Fageot Julien, Aziznejad Shayan, Unser Michael, Uhlmann Virginie
Biomedical Imaging Group, École polytechnique fédérale de Lausanne (EPFL), Station 17, 1015 Lausanne, Switzerland.
European Bioinformatics Institute (EMBL-EBI), Wellcome Genome Campus, Cambridge CB10 1SD, UK.
J Comput Appl Math. 2020 Apr;368:112503. doi: 10.1016/j.cam.2019.112503.
In this paper, we formally investigate two mathematical aspects of Hermite splines that are relevant to practical applications. We first demonstrate that Hermite splines are maximally localized, in the sense that the size of their support is minimal among pairs of functions with identical reproduction properties. Then, we precisely quantify the approximation power of Hermite splines for the reconstruction of functions and their derivatives. It is known that the Hermite and B-spline approximation schemes have the same approximation order. More precisely, their approximation error vanishes as when the step size goes to zero. In this work, we show that they actually have the same asymptotic approximation error constants, too. Therefore, they have identical asymptotic approximation properties. Hermite splines combine optimal localization and excellent approximation power, while retaining interpolation properties and closed-form expression, in contrast to existing similar functions. These findings shed a new light on the convenience of Hermite splines in the context of computer graphics and geometrical design.
在本文中，我们正式研究了埃尔米特样条在实际应用中相关的两个数学方面。我们首先证明，从具有相同再生特性的函数对中，埃尔米特样条的支撑集大小最小这一意义上来说，它具有最大局部性。然后，我们精确量化了埃尔米特样条在函数及其导数重构方面的逼近能力。众所周知，埃尔米特样条和B样条逼近方案具有相同的逼近阶数。更确切地说，当步长趋于零时，它们的逼近误差会消失。在这项工作中，我们还表明它们实际上也具有相同的渐近逼近误差常数。因此，它们具有相同的渐近逼近性质。与现有的类似函数相比，埃尔米特样条结合了最优局部性和出色的逼近能力，同时保留了插值性质和封闭形式表达式。这些发现为埃尔米特样条在计算机图形学和几何设计中的便利性提供了新的视角。
