A closed PP form of box splines via Green's function decomposition

For the class of non-degenerate box splines, we present a set construction scheme that separably decomposes the Green's function of a box spline, yielding its explicit piecewise polynomial form. While it is possible to use the well known recursive formulation to obtain these polynomial pieces, that procedure is quite expensive. We prove that, under certain conditions, our decomposition procedure is asymptotically orders of magnitude lower than the recursive procedure. This allows us to evaluate box splines with more direction vectors than what would be feasible under the recursive scheme. Finally, using the explicit polynomials in each region of the box spline, we show how to create fast evaluation schemes using this explicit characterization and a spatial data structure.