乘积拓扑与三维数字图像分析

发布时间：2018-02-01 13:05

本文关键词： 数字图像数字拓扑连续映射图像分类图像约化　出处：《河北师范大学》2017年硕士论文　论文类型：学位论文

【摘要】：本文立足于数字空间Z3上的拓扑理论,对3D数字图像进行理论分析和研究.首先,在数字空间Z3上建立一种拓扑结构:二维格点拓扑(GP2-拓扑)与Khalimsky线拓扑(K1-拓扑)的乘积拓扑,简称GK-拓扑.分析了在此种拓扑下每一点的最小开邻域的结构,根据最小开邻域结构的不同把数字空间Z3上的点进行了分类.其次,讨论了基于此种拓扑下的连续映射(称为GK-连续映射)和同胚(称为GK-同胚),并发现其在研究数字图像的旋转和分类等问题时存在局限性.为了克服此种局限性,引入了拓扑邻接邻域和拓扑邻接集的概念,从而建立了像素之间一种新的邻接关系.在此基础上定义了 GK-邻接映射和GK-A-映射,并给出了GK-A-映射保持连通性的证明.之后,通过具体例子分析了GK-邻接映射和GK-A-映射分别与GK-连续映射的异同并总结了上述三种映射在某些特定变换(旋转、平移等)下比较的结果.证明了 GK-连续映射一定是GK-A-映射,但反之未必.基于GK-A-映射是GK-连续映射的推广,在GK-拓扑下建立了两个新范畴GKAC和GKTC.论文也建立了GK-A-同构的概念,并证明了GK-同胚一定是GK-A-同构,但反之未必.通过GK-A-同构实现了对3D数字图像更广的一种等价分类.最后,基于GK-拓扑结构,本文提供了一种通过利用GK-A-收缩映射来细化或约化数字图像的方法,从而对计算机科学中的图像分析、图像处理提供帮助.
[Abstract]:Based on the topological theory of digital space Z3, this paper makes theoretical analysis and research on 3D digital image. A product topology is established on the digital space Z3: 2-D lattice topology (GP2- topology) and Khalimsky line topology (K1-topology). In this paper, the structure of the minimum open neighborhood of each point in this topology is analyzed, and the points on the digital space Z3 are classified according to the structure of the minimum open neighborhood. The continuous mapping (called GK-continuous mapping) and homeomorphism (called GK-homeomorphism) based on this topology are discussed. In order to overcome this limitation, the concepts of topological neighborhood and topological adjacency set are introduced. On the basis of this, we define GK-adjacent mapping and GK-A- map, and prove that GK-A- map maintains connectivity. The similarities and differences between GK-adjacent mapping and GK-A- mapping and GK-continuous mapping are analyzed through concrete examples, and the above three kinds of mappings are summarized in some special transformations (rotation). It is proved that GK-continuous mapping must be GK-A- mapping, but not necessarily vice versa. GK-A- mapping is a generalization of GK-continuous mapping based on GK-A- mapping. Two new categories of GKAC and GKTCare established under GK-topology. The concept of GK-A- isomorphism is also established and it is proved that GK-A- homomorphism must be GK-A- isomorphism. But not necessarily. Through GK-A- isomorphism to realize a more extensive equivalent classification of 3D digital images. Finally, based on GK- topology. In this paper, a method of thinning or reducing digital images by using GK-A-shrinkage mapping is provided, which can be helpful to image analysis and image processing in computer science.
【学位授予单位】：河北师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O189;TP391.41

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