Efficient computation of partition of unity interpolants through a block-based searching technique

TL;DR

This paper introduces a new block-based space partitioning method to efficiently compute partition of unity interpolants for large scattered datasets, significantly improving search efficiency in multidimensional interpolation tasks.

Contribution

A novel block-based data structure for domain partitioning enhances the efficiency of partition of unity interpolation with RBFs for large datasets.

Findings

Reduced computational complexity demonstrated in numerical experiments

Effective in 2D and 3D geometric modeling applications

Software implementation in MATLAB available for community use

Abstract

In this paper we propose a new efficient interpolation tool, extremely suitable for large scattered data sets. The partition of unity method is used and performed by blending Radial Basis Functions (RBFs) as local approximants and using locally supported weight functions. In particular we present a new space-partitioning data structure based on a partition of the underlying generic domain in blocks. This approach allows us to examine only a reduced number of blocks in the search process of the nearest neighbour points, leading to an optimized searching routine. Complexity analysis and numerical experiments in two- and three-dimensional interpolation support our findings. Some applications to geometric modelling are also considered. Moreover, the associated software package written in \textsc{Matlab} is here discussed and made available to the scientific community.