Dimensional properties of graphs and digital spaces

Abstract

Many applications of digital image processing now deal with three-and higher-dimensional images.One way to represent n-dimensional digital images is to use the specialization graphs of subspaces of the Alexandroff topological space Z n (where Z denotes the integers with the Khalimsky line topology). In this paper the dimension of any such graph is defined in three ways, and the equivalence of the three definitions is established. Two of the definitions have a geometric basis and are closely related to the topological definition of inductive dimension; the third extends the Alexandroff dimension to graphs. Diagrams are given of graphs that are dimensionally correct discrete models of Euclidean spaces, n-dimensional spheres, a projective plane and a torus. New characterizations of n-dimensional (digital) surfaces are presented. Finally, the local structure of the space Z n is analyzed, and it is shown that Z n is an n-dimensional surface for all n > 1.

Topology Preserving Discretization Schemes for Digital Image Segmentation and Digital Models of the Plane

“…Thus, a digital 1-surface is either a digital 1-sphere or a digital line. sphere [13]. A digital 2-surface with a finite number of points is called a digital closed 2-surface.…”

Graph Theoretical Models of Discrete Spaces with Locally Non-spherical Topology

“…Graph-theoretic approach equips a digital image with a graph structure based on the local adjacency relations of points (see e.g. [2,3,4]). In paper [4], digital n-surfaces were defined as simple undirected graphs and basic properties of n-surfaces were studied.…”

“…[2,3,4]). In paper [4], digital n-surfaces were defined as simple undirected graphs and basic properties of n-surfaces were studied. Properties of digital n-manifolds were investigated in ( [5,6,9,10]).…”

Periodicity of Generalized Fibonacci-like Sequences

“…Traditionally, a digital image has a graph structure (see [2,3,5]). A digital space G is a simple undirected graph G=(V,W) where V=(v 1 ,v 2 ,...v n ,…) is a finite or countable set of points, and W = ((v р v q ),....) is a set of edges.…”

“…A digital space G is a simple undirected graph G=(V,W) where V=(v 1 ,v 2 ,...v n ,…) is a finite or countable set of points, and W = ((v р v q ),....) is a set of edges. The induced subgraph ( )containing point v and all points adjacent to v is called the ball of point v in G. Graphs that are digital counterparts of continuous n-dimensional manifolds were studied in [5,6,7,8]. Consider the numerical solutions of the initial value problem for the wave equation on a graph G(v 1 , v 2 , … v s ).…”