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| {{Multiple issues|refimprove=July 2010|expert=Geometry|date=March 2011}}
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| '''Digital geometry''' deals with [[discrete space|discrete]] sets (usually discrete [[Point (geometry)|point]] sets) considered to be [[digitizing|digitized]] [[scale model|models]] or [[image]]s of objects of the 2D or 3D [[Euclidean space]].
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| Simply put, '''digitizing''' is replacing an object by a discrete set of its points. The images we see on the TV screen, the [[raster graphics|raster]] display of a computer, or in newspapers are in fact [[Digital data|digital]] images.
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| Its main application areas are [[computer graphics]] and [[image analysis]].
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| Main aspects of study are:
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| * Constructing digitized representations of objects, with the emphasis on precision and efficiency (either by means of synthesis, see, for example, [[Bresenham's line algorithm]] or digital disks, or by means of digitization and subsequent processing of digital images).
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| * Study of properties of digital sets; see, for example, [[Pick's theorem]], digital convexity, digital straightness, or digital planarity.
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| * Transforming digitized representations of objects, for example (A) into simplified shapes such as (i) skeletons, by repeated removal of simple points such that the [[digital topology]] of an image does not change, or (ii) medial axis, by calculating local maxima in a distance transform of the given digitized object representation, or (B) into modified shapes using [[mathematical morphology]].
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| * Reconstructing "real" objects or their properties (area, length, curvature, volume, surface area, and so forth) from digital images.
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| * Study of digital curves, [[digital surface]]s, and [[digital manifold]]s.
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| * Designing tracking algorithms for digital objects.
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| * Functions on digital space.
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| Digital geometry heavily overlaps with [[discrete geometry]] and may be considered as a part thereof.
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| ==Digital space== | |
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| A 2D digital space usually means a 2D grid space that only contains integer points in 2D Euclidean space. A 2D image is a function on a 2D digital space (See [[image processing]]).
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| In Rosenfeld and Kak's book, digital connectivity are defined as the relationship among elements in digital space. For example, 4-connectivity and 8-connectivity in 2D. Also see [[pixel connectivity]]. A digital space and its (digital-)connectivity determine a [[digital topology]]. | |
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| In digital space, the digitally continuous function (A. Rosenfeld, 1986) and the [[Gradually varied surface|gradually varied function]] (L. Chen, 1989) were proposed, independently.
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| A digitally continuous function means a function in which the value (an integer) at a digital point is the same or off by at most 1 from its neighbors. In other words, if ''x'' and ''y'' are two adjacent points in a digital space, |''f''(''x'') − ''f''(''y'')| ≤ 1.
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| A gradually varied function is a function from a digital space <math>\Sigma</math> to <math>\{ A_1, \dots,A_m \}</math> where <math> A_1< \cdots <A_m </math> and <math> A_i</math> are real numbers. This function possesses the following property: If ''x'' and ''y'' are two adjacent points in <math>\Sigma</math>, assume <math>f(x)=A_i</math>, then <math>f(y)=A_{i}</math>, <math>f(x)=A_{i+1}</math>, or <math>A_{i-1}</math>. So we can see that the gradually varied function is defined to be more general than the digitally continuous function.
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| An extension theorem related to above functions was mentioned by A. Rosenfeld (1986) and completed by L. Chen (1989). This theorem states: Let <math>D \subset \Sigma</math> and <math>f: D\rightarrow \{ A_1, \dots,A_m \}</math>. The necessary and sufficient condition for the existence of the gradually varied extension <math>F</math> of <math>f</math> is : for each pair of points <math>x</math> and <math>y</math> in <math>D</math>, assume <math>f(x)=A_i</math> and <math>f(y)=A_j</math>, we have <math>|i-j|\le d(x,y)</math>, where <math>d(x,y)</math> is the (digital) distance between <math>x</math> and <math>y</math>.
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| == See also ==
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| *[[Computational geometry]]
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| *[[Digital topology]]
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| *[[Discrete geometry]]
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| *[[Combinatorial geometry]]
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| *[[Tomography]]
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| ==References==
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| *A. Rosenfeld, `Continuous' functions on digital pictures, Pattern Recognition Letters, v.4 n.3, p. 177–184, 1986.
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| *L. Chen, The necessary and sufficient condition and the efficient algorithms for gradually varied fill, Chinese Sci. Bull. 35 (10), pp 870–873, 1990.
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| == Further reading ==
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| *{{cite book | author=[[Azriel Rosenfeld|Rosenfeld, Azriel]] | title=Picture Processing by Computer | publisher=Academic Press | year= 1969 | id=ISBN ???}}
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| *{{cite book |author=[[Azriel Rosenfeld|Rosenfeld, Azriel]] |title=Digital picture analysis |publisher=Springer-Verlag |location=Berlin |year=1976 |pages= |isbn=0-387-07579-8 |oclc= |doi= }}
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| *{{cite book |author=[[Azriel Rosenfeld|Rosenfeld, Azriel]]; Kak, Avinash C.|title=Digital picture processing |publisher=Academic Press |location=Boston |year=1982 |pages= |isbn=0-12-597301-2 |oclc= |doi= }}
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| *{{cite book | author=[[Azriel Rosenfeld|Rosenfeld, Azriel]] | title=Picture Languages | publisher=Academic Press | year= 1979 | isbn=0-12-597340-3}}
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| *{{cite book | author=Chassery, J., and A. Montanvert. | title=Geometrie discrete en analyze d’images | publisher= Hermes | year= 1991 | isbn=2-86601-271-2}}
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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= Herman, G. T. | authorlink=Gabor Herman | title=Geometry of Digital Spaces | publisher=Birkhauser | year=1998 | isbn=0-8176-3897-0 }}
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| *{{cite book | author=Marchand-Maillet, S., and Y. M. Sharaiha | title=Binary Digital Image Processing | publisher=Academic Press | year=2000 | isbn=0-12-470505-7}}
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| *{{cite book | author= Soille, P. | title=Morphological Image Analysis: Principles and Applications | publisher=Springer | year=2003 | isbn=3-540-42988-3}}
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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=[[Azriel Rosenfeld|Rosenfeld, Azriel]]; Klette, Reinhard |title=Digital Geometry: Geometric Methods for Digital Image Analysis (The Morgan Kaufmann Series in Computer Graphics) |url=http://www.mi.auckland.ac.nz/index.php?option=com_content&view=article&id=49&Itemid=49 |publisher=Morgan Kaufmann |location=San Diego |year=2004 |pages= |isbn=1-55860-861-3 |oclc= |doi= |accessdate=}}
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| == External links ==
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| * [http://www.cb.uu.se/~tc18/ IAPR Technical Committee on Discrete Geometry]
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| * [http://www.mi.auckland.ac.nz/index.php?option=com_content&view=article&id=50&Itemid=66/ Website on digital geometry and topology]
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| * [http://www.math.uu.se/~kiselman/dgmm2004.html Course on digital geometry and mathematical morphology (Ch. Kiselman)]
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| * [http://libdgtal.org DGtal: Open Source Digital Geometry Toolbox and Algorithms library]
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| {{DEFAULTSORT:Digital Geometry}}
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| [[Category:Digital geometry| ]]
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| [[fr:Géométrie discrète]]
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