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| '''Digital topology''' deals with properties and features of [[two-dimensional]] (2D) or [[Three-dimensional space|three-dimensional]] (3D) [[digital images]]
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| that correspond to [[Topology|topological]] properties (e.g., [[connectedness]]) or topological features (e.g., [[Boundary (topology)|boundaries]]) of objects.
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| Concepts and results of digital topology are used to specify and justify important (low-level) [[image analysis]] algorithms,
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| including algorithms for [[Thinning (morphology)|thinning]], border or surface tracing, counting of components or tunnels, or region-filling.
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| == History ==
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| Digital topology was first studied in the late 1960s by the [[computer image analysis]] researcher [[Azriel Rosenfeld]] (1931–2004), whose publications on the subject played a major role in establishing and developing the field. The term "digital topology" was itself invented by Rosenfeld, who used it in a 1973 publication for the first time.
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| A related work called the [[grid cell topology]] appeared in Alexandrov-Hopf's book Topologie I (1935) can be considered as a link to classic [[combinatorial topology]]. Rosenfeld ''et al.'' proposed digital connectivity such as 4-connectivity and 8-connectivity in two dimensions as well as 6-connectivity and 26-connectivity in three dimensions. The labeling method for inferring a connected component was studied in 1970s. T. Pavlidis (1982) suggested the use of graph-theoretic algorithms such as the [[depth-first search]] method for finding connected components. V. Kovalevsky (1989) extended Alexandrov-Hopf's 2D grid cell topology to three and higher dimensions. He also proposed (2008) a more general axiomatic theory of [[Locally finite collection|locally finite topological spaces]] and [[abstract cell complexes]] formerly suggested by Steinitz (1908). It is the [[Alexandrov topology]]. The book of 2008 contains new definitions of topological balls and spheres independent of a metric and numerous applications to digital image analysis.
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| In early 1980s, [[digital surface]]s were studied. Morgenthaler and Rosenfeld (1981) gave a mathematical definition of surfaces in three-dimensional digital space. This definition contains a total of nine types of digital surfaces. The [[digital manifold]] was studied in 1990s. A recursive definition of the digital k-manifold was proposed intuitively by Chen and Zhang in 1993. Many applications were found in image processing and computer vision.
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| == Basic results ==
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| A basic (early) result in digital topology says that 2D binary images require the alternative use of 4- or 8-adjacency or "[[pixel connectivity]]" (for "object" or "non-object"
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| [[pixels]]) to ensure the basic topological duality of separation and connectedness. This alternative use corresponds to open or closed
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| sets in the 2D [[grid cell topology]], and the result generalizes to 3D: the alternative use of 6- or 26-adjacency corresponds
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| to open or closed sets in the 3D [[grid cell topology]]. Grid cell topology also applies to multilevel (e.g., color) 2D or 3D images,
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| for example based on a total order of possible image values and applying a 'maximum-label rule' (see book by Klette and Rosenfeld, 2004).
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| Digital topology is highly related to [[combinatorial topology]]. The main differences between them are: (1) digital topology mainly studies digital objects that are formed by grid cells,{{clarify|reason=How does that differ?|date=October 2011}} and (2) digital topology also deals with non-Jordan manifolds.
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| A combinatorial manifold is a kind of manifold which is discretization of a manifold. It usually means a [[piecewise linear manifold]] made by [[simplicial complexes]]. A [[digital manifold]] is a special kind of combinatorial manifold which is defined in digital space i.e. grid cell space.
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| A digital form of the [[Gauss–Bonnet theorem]] is: Let ''M'' be a closed digital 2D [[manifold]] in direct adjacency (i.e. a (6,26)-surface in 3D).
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| The formula for genus is
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| : <math> g = 1 + (M_{5} + 2 M_{6} - M_{3}) / 8 ,\! </math>
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| where ''M''<sub>''i''</sub> indicates the set of surface-points each of which has ''i'' adjacent points on the surface (Chen and Rong, ICPR 2008).
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| If ''M'' is simply connected, i.e. ''g'' = 0, then ''M''<sub>3</sub> = 8 + ''M''<sub>5</sub> + 2''M''<sub>6</sub>. (See also [[Euler characteristic]].)
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| ==See also==
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| *[[Digital geometry]]
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| *[[Combinatorial topology]]
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| *[[Computational geometry]]
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| *[[Computational topology]]
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| *[[Topological data analysis]]
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| *[[Topology]]
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| *[[Discrete mathematics]]
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| ==References==
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| *{{cite book | author=Herman, G.T. | title=Geometry of Digital Spaces | publisher=Birkhäuser | year=1998 | isbn=978-0-8176-3897-9}}
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| *{{cite book | author=Kong, T.Y., and A. Rosenfeld (editors) | title=Topological Algorithms for Digital Image Processing | publisher=Elsevier | year=1996 | isbn=0-444-89754-2}}
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| *{{cite book | author=Voss, K. | title= Discrete Images, Objects, and Functions in Zn | publisher= Springer | year= 1993 | isbn=0-387-55943-4}}
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| *{{cite book | author=Chen, L.| title=Discrete Surfaces and Manifolds: A Theory of Digital-Discrete Geometry and Topology | publisher=SP Computing| year=2004 | isbn=0-9755122-1-8 }}
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| *{{cite book | author=Klette, R., and A. Rosenfeld | title=Digital Geometry | url=http://www.mi.auckland.ac.nz/index.php?option=com_content&view=article&id=49&Itemid=49 | publisher=Morgan Kaufmann | year=2004 | isbn=1-55860-861-3}}
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| *{{cite book | author=Pavlidis, T. | title=Algorithms for graphics and image processing | publisher=Computer Science Press | year=1982 | isbn=0-914894-65-X}}
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| *{{cite book | author=Kovalevsky, V. | title=Geometry of Locally Finite Spaces |publisher=Publishing House Dr. Baerbel Kovalevski, Berlin | year=2008 | isbn=978-3-9812252-0-4}}
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| [[Category:Digital topology| ]]
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