We analyse the convergence of nonlinear Riemannian analogues of linear subdivision processes operating on data in the sphere. We show how for curve subdivision rules we can derive bounds guaranteeing convergence if the density of input data is below that threshold. Previous results only yield thresholds that are several magnitudes smaller and are thus useless for a priori checking of convergence. It is the first time that such a result has been shown for a geometry with positive curvature and for subdivision rules not enjoying any special properties like being interpolatory or having non-negative mask.

中文翻译：

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球面细分过程的收敛性分析

我们分析了对球体中的数据进行操作的线性细分过程的非线性黎曼类比的收敛性。我们展示了对于曲线细分规则，如何在输入数据的密度低于该阈值的情况下得出保证收敛的边界。先前的结果仅产生了小几个数量级的阈值，因此对先验收敛没有用。对于具有正曲率的几何图形和细分规则不具有任何特殊属性（例如内插或非负蒙版）的情况，这是首次显示这种结果。