Mehler-Fock变换及其在纹理分析和颜色处理中的应用
The Mehler-Fock Transform and some Applications in Texture Analysis and Color Processing
摘要：
许多随机过程定义在特殊的几何对象上，如球体和圆锥体。我们描述了如何从调和分析，即傅立叶分析组的工具，可以用来调查概率密度函数（PDF）的群体和齐次空间。我们考虑了洛伦兹群SU（1，1）和单位圆盘及其双曲几何的特殊情况，但这个过程可以推广到更广泛的李群。我们主要集中在Mehler-Fock变换，这是径向部分的傅里叶变换的磁盘。这种变换的一些特征是与群卷积的关系、信号与变换空间之间的等距、与拉普拉斯-贝尔特拉米算子的关系以及与群表示理论的关系。我们将概述这些性质及其在信号处理中的应用。我们将从低层次视觉和彩色图像处理两个例子来说明理论。
英文摘要：
Many stochastic processes are defined on special geometrical objects like spheres and cones. We describe how tools from harmonic analysis, i.e. Fourier analysis on groups, can be used to investigate probability density functions (pdfs) on groups and homogeneous spaces. We consider the special case of the Lorentz group SU(1,1) and the unit disk with its hyperbolic geometry, but the procedure can be generalized to a much wider class of Lie-groups. We mainly concentrate on the Mehler-Fock transform which is the radial part of the Fourier transform on the disk. Some of the characteristic features of this transform are the relation to group-convolutions, the isometry between signal and transform space, the relation to the Laplace-Beltrami operator and the relation to group representation theory. We will give an overview over these properties and their applications in signal processing. We will illustrate the theory with two examples from low-level vision and color image processing.
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