Numerical methods for checking the regularity of subdivision schemes

TL;DR

This paper develops and compares numerical methods based on spectral radius concepts to determine the regularity of multivariate subdivision schemes, with applications in computer graphics and animation.

Contribution

It introduces and analyzes numerical schemes derived from joint and restricted spectral radius approaches for checking subdivision regularity, highlighting differences in multivariate cases.

Findings

Spectral radius methods characterize regularity of subdivision schemes.

Univariate scalar and vector cases yield similar upper estimates.

Multivariate cases show distinct numerical schemes for regularity estimation.

Abstract

In this paper, motivated by applications in computer graphics and animation, we study the numerical methods for checking $C^k-$regularity of vector multivariate subdivision schemes with dilation 2I. These numerical methods arise from the joint spectral radius and restricted spectral radius approaches, which were shown in Charina (Charina, 2011) to characterize $W^k_p-$regularity of subdivision in terms of the same quantity. Namely, the $(k,p)-$joint spectral radius and the $(k,p)-$restricted spectral radius are equal. We show that the corresponding numerical methods in the univariate scalar and vector cases even yield the same upper estimate for the $(k,\infty)-$joint spectral radius for a certain choice of a matrix norm. The difference between the two approaches becomes apparent in the multivariate case and we confirm that they indeed offer different numerical schemes for estimating the regularity of subdivision. We illustrate our results with several examples.