Hexagonal Versus Orthogonal Lattices: A New Comparison Using Approximation Theory

We provide a new comparison between hexagonal and orthogonal lattices, based on approximation theory. For each of the lattices, we select the “natural” spline basis function as generator for a shift-invariant function space; i.e., the tensor-product B-splines for the orthogonal lattice and the non-separable hex-splines for the hexagonal lattice. For a given order of approximation, we compare the asymptotic constants of the error kernels, which give a very good indication of the approximation quality. We find that the approximation quality on the hexagonal lattice is consistently better, when choosing lattices with the same sampling density. The area sampling gain related to these asymptotic constants quickly converges when the order of approximation of the basis functions increases. Surprisingly, nearest-neighbor interpolation does not allow to profit from the hexagonal grid. For practical purposes, the second-order hex-spline (i.e., constituted by linear patches) appears as a particularly useful candidate to exploit the advantages of hexagonal lattices when representing images on them.