Robert J. Renka

Multivariate interpolation of large sets of scattered data

This paper presents a method of constructing a smooth function of two or more variables that interpolates data values at arbitrarily distributed points. Shepards method for fitting a surface to data values at scattered points in the plane has the advantages of a small storage requirement and an easy generalization to more than two independent variables, but suffers from low accuracy and a high computational cost relative to some alternative methods. Localizations of this method have reasonably low computational costs, but remain relatively inaccurate. We describe a modified Shepards method that, without sacrificing the advantages, has accuracy comparable to other local methods. Computational efficiency is also improved by using a cell method for nearest-neighbor searching. Test results for two and three independent variables are presented.

Algorithm 772: STRIPACK: Delaunay triangulation and Voronoi diagram on the surface of a sphere

STRIPACK is a Fortran 77 software package that employs an incremental algorithm to construct a Delaunay triangulation and, optionally, a Voronoi diagram of a set of points (nodes) on the surface of the unit sphere. The triangulation covers the convex hull of the nodes, which need not be the entire surface, while the Voronoi diagram covers the entire surface. The package provides a wide range of capabilities including an efficient means of updating the triangulation with nodal additions or deletions. For <italic>N</italic> nodes, the storage requirement for the triangulation is 13<italic>N</italic> integer storage locations in addition to 3<italic>N</italic> nodal corrdinates. Using an off-line algorithm and work space of size 3<italic>N</italic>, the triangulation can be constructed with time complexity <italic>O(NlogN)</italic>.

Interpolation of data on the surface of a sphere

The problem treated is that of constructing a C ~ interpolant of data values associated with arbitrarily distributed nodes on the surface of a sphere. A local interpolation method that has proved very successful for fitting data on the plane consists of generating a triangulation of the nodes, estimating gradients at the nodes, and constructing a triangle-based mterpolant of the data and gradient estamates Methods and software that extend thas solution procedure to the surface of the sphere are described, and test results are presented. The method is shown to be quite efficient and accurate for data taken from a variety of test functions.

Algorithm 751: TRIPACK: a constrained two-dimensional Delaunay triangulation package

TRIPACK is a Fortran 77 software package that employs an incremental algorithm to construct a constrained Delaunay traingulation of a set of points in the plane (nodes). The triangulation covers the convex hull of the nodes but may include polygonal constraint regions whose triangles are distinguishable from those in the remainder of the triangulation. This effectively allows for a nonconvex or multiply connected triangulation (the complement of the union of constraint regions) while retaining the efficiency of searching and updating a convex triangulation. The package provides a wide range of capabilities including an efficient means of updating the triangulation with nodal additions or deletions. For N nodes, the storage requirement is 13N integer storage locations in addition to the 2N nodal coordinates.

Algorithm 624: Triangulation and Interpolation at Arbitrarily Distributed Points in the Plane

This algorithm is a 1966 American National Standard FORTRAN implementation of the methods discussed in [1] and [2]. The software consists of a set of triangulation modules (which have application in problems other than interpolation) and a set of interpolation modules that require the triangulation routines. The triangulation software constructs a Thiessen triangulatiofi of a set of N nodes (x,, y,), i = 1 . . . . . N, arbitrarily distributed in the x-y plane. Only 7N storage locations are required to represent the triangulation and the software provides the capability of updating the data structure with the addition of a new node. Given data values z, associated with the nodes, the interpolation problem is to construct a C 1 function F such that

Interpolatory tension splines with automatic selection of tension factors

A powerful and versatile method of shape-preserving interpolation is developed in terms of piecewise exponential functions with a tension factor associated with each interval. Knots coincide with data points, and the interpolant is formulated in terms of its values and first derivatives at these points. For a given set of derivatives, this enables the efficient computation of the minimum tension factor for which the interpolant satisfies locally defined properties such as monotonicity and convexity, as well as more general bounds on function values and derivatives, in each interval. A local derivative-estimation procedure results in a

Algorithm 660: QSHEP2D: Quadratic Shepard Method for Bivariate Interpolation of Scattered Data

C^1

Algorithm 792: accuracy test of ACM algorithms for interpolation of scattered data in the plane

interpolant satisfying the constraints with minimum tension, and an iterative procedure can be used to obtain a

Gridding-based direct Fourier inversion of the three-dimensional ray transform

C^2

Algorithm 716: TSPACK: tension spline curve-fitting package

spline fit which satisfies the constraints. Test results are presented which show both methods to produce visually pleasing interpolants to various data sets.