Smoothing nonlinear subdivision schemes by averaging

Abstract

In the theory of linear subdivision algorithms, it is well-known that the regularity of a linear subdivision scheme can be elevated by one order (say, from Ck to Ck+1) by composing it with an averaging step (equivalently, by multiplying to the subdivision mask a(z) a (1 + z) factor. In this paper, we show that the same can be done to nonlinear subdivision schemes: by composing with it any nonlinear, smooth, 2-point averaging step, the lifted nonlinear subdivision scheme has an extra order of regularity than the original scheme. A notable application of this result shows that the classical Lane-Riesenfeld algorithm for uniform B-Spline, when extended to Riemannian manifolds based on geodesic midpoint, produces curves with the same regularity as their linear counterparts. (In particular, curvature does not obstruct the nonlinear Lane-Riesenfeld algorithm to inherit regularity from the linear algorithm.) Our main result uses the recently developed technique of differential proximity conditions.