Homological Methods for the Classification of Discrete Euclidean Structures
01 Jul 1977-Siam Journal on Applied Mathematics (Society for Industrial and Applied Mathematics)-Vol. 33, Iss: 1, pp 51-54
TL;DR: This paper presents 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 computations of the $(n - 1)$st homology group of aructure imbedding in an n-dimensional geometry.
Abstract: 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.
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01 Jan 1997
TL;DR: 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.
Abstract: 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.
27 citations
TL;DR: This work appears to be the first one on digital manifolds based on a graph-theoretical definition of dimension, and provides (in particular) a general theoretical basis for curve or surface tracing in picture analysis.
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 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.
25 citations
01 Dec 2004
TL;DR: This work appears to be the first one on digital manifolds where the definitions involve the notion of dimension, and a uniform approach for studying good pairs defined by separations and obtain a clssification of good pairs in arbitrary dimension.
Abstract: 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.
23 citations
Cites background from "Homological Methods for the Classif..."
...The approximation of n-dimensional manifolds by graphs is studied in [29, 30], with a special focus on topological properties of such graphs defined by homotopy and on homology or cohomology groups....
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TL;DR: The main result is a characterization of a simple point p in terms of the homology groups of the (3/sup d/-1) neighborhood of p for arbitrary, finite dimensions d.
Abstract: 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.
19 citations
Cites background or result from "Homological Methods for the Classif..."
...related to that of Tourlakis and Mylopoulos [8], [9], but associates a -dimensional unitary cube (for a -dimensional digital...
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...The approach for simple point detection proposed in Section II is rooted in algebraic topology and most closely related to the work of Tourlakis and Mylopoulos [9] to which it will be compared in the remainder of this section....
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...Our approach is most closely related to that of Tourlakis and Mylopoulos [8], [9], but associates a -dimensional unitary cube (for a -dimensional digital image) with every discrete picture element, instead of a point 1057-7149/$20.00 © 2006 IEEE in the -dimensional Euclidean space ....
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...Section III relates the approach presented in this paper to previous work and compares it in particular to the approach by Tourlakis and Mylopoulos [8], [9]....
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References
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TL;DR: It is shown that two different methods of approximating an n-dimensional closed manifold with boundary by a graph of the type studied in this paper lead to graphs whose corresponding homology groups are isomorphic.
Abstract: It is the object of this paper to study the topological properties of finite graphs that can be embedded in the n-dimensional integral lattice (denoted Nn). The basic notion of deletability of a node is first introduced. A node is deletable with respect to a graph if certain computable conditions are satisfied on its neighborhood. An equivalence relation on graphs called reducibility and denoted by “∼” is then defined in terms of deletability, and it is shown that (a) most important topological properties of the graph (homotogy, homology, and cohomology groups) are ∼-invariants; (b) for graphs embedded in N3, different knot types belong to different ∼-equivalence classes; (c) it is decidable whether two graphs are reducible to each other in N2 but this problem is undecidable in Nn for n ≥ 4. Finally, it is shown that two different methods of approximating an n-dimensional closed manifold with boundary by a graph of the type studied in this paper lead to graphs whose corresponding homology groups are isomorphic.
40 citations