图像块流形维数及其证明方法研究  

Research on image patch manifold dimension and its proof methods

作　　者：何瑞强[1] HE Rui-qiang(Department of Mathematics,Xinzhou Teachers University,Xinzhou 034000,China)

机构地区：[1]忻州师范学院数学系,山西忻州034000

出　　处：《吉林师范大学学报（自然科学版）》2021年第2期71-74,共4页Journal of Jilin Normal University:Natural Science Edition

基　　金：国家自然科学基金项目(61941111);忻州师范学院科研基金项目(2019KY07);山西省应用基础研究青年科技研究基金项目(201801D221033)。

摘　　要：在概述流形理论的基础上,首先给出等间距地嵌套在高维空间中的图像块流形维数公式,然后借助微分几何的一些基本工具,从两个不同的角度给出该公式的详细证明过程.一方面,将函数在流形上的梯度看作环绕空间中的向量来计算流形维数;另一方面,利用流形在任意一点处切空间的标准正交基来导出流形维数公式.图像块流形维数公式的推导可以为流形的深入学习和相关应用研究提供理论支撑.On the basis of an overview of manifold theory,firstly the dimensionality formula of image patch manifold nested in a high-dimensional space at equal intervals was given,and then the detailed proof process of the formula was presented from two different perspectives with the help of some basic tools of differential geometry.On the one hand,the gradient of the function on the manifold was regarded as a vector in the surrounding space to calculate the manifold dimension;on the other hand,the manifold dimension formula was derived by using the orthonormal basis of the tangent space of the manifold at any point.The derivation of manifold dimension formula of image patch provides theoretical support for further study of manifold and related application research.

关 键 词：流形 维数 梯度算子 图像处理 标准正交基 

分 类 号：TP391.4[自动化与计算机技术—计算机应用技术]