Stationary subdivision schemes have been extensively studied and have numerous applications in CAGD and wavelet analysis. To have high-order smoothness of the scheme, it is usually inevitable to enlarge the support of the mask that is used, which is a major difficulty with stationary subdivision schemes due to complicated implementation and dramatically increased special subdivision rules at extraordinary vertices. In this paper, we introduce the notion of a multivariate quasi-stationary subdivision scheme and fully characterize its convergence and smoothness. We will also discuss the general procedure of designing interpolatory masks with short support that yields smooth quasi-stationary subdivision schemes. Specifically, using the dyadic dilation of both triangular and quadrilateral meshes, for each smoothness exponent $m=1,2$, we obtain examples of $C^m$-convergent quasi-stationary $2I_2$-subdivision schemes with bivariate symmetric masks having at most $m$-ring stencils. Our examples demonstrate the advantage of quasi-stationary subdivision schemes, which can circumvent the difficulty above with stationary subdivision schemes.

翻译：平稳细分格式已被广泛研究，并在计算机辅助几何设计与小波分析中具有众多应用。为使格式具有高阶光滑性，通常不可避免地需要扩大所用掩模的支集，这是平稳细分格式的主要难点，原因在于实现复杂且在奇异顶点处特殊细分规则急剧增加。本文引入多元拟平稳细分格式的概念，并完整刻画其收敛性与光滑性。我们还将讨论设计具有短支集的插值掩模以产生光滑拟平稳细分格式的一般流程。具体而言，利用三角形与四边形网格的二进膨胀，对每个光滑指数$m=1,2$，我们获得了具有双变量对称掩模（至多$m$环模板）的$C^m$收敛拟平稳$2I_2$细分格式的实例。我们的算例证明了拟平稳细分格式的优势——能够规避上述平稳细分格式所面临的困难。