New optimized spline functions for interpolation on the hexagonal lattice

Abstract

We propose new discrete-to-continuous interpolation models for hexagonally sampled data, that generalize two families of splines developed in the literature for the hexagonal lattice, to say the hexsplines and three directional box-splines. This extension is inspired by the construction of MOMS functions in 1-D, that generalize and outperform classical 1-D B-splines [1]. Our new generators have optimal approximation theoretic performances, for exactly the same computation cost as their spline counterparts.

Refinability of splines derived from regular tessellations

“…That paper compares hex-splines to tensor-product splines and uses the Fourier transform of hex-splines to derive, for low frequencies, the L 2 approximation order, as a combination of the projection into the hex-spline space and a quasi-interpolation error. [CvB05] derived quasi-interpolation formulas and showed promising results when applying hex-splines to the reconstruction of images (see also [CvU06,Cv07,Cv08]). Van De Ville et al [vBU + 04].…”

Solving the guiding-center model on a regular hexagonal mesh

“…The derivatives along the three directions can also be nicely computed using the regular finite difference method along the three directions. And last but not least, there is a spline construction on this mesh, called Box-spline [13]. These splines have a hexagonal support and are invariant by translations along the three directions of the mesh.…”