广义球面Radon变换及其在纹理分析中的应用
The Generalized Spherical Radon Transform and Its Application in Texture Analysis
摘要：
广义球面Radon变换将球面环面上的平均值与定义在$\mathbb{S}^3 \subset \mathbb{H}$上的函数$f$相关联，其中$\mathbb{S}^3$的元素被认为是表示旋转的四元数。将其引入晶体学择优取向的分析中，并用与角分布函数$W$对应的概率密度函数进行了识别。最终，该通信提出了一种新的方法来从数据采样$W$恢复$f$的近似。与此同时，它提供了额外的澄清最近建议的方法，应用再生核和径向基函数的启发性的洞察力，在其涉及的几何形状。重点是几何和组特征的对应关系，但不是函数及其空间的映射。
英文摘要：
The generalized spherical Radon transform associates the mean values over spherical tori to a function $f$ defined on $\mathbb{S}^3 \subset \mathbb{H}$, where the elements of $\mathbb{S}^3$ are considered as quaternions representing rotations. It is introduced into the analysis of crystallographic preferred orientation and identified with the probability density function corresponding to the angle distribution function $W$. Eventually, this communication suggests a new approach to recover an approximation of $f$ from data sampling $W$. At the same time it provides additional clarification of a recently suggested method applying reproducing kernels and radial basis functions by instructive insight in its involved geometry. The focus is on the correspondence of geometrical and group features but not on the mapping of functions and their spaces.
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