几何细分与多尺度变换
Geometric subdivision and multiscale transforms
摘要：
应用于数据的任何程序以及从数据导出的任何量都需要尊重数据的性质和对称性。这个公理适用于细化过程和多分辨率变换，以及更基本的操作，如平均值。本章讨论不同种类的几何结构，如度量空间、黎曼流形和群，以及我们如何使初等运算具有几何意义。一个很好的例子是与正定矩阵空间自然联系在一起的黎曼度量和由其导出的正定矩阵上的内在运算。我们首先讨论平均值，然后进行细化运算（细分）和多尺度变换。特别是，我们报告目前的知识，收敛性和平滑性。
英文摘要：
Any procedure applied to data, and any quantity derived from data, is required to respect the nature and symmetries of the data. This axiom applies to refinement procedures and multiresolution transforms as well as to more basic operations like averages. This chapter discusses different kinds of geometric structures like metric spaces, Riemannian manifolds, and groups, and in what way we can make elementary operations geometrically meaningful. A nice example of this is the Riemannian metric naturally associated with the space of positive definite matrices and the intrinsic operations on positive definite matrices derived from it. We disucss averages first and then proceed to refinement operations (subdivision) and multiscale transforms. In particular, we report on the current knowledge as regards convergence and smoothness.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
