How Data Dependent is a Nonlinear Subdivision Scheme? A Case Study Based on Convexity Preserving Subdivision

Abstract

The regularity of the limit function of a linear subdivision scheme is essentially irrelevant to the initial data. How data dependent, then, is the regularity of the limit of a nonlinear subdivision scheme? The answer is the most obvious---it depends. In this paper, we prove that the nonlinear convexity preserving subdivision scheme developed independently by Floater and Micchelli [M. S. Floater and C. A. Micchelli, Approximation Theory, Marcel Dekker, New York, 1998, pp. 209-224] and Kuijt and van Damme [F. Kuijt and R. van Damme, Constr. Approx., 14 (1998), pp. 609-630] exhibits a rather strong nonlinear, data-dependent, behavior: For any $\nu \in (1,2)$, there exists initial convex data such that the critical Holder regularity of the limit curve is exactly $\nu$. We also show that the limit function of any initial data always has Holder regularity less than 2, unless if restricted to a subset of the domain at which the initial data is sampled from the convex branch of a rational polynomial of degree 2 over degree 1. This result stands in contrast to what is reported in several recent publications on nonlinear subdivision schemes [I. Ur Rahman, I. Drori, V. C. Stodden, D. L. Donoho, and P. Schr{oder, SIAM J. Multiscale Modeling and Simulation, submitted, 2005; G. Xie and T. P.-Y. Yu, Constr. Approx., 22 (2005), pp. 219-254; G. Xie and T. P.-Y. Yu, Advances in Constructive Approximation, 2004, pp. 519-533; I. Daubechies, O. Runborg, and W. Sweldens, Constr. Approx., 3 (2004), pp. 399-463; T. P.-Y. Yu, Cutting corners on the sphere, preprint available at http://www.rpi.edu/~yut/Papers/CuttingCorners.pdf, 2005], in which various families of nonlinear subdivision schemes are either proved or empirically observed to have rather weak nonlinearity in the sense that they produce limit curves with smoothness insensitive to initial data.