Analysis of symmetry groups of box-splines for evaluation on GPUs

In this paper we analyze the symmetry groups of box-splines for efficient analytic evaluation of splines and their derivatives on GPUs (Graphics Processing Units). Given a box-spline, we first analyze its polynomial structure and find its space group which is composed of a point group and a translational group on the domain lattice. To evaluate a spline generated by the box-spline (or its derivative) function, the point group is decomposed into right cosets such that all the polytopes in the same coset share the same analytic polynomial formula. Moreover, by leveraging their symmetries, sufficient number of linearly independent derivative functions of the same order are chosen such that they have a change-of-variables relation with each other. Our OpenCL implementations show that our method is at least ≈ 30% faster but the kernel is at least ≈ 30% smaller compared with the other techniques.