Homology Group Of Digital Image Based On GK-adjacency

In digital image analysis,the topological adjacency relation between pixels can be used to establish a finite simplicial complex and to introduce simplicial homology groups of digital image.The simplicial homology groups play an important role in the classification of digital image.Based on the GK-topology on the digital space of Z3,we first review concepts of minimum open neighborhoods and neighborhood structure of pixels,GKA-mapping of keeping GK-adjacency neighborhood between digital images and category GKAC.Sec-ondly,we give the concepts of GKA-simplex,GKA-simplicial complex,GKA-edge ho-momorphism,GKA-simplicial homology group,GKA-surface with boundary,and prove the GKA-simplicial homology group is an isomorphic invariant of a digital image in GKAC,but it no longer maintains the invariance of the homotopy.We also calculate the GKA-simplicial homology groups of simplicial complexes associated with some digital im-ages,and compare the GKA-simplicial homology group with the digital homology group defined by other scholars based on the k-adjacency relation,illustrating the difference be-tween the two kinds of homology groups.In addition,the concept of GKA-wedge sum is introduced,and its properties are studied.Finally,the concept of GKA-Euler charac-teristic of digital image in GKAC is given,and the relationship between the GKA-Euler characteristic and the GKA-homology group is obtained,and some properties of Euler characteristic of digital image are discussed.