by Brimkov, VE and Klette, R
Abstract:
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 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 approachis fully based on the concept of adjacency relation and complements the concept of dimension ascommon in combinatorial topology. This work appears to be the first one on digital manifolds based ona graph-theoretical definition of dimension. In particular, in the n-dimensional digital space, a digitalcurve is a one-dimensional object and a digital hypersurface is an (n-1)-dimensional object, as it isin 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. © 2008 IEEE.
Reference:
Border and surface tracing – Theoretical foundations (Brimkov, VE and Klette, R), In IEEE Transactions on Pattern Analysis and Machine Intelligence, volume 30, 2008.
Bibtex Entry:
@article{brimkov2008borderfoundations,
author = "Brimkov, VE and Klette, R",
journal = "IEEE Transactions on Pattern Analysis and Machine Intelligence",
month = "Apr",
pages = "577--590",
title = "Border and surface tracing - Theoretical foundations",
volume = "30",
year = "2008",
abstract = "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 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 approachis fully based on the concept of adjacency relation and complements the concept of dimension ascommon in combinatorial topology. This work appears to be the first one on digital manifolds based ona graph-theoretical definition of dimension. In particular, in the n-dimensional digital space, a digitalcurve is a one-dimensional object and a digital hypersurface is an (n-1)-dimensional object, as it isin 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. © 2008 IEEE.",
doi = "10.1109/TPAMI.2007.70725",
issn = "0162-8828",
issue = "4",
keyword = "Digital curve",
keyword = "Digital geometry",
keyword = "Digital hypersurface",
keyword = "Digital manifold",
keyword = "Digital topology",
keyword = "Discrete dimension",
keyword = "Good pair",
language = "eng",
}