Homotopy And Fixed Point Properties Of Mapping In Category Related To G-Topology

In 2D image analysis,two important categories related to lattice topology have been established,one is 2D lattice topology category GTC,the other is 2D lattice adjacency category GAC.This thesis generalizes these two categories to 3D and higher-dimensional digital space and discusses their applications in image analysis.In 3D lattice topology category GTC,we define a G-continuous mapping and a G-homotopy,give several equivalent descriptions of the G-homotopy.We also define a G-homotopy equivalence between two digital images,and prove that these two relationships are equivalence relationship.Then we generalize the G-contractibility to high-dimensional digital space and prove that a simple G-path has G-contractibility and a connected proper subset of the simple closed G-curve SCG3,l(l?4)is G-contractible.However,the simple closed G-curve SC3,4 is not G-contractible.Then we discuss a local G-contractibility of high-dimensional digital images and a relationship between the local G-contractibility and G-contractibility.We prove that any digital space in GTC is locally G-contractible,that the G-contractibility can induce the local G-contractibility and conversely it is not true.Finally,we define a fixed point property and a retract kernel of higher-dimensional digital images in GTC,and prove that the smallest open neighborhood of a point in 3D digital space has the fixed point property and the retract kernel of digital images keeps a fixed point property of digital images.In 3D lattice adj acency category GAC,we prove that a GA-mapping is equivalent to a G-connect.edness-preserving mapping.We discuss GA-homotopy of two GA-mappings and some properties of GA-homotopy equivalence between two digital images,analyze the relationship between GA-homotopy and G-homotopy and show that G-homotopy must be GA-homotopy and conversely,it is not valid.We also discuss a GA-contractibility and prove that the simple closed GA-curve SCGA3,4 is GA-contractible,and SCGA3,l is not GA-contractible for l>4.Based on this result,it is proved that a simple closed G-curve SCG3,l(l?4)is not G-contractible,and that SCG3,l1 is equivalent to SCG3,l2 up to G-homotopy if and only if l1=l2.