A 3D Efficient Procedure for Shepard Interpolants on Tetrahedra

Abstract

The need of scattered data interpolation methods in the multivariate framework and, in particular, in the trivariate case, motivates the generalization of the fast algorithm for triangular Shepard method. A block-based partitioning structure procedure was already applied to make the method very fast in the bivariate setting. Here the searching algorithm is extended, it allows to partition the domain and nodes in cubic blocks and to find the nearest neighbor points that need to be used in the tetrahedral Shepard interpolation.

Vector-Valued Shepard Processes: Approximation with Summability

“…The Shepard operators, initially defined by Donald Shepard in 1968 [1], are effectively employed in a wide array of fields, ranging from mathematics to engineering, and from geographical mapping systems to mining, owing to their interpolation capabilities as well as their ability to approximate functions more rapidly. These operators are quite successful not only in scattered data interpolation problems (see [2][3][4][5][6][7][8]) but also in the classical approximation theory (see [9][10][11][12][13][14][15]). Recently in [16,17], we investigated the approximation behavior of the Shepard operators in complex cases.…”