On the Approximation Power of Splines: Orthogonal Versus Hexagonal Lattices

Recently, we have proposed a novel family of bivariate, non-separable splines. These splines, called "hexsplines" have been designed to deal with hexagonally sampled data. Incorporating the shape of the Voronoi cell of a hexagonal lattice, they preserve the twelve-fold symmetry of the hexagon tiling cell. Similar to B-splines, we can use them to provide a link between the discrete and the continuous domain, which is required for many fundamental operations such as interpolation and resampling. The question we answer in this paper is "How well do the hex-splines approximate a given function in the continuous domain?" and more specifically "How do they compare to separable B-splines deployed on a lattice with the same sampling density?"