Summary

Hex-splines are a new type of bivariate splines especially designed for hexagonal lattices. Inspired by the indicator function of the Voronoi cell, they are able to preserve the isotropy of the hexagonal lattice (as opposed to their B-spline counterparts). They can be constructed for any order and are piecewise polynomial (on a triangular mesh). Analytical formulas have been worked out in both spatial and Fourier domains. For orthogonal lattices, the hex-splines revert to the classical tensor-product B-splines. While the standard approach to represent two-dimensional data uses orthogonal lattices, hexagonal lattices provide several advantages, including a higher degree of symmetry and a better packing density.

Main Contribution

We present a thorough mathematical analysis of this new bivariate spline family. In particular, we derive a closed form for a hex-spline of arbitrary order. We also discuss important properties, such as their Fourier transform, the fact they form a Riesz basis, approximation order, and so on. Additionally, we discuss how to apply them advantageously for image processing. We show examples of interpolation and least-squares resampling.

The following page contains some examples and a Maple 7 worksheet to generate the analytical form of any hex-spline.