Classification of digital n-manifolds

A digital approach to geometry and topology plays an important role in analyzing n-dimensional digitized images arising in computer graphics as well as in many areas of science including neuroscience, medical imaging, industrial inspection, geoscience and fluid dynamics. Concepts and results of the digital approach are used to specify and justify some important low-level image processing algorithms, including algorithms for thinning, boundary extraction, object counting, and contour filling. Usually, a digital object is equipped with a graph structure based on the local adjacency relations of digital points [8]. In papers [9-10], a digital n-surface was defined as a simple undirected graph and basic properties of n-surfaces were studied. Paper [9] analyzes a local structure of the digital space Z n. It is shown that Z n is an n-surface for all n>0. In paper [10], it is proven that if A and B are n-surfaces and AB, then A=B. X. Daragon et al. [6-7] studied partially ordered sets in connection with the notion of n-surfaces. In particular, it was proved that (in the framework of simplicial complexes) any n-surface is an n-pseudomanifold, and that any n-dimensional combinatorial manifold is an n-surface. In paper [23], M. Smyth et al. defined dimension at a vertex of a graph as basic dimension, and the dimension of a graph as the sup over its vertices. They proved that dimension of a strong product G × H is dim (G) + dim (H) (for non-empty graphs G and H). An interesting method using cubical images with direct adjacency for determining such topological invariants as genus and the Betti numbers was designed and studied by L. Chen et al. [3]. E. Melin [22] studies the join operator, which combines two digital spaces into a new space. Under the natural assumption of local finiteness, he shows that spaces can be uniquely decomposed as a join of indecomposable spaces. In papers [2,14], digital covering spaces were classified by using the conjugacy class corresponding to a digital covering space. A digital n-manifold, which we regard in this paper, is a special case of a digital n-surface. We define the complexity of digital n-manifolds similar to the notion of complexity of continuous 3-and 4-manifolds studied in [20-21]. We introduce compressed digital n-manifolds and show that any n-manifold can be transformed to a compressed one by transformations retaining the connectedness and the dimension of the given n-manifold. We show that a …