This paper presents a method of constructing a smooth function of two or more variables that interpolates data values at arbitrarily distributed points. Shepard's 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 Shepard's 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.

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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 N nodes, the storage requirement for the triangulation is 13 N integer storage locations in addition to 3 N nodal corrdinates. Using an off-line algorithm and work space of size 3 N , the triangulation can be constructed with time complexity O(NlogN) .

We describe a fast and accurate direct Fourier method for reconstructing a function f of three variables from a number of its parallel beam projections. The main application of our method is in single particle analysis, where the goal is to reconstruct the mass density of a biological macromolecule. Typically, the number of projections is extremely large, and each projection is extremely noisy. The projection directions are random and initially unknown. However, it is possible to determine both the directions and f by an iterative procedure; during each stage of the iteration, one has to solve a reconstruction problem of the type considered here. Our reconstruction algorithm is distinguished from other direct Fourier methods by the use of gridding techniques that provide an efficient means to compute a uniformly sampled version of a function g from a nonuniformly sampled version of Fg, the Fourier transform of g, or vice versa. We apply the two-dimensional reverse gridding method to each available projection of f, the function to be reconstructed, in order to obtain Ff on a special spherical grid. Then we use the three-dimensional gridding method to reconstruct f from this sampled version of Ff. This stage requires a proper weighting of the samples of Ff to compensate for their nonuniform distribution. We use a fast method for computing appropriate weights that exploits the special properties of the spherical sampling grid for Ff and involves the computation of a Voronoi diagram on the unit sphere. We demonstrate the excellent speed and accuracy of our method by using simulated data.

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.