Fast Summation of Functions on the Rotation Group
Fast Summation of Functions on the Rotation Group
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DOI：
10.1007/s11004-010-9281-x
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发表时间：
2010-06
影响因子：
2.6
通讯作者：
R. Hielscher;Jürgen Prestin;A. Vollrath
中科院分区：
文献类型：
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作者：
R. Hielscher;Jürgen Prestin;A. Vollrath
用旋转群上的函数进行计算是在各种应用领域中执行的任务。当涉及到近似时，基于内核的方法是处理这些函数的合适工具。在本文中，我们提出了一个算法，它允许我们评估的旋转组上的函数的线性组合，以及一个真正的快速算法来总结旋转组上的径向函数。这些方法基于SO（3）上的非等间距FFT，分别对任意分布的Mand N个源节点和目标节点进行算术运算。在本文中，我们研究了径向函数的选择，并给出明确的理论误差界，以及数值例子的逼近误差。此外，我们提供了我们的方法的应用，即核密度估计从电子背散射衍射（EBSD）数据，在纹理分析相关的问题。
Computing with functions on the rotation group is a task carried out in various areas of application. When it comes to approximation, kernel based methods are a suitable tool to handle these functions. In this paper, we present an algorithm which allows us to evaluate linear combinations of functions on the rotation group as well as a truly fast algorithm to sum up radial functions on the rotation group. These approaches based on nonequispaced FFTs on SO(3) takearithmetic operations forMandNarbitrarily distributed source and target nodes, respectively. In this paper, we investigate a selection of radial functions and give explicit theoretical error bounds, as well as numerical examples of approximation errors. Moreover, we provide an application of our method, namely the kernel density estimation from electron back scattering diffraction (EBSD) data, a problem relevant in texture analysis.