A General Proximity Analysis of Nonlinear Subdivision Schemes
A General Proximity Analysis of Nonlinear Subdivision Schemes
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DOI：
10.1137/09075963x
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发表时间：
2010-03
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影响因子：
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通讯作者：
P. Grohs
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文献类型：
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作者：
P. Grohs
在最近的工作中，利用所谓的接近条件来限定线性格式与非线性格式之间的区别，成功地分析了作用于流形值数据的非线性细分格式。这种方法的主要困难在于对这些条件的验证。在本文中，我们非常清楚地认识到非线性格式必须满足哪些性质才能满足邻近条件。为此，我们引入了线性细分格式的一个新的多项式生成性质，并通过掩模系数矩的简单乘法性质得到了该性质的表征。作为我们结果的一个主要应用，我们证明了用黎曼质心代替线性平均定义的线性细分格式的黎曼模拟满足任意阶的接近条件。作为一个推论，我们得出黎曼模拟总是产生极限曲线在…
In recent work nonlinear subdivision schemes which operate on manifold-valued data have been successfully analyzed with the aid of so-called proximity conditions bounding the difference between a linear scheme and the nonlinear one. The main difficulty with this method is the verification of these conditions. In the present paper we obtain a very clear understanding of which properties a nonlinear scheme has to satisfy in order to fulfill proximity conditions. To this end we introduce a novel polynomial generation property for linear subdivision schemes and obtain a characterization of this property via simple multiplicativity properties of the moments of the mask coefficients. As a main application of our results we prove that the Riemannian analogue of a linear subdivision scheme which is defined by replacing linear averages by the Riemannian center of mass satisfies proximity conditions of arbitrary order. As a corollary we conclude that the Riemannian analogue always produces limit curves which are at...