Comparison of 2D Regular Lattices for the CPWL Approximation of Functions

TL;DR

This paper compares different 2D regular lattices for CPWL function approximation, finding that hexagonal lattices provide the minimal asymptotic error, thus offering an optimal structure for such approximations.

Contribution

It introduces a systematic comparison of 2D regular lattices for CPWL approximation, highlighting the optimality of hexagonal lattices in minimizing approximation error.

Findings

Hexagonal lattices minimize asymptotic approximation error.

Approximation error depends on lattice stepsize and angles.

Hexagonal lattice is optimal among tested regular lattices.

Abstract

We investigate the approximation error of functions with continuous and piecewise-linear (CPWL) representations. We focus on the CPWL search spaces generated by translates of box splines on two-dimensional regular lattices. We compute the approximation error in terms of the stepsize and angles that define the lattice. Our results show that hexagonal lattices are optimal, in the sense that they minimize the asymptotic approximation error.