Elementary factorisation of Box spline subdivision

Abstract

When a subdivision scheme is factorised into lifting steps, it admits an in-place and invertible implementation, and it can be the predictor of many multiresolution biorthogonal wavelet transforms. In the regular setting where the underlying lattice hierarchy is defined by Z s and a dilation matrix M , such a factorisation should deal with every vertex of each subset in Z s /M Z s in the same way. We define a subdivision scheme which admits such a factorisation as being uniformly elementary factorable. We prove a necessary and sufficient condition on the directions of the Box spline and the arity of the subdivision for the scheme to admit such a factorisation, and recall some known keys to construct it in practice.

Completeness characterization of Type-I box splines

“…Truncated hierarchical type-I box splines were considered in [22] and [19] in connection to isogeometric analysis applications. Other subdivision schemes has been explored in [15]. A C 1 -continuous scheme based on cubic half-box splines was presented in [1].…”