have been segmented into a number of different classes is presented. The problem generalizes the concept of an isosurface which separates data points that have been segmented into only two classes: those for which data function values are above the threshold and those which are below the threshold value. The algorithm is very simple, easy to implement and applies without limit to the number of classes.

Least Squares Surface Approximation to Scattered Data Using Multiquadric Functions

The paper documents an investigation into some methods for fitting surfaces to scattered data. The form of the fitting function is a multiquadratic function with the criteria for the fit being the least mean squared residual for the data points. The principal problem is the selection of knot points (or base points for the multiquadratic basis functions), although the selection of the multiquadric parameter also plays a nontrivial role in the process. We first describe a greedy algorithm for knot selection, and this procedure is used as an initial step in what follows. The minimization including knot locations and the multiquadric parameter is explored, with some unexpected results in terms of “near repeated” knots. This phenomenon is explored, and leads us to consider variable parameter values for the basis functions. Examples and results are given throughout.

A BIBLIOGRAPHY OF MULTIVARIATE APPROXIMATION

Publisher Summary This chapter presents a bibliography of multivariate approximation. Multivariate interpolation and approximation have always been an important part of general Approximation Theory, however, in the past ten years or so, there has been a significant increased interest in multivariate problems. A very substantial number of both theoretical and practical results have been obtained on a variety of topics, ranging from the classical tensor-product methods to the newfangled multivariate splines. In addition to their applicability to such classical problems as data fitting and numerical solution of operator equations, modern multivariate methods have become important in several new applied areas, such as computer-aided design, robotics, and computer-aided tomography.

Least squares surface approximation using multiquadrics and parametric domain distortion

A method for approximating scattered data using multiquadric approximation on a transformed domain region is given. With the imposition of a one-to-one constraint on the domain transformation, all properties of the approximating space carry over. Such a domain transformation is defined in terms of a biquadratic Bezier representation. The parameters in the transformation, and the location of the base points (or knots) in the multiquadric approximation are simultaneously determined using a nonlinear least squares paradigm. The results of some comparisons with previous approaches are given and discussed.

Sources of Error in Objective Analysis

Abstract The errors in objective analysis methods that are based on corrections to first-guess fields are considered. An expression that gives a decomposition of an error into three independent components is derived. To test the magnitudes of the contributions of each component, a series of computer simulations was conducted. The grid-to-observation-point interpolation schemes considered ranged from simple piecewise linear functions to highly accurate spline functions. The observation-to-grid interpolation methods considered included most of those in present meteorological use, such as optimum interpolation and successive corrections, as well as proposed schemes such as thin plate splines, and several variations of these schemes. The results include an analysis of cost versus skill; this information is summarized in plots for most combinations. The degradation in performance due to inexact parameter specification in statistical observation-to-grid interpolation schemes is addressed. The efficacy of the me...