An Efficient Trivariate Algorithm for Tetrahedral Shepard Interpolation

Abstract

In this paper we present a trivariate algorithm for fast computation of tetrahedral Shepard interpolants. Though the tetrahedral Shepard method achieves an approximation order better than classical Shepard formulas, it requires to detect suitable configurations of tetrahedra whose vertices are given by the set of data points. In doing that, we propose the use of a fast searching procedure based on the partitioning of domain and nodes in cubic blocks. This allows us to find the nearest neighbor points associated with each ball that need to be used in the 3D interpolation scheme. Numerical experiments show good performance of our interpolation algorithm.

A 3D Efficient Procedure for Shepard Interpolants on Tetrahedra

“…We also denote by e k,ℓ = x j k − x j ℓ , with k, ℓ = 1, 2, 3, 4, the edge vectors of the tetrahedron h j . Then, the following result holds (for the proof see [4]).…”

Numerical differentiation on scattered data through multivariate polynomial interpolation

“…The previous results are useful to estimate the error of approximation in several processes of scattered data interpolation which use polynomials as local interpolants, like for example triangular Shepard [4,11], hexagonal Shepard [10] and tetrahedral Shepard methods [5]. They are also crucial to realize extensions of those methods to higher dimensions [9].…”

Numerical differentiation on scattered data through multivariate polynomial interpolation

“…Remark 2.8. The previous results are useful to estimate the error of approximation in several processes of scattered data interpolation which use polynomials as local interpolants, like for example triangular Shepard [10,3], hexagonal Shepard [9] and tetrahedral Shepard methods [4]. They are also crucial to realize extensions of those methods to higher dimensions [8].…”