Numerical methods for checking the regularity of subdivision schemes
For researchers in computer graphics and animation, this work provides numerical tools to assess subdivision regularity, though it is an incremental extension of existing theoretical results.
The paper develops numerical methods to check C^k-regularity of multivariate subdivision schemes, showing that joint and restricted spectral radius approaches yield different numerical schemes in the multivariate case, with examples confirming the theory.
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.