Refinability of splines from lattice Voronoi cells
This addresses a theoretical limitation in spline construction for computational geometry and approximation theory, identifying non-refinable cases that may degrade performance in applications like image processing or numerical analysis.
The paper tackled the problem of determining which spline families constructed from lattice Voronoi cells are refinable, finding that only box splines and tensor-product splines are refinable, while others like hex-splines are not, and demonstrated that non-refinable splines can lead to increased approximation error upon lattice refinement.
Splines can be constructed by convolving the indicator function of the Voronoi cell of a lattice. This paper presents simple criteria that imply that only a small subset of such spline families can be refined: essentially the well-known box splines and tensor-product splines. Among the many non-refinable constructions are hex-splines and their generalization to non-Cartesian lattices. An example shows how non-refinable splines can exhibit increased approximation error upon refinement of the lattice.