http://www.mi.auckland.ac.nz/DATA/CCV/books/KR/FrontPages.pdf

Geometric Methods for Digital Picture Analysis

Digital geometry deals with geometrical properties of "digital objects", which are usually taken to be sets of lattice points in the discrete space Zn. Such objects are often the result of applying a "digitization" process to objects in the Euclidean space Rn. A central theme in digital geometry is how to characterize digital objects that could be the digitizations of "real" objects that have given geometric properties. The literature on digital geometry dates back to the late 1960's. The report includes a bibliography of more then 900 papers on the subject, organized by topic. It outlines the main lines of development of the field, and indicates areas in which interesting problems remain open. * Center for Automation Research, University of Maryland, College Park, MD 20742-3275, USA This Material was presented at the "Digital Geometry Day 1997", 29 January 1997, University of Auckland, organised by the CITR at Tamaki and SMIS.

Digital Geometry: Introduction and Bibliography

In this paper, we define and study digital manifolds of arbitrary dimension, and provide (in particular) a general theoretical basis for curve or surface tracing in picture analysis. The studies involve properties such as the one-dimensionality of digital curves and (n - 1)-dimensionality of digital hypersurfaces that makes them discrete analogs of corresponding notions in continuous topology. The presented approach is fully based on the concept of adjacency relation and complements the concept of dimension, as common in combinatorial topology. This work appears to be the first one on digital manifolds based on a graph-theoretical definition of dimension. In particular, in the n-dimensional digital space, a digital curve is a one-dimensional object and a digital hypersurface is an (n - 1)-dimensional object, as it is in the case of curves and hypersurfaces in the Euclidean space. Relying on the obtained properties of digital hypersurfaces, we propose a uniform approach for studying good pairs defined by separations and obtain a classification of good pairs in arbitrary dimension. We also discuss possible applications of the presented definitions and results.

Border and SurfaceTracing - Theoretical Foundations

In this paper we propose several equivalent definitions of digital curves and hypersurfaces in arbitrary dimension. The definitions involve properties such as one-dimensionality of curves and (n – 1)-dimensionality of hypersurfaces that make them discrete analogs of corresponding notions in topology. Thus this work appears to be the first one on digital manifolds where the definitions involve the notion of dimension. In particular, a digital hypersurface in nD is an (n – 1)-dimensional object, as it is in the case of continuous hypersurfaces. Relying on the obtained properties of digital hypersurfaces, we propose a uniform approach for studying good pairs defined by separations and obtain a clssification of good pairs in arbitrary dimension.

Curves, hypersurfaces, and good pairs of adjacency relations

Simple point detection is an important task for several problems in discrete geometry, such as topology preserving thinning in image processing to compute discrete skeletons. In this paper, the approach to simple point detection is based on techniques from cubical homology, a framework ideally suited for problems in image processing. A (d-dimensional) unitary cube (for a d-dimensional digital image) is associated with every discrete picture element, instead of a point in /spl epsi//sup d/ (the d- dimensional Euclidean space) as has been done previously. A simple point in this setting then refers to the removal of a unitary cube without changing the topology of the cubical complex induced by the digital image. The main result is a characterization of a simple point p (i.e., simple unitary cube) in terms of the homology groups of the (3/sup d/-1) neighborhood of p for arbitrary, finite dimensions d.

/pdf/on-the-detection-of-simple-points-in-higher-dimensions-using-2b6jjajroc.pdf

On the detection of simple points in higher dimensions using cubical homology

Homology groups are important topological invariants of discrete structures. In this paper we present efficient algorithms for the computation of all the homology groups of a structure imbedded in a 3-dimensional Euclidean space, as well as an efficient algorithm for the computation of the $(n - 1)$st homology group of a structure imbedded in an n-dimensional Euclidean space.

George Tourlakis

Papers

Homological Methods for the Classification of Discrete Euclidean Structures