Cohomology And Fundamental Group Of Digital Image Based On GK~n-topology

A topological method is one of important methods of digital image analysis.Its basic idea is to introduce effective topological structure into digital space,establish topological adjacency relationship between pixels,and then analyze digital images.Based on the GKn-topological structure in digital space,this thesis defines the GKn-adjacency category,and then introduces GKnA-cohomology group,GKnA-relative cohomology group,GKnA-cohomology ring,GKnA-fundamental group and covering space of digital images,and discusses their propertiesFirstly,it is proved that,in the GKn-adjacency category,the GKnA-cohomology group and cohomology ring of digital image are isomorphic invariants of digital image,but not homotopy invariants.The function of multiplication in ring structure is illustrated by an example.The exact coincidence of GKnA-cohomology sequences,the excision theorem and M-V sequences are proved.Secondly,it is proved that the GKnA-fundamental group of digital image is not only isomorphic invariant,but also homotopy invariant.By introducing the covering space and covering mapping in the GKn-adjacency category,the GKnA-fundamental group of simple closed curves is calculated.At the same time,a basic theorem for calculating GKnA-fundamental group is proved,which is another version of Van Kampen theorem in GKn-adjacency category.A new relation between GKnAfundamental group and GKnA-homology group is obtained.In order to calculate the above invariants by a computer,we give algorithms to calculate the GKnA-cohomology groups and free generators of GKnA-fundamental group.We also discuss several operations on digital images and their effects on computing cohomology groups and the fundamental group.Finally,we compare the difference between the GKn-adjacency category and the digital k-adjacency category.In the course of study,we have found defects or errors of some results from other literatures,and supplemented or corrected the results.