Nonclassical light - Revision history

74.202.184.2 at 21:40, 6 May 2014

2014-05-06T21:40:00Z

← Older revision		Revision as of 23:40, 6 May 2014
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	~~'''Digital topology''' deals with properties and features~~ of ~~[[two-dimensional]] (2D) or [[Three-dimensional space\|three-dimensional]] (3D) [[digital images]]~~		The name of the writer is Nestor. One of his favorite hobbies is taking part in crochet but he hasn't made a dime with it. My job is a manufacturing and distribution officer and I'm performing pretty great financially. Arizona is her birth location and she will never transfer.<br><br>My blog ... [http://kaonthana.com/index.php?mod=users&action=view&id=5336 extended car warranty]
	~~that correspond to [[Topology\|topological]] properties (e.g., [[connectedness]]) or topological features (e.g., [[Boundary (topology)\|boundaries]]) of objects~~.

	~~Concepts and results~~ of ~~digital topology are used to specify and justify important (low-level) [[image analysis]] algorithms,~~
	~~including algorithms for [[Thinning (morphology)\|thinning]], border or surface tracing, counting of components or tunnels, or region-filling.~~

	~~== History ==~~
	~~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~~.

	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.~~

	~~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.

	~~== Basic results ==~~
	~~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"~~
	~~[[pixels]]) to ensure the basic topological duality of separation and connectedness. This alternative use corresponds to open or closed~~
	~~sets in the 2D [[grid cell topology]], and the result generalizes to 3D: the alternative use of 6- or 26-adjacency corresponds~~
	~~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,~~
	~~for example based on a total order of possible image values and applying a~~ '~~maximum-label rule' (see book by Klette and Rosenfeld, 2004)~~.

	~~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.~~

	~~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.

	~~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).~~
	~~The formula for genus is~~
	~~: <math> g = 1 + (M_{5} + 2 M_{6} - M_{3}) / 8 ,\! </math>~~
	~~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)~~.
	~~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]]~~.)

	~~==See also==~~
	*[~~[Digital geometry]]~~
	*[[Combinatorial topology]]
	*[[Computational geometry]]
	*[[Computational topology]]
	*[[Topological data analysis]]
	*[[Topology]]
	*[[Discrete mathematics]]

	~~==References==~~

	*{{cite book \| author=Herman, G.T. \| title=Geometry of Digital Spaces \| publisher=Birkhäuser \| year=1998 \| isbn=978-0-8176-3897-9}}

	*{{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}}

	*{{cite book \| author=Voss, K. \| title= Discrete Images, Objects, and Functions in Zn \| publisher= Springer \| year= 1993 \| isbn=0-387-55943-4}}

	*{{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 }}

	*{{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}}~~

	*{{cite book \| author=Pavlidis, T. \| title=Algorithms for graphics and image processing \| publisher=Computer Science Press \| year=1982 \| isbn=0-914894-65-X}}

	*{{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}}


	~~[[Category:Digital topology\| ]~~]

71.13.151.210: /* General references */

2013-05-14T15:17:52Z

General references

← Older revision		Revision as of 17:17, 14 May 2013
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	~~Hi there~~. ~~My name is Garland even though~~ it is ~~not~~ the ~~title on my beginning certificate~~. The ~~preferred pastime~~ for ~~him~~ and ~~his kids~~ is to ~~perform badminton but he~~ is ~~struggling to find time for it~~. ~~I am~~ a ~~manufacturing~~ and ~~distribution officer~~. ~~Delaware~~ is ~~our birth place~~.<br><br>~~My web blog~~ :~~: [~~http://~~Www~~.~~evil-Esports~~.~~de/cs~~/index.php?~~mod~~=~~users~~&~~action~~=~~view~~&id=~~5958 http://Www~~.~~evil~~-~~Esports~~.~~de/~~]		'''Digital topology''' deals with properties and features of [[two-dimensional]] (2D) or [[Three-dimensional space\|three-dimensional]] (3D) [[digital images]]
			that correspond to [[Topology\|topological]] properties (e.g., [[connectedness]]) or topological features (e.g., [[Boundary (topology)\|boundaries]]) of objects.

			Concepts and results of digital topology are used to specify and justify important (low-level) [[image analysis]] algorithms,
			including algorithms for [[Thinning (morphology)\|thinning]], border or surface tracing, counting of components or tunnels, or region-filling.

			== History ==
			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.

			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.

			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.

			== Basic results ==
			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"
			[[pixels]]) to ensure the basic topological duality of separation and connectedness. This alternative use corresponds to open or closed
			sets in the 2D [[grid cell topology]], and the result generalizes to 3D: the alternative use of 6- or 26-adjacency corresponds
			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,
			for example based on a total order of possible image values and applying a 'maximum-label rule' (see book by Klette and Rosenfeld, 2004).

			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.

			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.

			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).
			The formula for genus is
			: <math> g = 1 + (M_{5} + 2 M_{6} - M_{3}) / 8 ,\! </math>
			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).
			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]].)

			==See also==
			*[[Digital geometry]]
			*[[Combinatorial topology]]
			*[[Computational geometry]]
			*[[Computational topology]]
			*[[Topological data analysis]]
			*[[Topology]]
			*[[Discrete mathematics]]

			==References==

			*{{cite book \| author=Herman, G.T. \| title=Geometry of Digital Spaces \| publisher=Birkhäuser \| year=1998 \| isbn=978-0-8176-3897-9}}

			*{{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}}

			*{{cite book \| author=Voss, K. \| title= Discrete Images, Objects, and Functions in Zn \| publisher= Springer \| year= 1993 \| isbn=0-387-55943-4}}

			*{{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 }}

			*{{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}}

			*{{cite book \| author=Pavlidis, T. \| title=Algorithms for graphics and image processing \| publisher=Computer Science Press \| year=1982 \| isbn=0-914894-65-X}}

			*{{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}}


			[[Category:Digital topology\| ]]

99.1.33.41 at 17:24, 18 August 2012

2012-08-18T17:24:47Z

New page

Hi there. My name is Garland even though it is not the title on my beginning certificate. The preferred pastime for him and his kids is to perform badminton but he is struggling to find time for it. I am a manufacturing and distribution officer. Delaware is our birth place.<br><br>My web blog :: [http://Www.evil-Esports.de/cs/index.php?mod=users&action=view&id=5958 http://Www.evil-Esports.de/]