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

Abstract

In this paper, discretization schemes are defined and studied that allow us to build digital models of 2-dimensional continuous objects with the same topological properties as their continuous counterparts. In particular, it is shown that the digital model of the plane is necessarily the digital 2-surface defined and studied in the context of digital topology. It is shown that in proposed discretization schemes, any required resolution can be used in any area of the object of interest. Finally, we describe a method, which transforms regions of interest produced by the image acquisition process into digital spaces with topological features of the regions.

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

“…Digital n-dimensional spaces with a locally spherical topology are graph theoretical models of continuous n-dimensional spaces as it is shown in [7,11]. We discuss the particular case when the graph is a digital n-dimensional surface with a locally non-spherical topology.…”