Product Topology And 3D Digital Image Analysis

This thesis focuses on the analysis of 3D digital images through the topology theory on the digital space Z3.First of all,a topological structure on Z3 is established.It is called GK-topology which is the product topology of the grid point topology(briefly,GP2-topology)on the 2D digital space Z2 and the Khalimsky line topology(briefly,K1-topology).The smallest open neighborhood structure of every point related to GK-topology is analyzed.The points in digital space Z3 are classified according to the smallest open neighborhood structures.Then some drawbacks of GK-continuous map and GK-homeomorphism are dis-cussed in rotating and classifying digital images.In order to overcome these drawback-s,the thesis introduces the notions of GK-adjacency neighborhood and GK-adjacency set,and so establishes a new relation among pixels.Furthermore,two new maps,GK-adjacency map and GK-A-map,are defined and it is proved that the GK-A-map is a connectedness-preserving map.The difference between a GK-A-map and a GK-continuous map is made clear by comparison and so do GK-adjacency map and GK-continuous map.In fact,a GK-continuous map must be a GK-A-map but the converse is not all true.Based on the GK-A-map and GK-continuous map,two new categories,GK-adjacency category(GKAC)and GK-topology category(GKTC),are established.The thesis al-so introduces a GK-A-isomorphism,and proves that a GK-homeomorphism must be a GK-A-isomorphism,but not vice versa.A classification of 3D digital image is realized by means of a GK-A-isomorphism.Finally,the thesis provides a method of thinning or compressing digital images through GK-A-retraction,which is based on the GK-topology.The method could be used in computer image analysis and image processing.