Topological And Combinatorial Method Based On Digital Image Analysis And Applications

Digital image analysis is the theoretical basis of digital image processing.The present paper,by topological and combinatorial methods,discusses some problems related to the digital image analysis in non-orthogonal grid models,which breaks through limitations of traditional orthogonal grid models in digital image analysis.The main contents include the following parts:In the first part,we mainly discuss the complementary descriptions of Euler charac-teristics of a simplicial incidence pseudograph.We obtain the matching theorems for a closed subset or an open subset in a regular incidence pseudograph with a poset topology.The9)-skeleton of the closure complex,as the geometric realization of an(8),9))-complete incidence pseudograph,plays an important role in the discussion.For our purpose,from the viewpoints of homology,Laplacian operator and homotopy,we study algebraic topol-ogy properties of the9)-skeleton of a closure complex.By embedding an9)-simplicial incidence pseudograph into an(8),9))-complete incidence pseudograph,we provide a new method of calculating the Euler characteristic of the9)-simplicial incidence pseudographthrough a subset of its boundary in the(8),9))-complete incidence pseudograph.There-fore,it may simplify,to an extent,the complexity of computing the Euler characteristic of.In addition,we give an efficient algorithm for computing the invariance.In the second part,we introduce a strong homotopy(i.e.,-homotopy)in the finite topological adjacency category and discuss its properties.The tool from the homotopy allows the possibility of reducing an image to a‘simple'image,and of classifying images up to the homotopy equivalence.The digital6)-homotopy and the-homotopy in Khalimsky topological spaces have been studied in previous works.In consideration of shortcomings of the-homotopy(see the third chapter),we investigate the-homotopy in a finite topological adjacency category,which is more faithfully exhibits some relevant features of a continuous original than-homotopy.We also discuss properties of-homotopy,and prove that all-strong deformation retracts of a finite₀-space(3 could be obtained by removing trivial points one by one.We also prove that two finite₀-spaces are homotopy equivalent if and only if their cores are isomorphic.As an application,we prove that all simple closed-curves are not-contractible.Moreover,in the sense of-homotopy we answer two questions having close relationships with that posed by Laurence Boxer who is a well-known expert in image analysis.In the third part,based on the adjacency neighborhood description of-homotopy in the finite topological adjacency category,we generalize the notion of-homotopy from the finite topological adjacency category to the graph category,and discuss its properties in the graph category.It turns out that some of properties are similar to that in the finite topological adjacency category.The graph homotopy and-homotopy have been discussed in terms of combinatorics by deleting or gluing vertices.Comparing-homotopy,which is defined in terms of the mapping form,with the graph homotopy and the-homotopy,we show the advantages of-homotopy over the graph homotopy and the-homotopy.As an application,we investigate the symmetry of graphs using the mapping class groups up to-homotopy.Finally,it is pointed out that the theoretical research model and method in this paper are suitable not only for image data analysis,but also for other general types of data analysis.