检查细分格式正则性的数值方法
Numerical methods for checking the regularity of subdivision schemes
AI总结：
针对计算机图形学和动画中的细分格式，本文利用联合谱半径和受限谱半径方法，提出并比较了检查向量多变量细分格式C^k正则性的数值方案。
摘要：
本文受计算机图形学和动画应用的启发，研究了检查具有膨胀矩阵2I的向量多变量细分格式$C^k$正则性的数值方法。这些数值方法源于联合谱半径和受限谱半径方法，Charina (Charina, 2011) 已证明这两种方法通过相同的量刻画细分的$W^k_p$正则性，即$(k,p)$-联合谱半径与$(k,p)$-受限谱半径相等。我们表明，在单变量标量和向量情形下，对于特定矩阵范数的选择，相应的数值方法甚至给出$(k,\infty)$-联合谱半径相同的上界估计。两种方法的差异在多变量情形下变得明显，我们确认它们确实提供了估计细分正则性的不同数值方案。我们通过几个例子说明了结果。
英文摘要：
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.
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