Digital geometry deals with discrete sets (usually discrete point sets) considered to be digitized models or images of objects of the 2D or 3D Euclidean space. The word discrete comes from the Latin word discretus which means separate. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... Digitizing, or digitization, is the process of turning an analog signal into a digital representation of that signal. ... A scale model is a representation or copy of an object that is larger or smaller than the actual size of the object being represented. ... For images in Wikipedia, see Wikipedia:Images. ...

Simply put, digitizing is replacing an object by a discrete set of its points. The images we see on the TV screen, the raster display of a computer, or in newspapers are in fact digital images. Suppose the smiley face in the top left corner is an RGB bitmap image. ... A digital system is one that uses discrete numbers, especially binary numbers, or non-numeric symbols such as letters or icons, for input, processing, transmission, storage, or display, rather than a continuous spectrum of values (an analog system). ...

Its main application areas are computer graphics and image analysis. Computer graphics (CG) is the field of visual computing, where one utilizes computers both to generate visual images synthetically and to integrate or alter visual and spatial information sampled from the real world. ... Image analysis is the extraction of useful information from images; mainly from digital images by means of digital image processing techniques. ...

Main aspects of study are:

• 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).
• Study of properties of digital sets; see, for example, Pick's theorem, digital convexity, digital straightness, or digital planarity.
• 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.
• Reconstructing "real" objects or their properties (area, length, curvature, volume, surface area, and so forth) from digital images.

Digital geometry heavily overlaps with discrete geometry and may be considered as a part thereof. Bresenhams line algorithm is an algorithm that determines which points on a 2-dimensional raster should be plotted in order to form a close approximation to a straight line between two given points. ... Given a simple polygon constructed on a grid of equal-distanced points (i. ... Digital topology deals with properties and features of two-dimensional (2D) or three-dimensional (3D) digital images that correspond to topological properties (e. ... Mathematical morphology (MM) is a theoretical model for digital images built upon lattice theory and topology. ... Discrete geometry or combinatorial geometry may be loosely defined as study of geometrical objects and properties that are discrete or combinatorial, either by their nature or by their representation; the study that does not essentially rely on the notion of continuity. ...

See also: computational geometry, digital topology, tomography. In computer science, computational geometry is the study of algorithms to solve problems stated in terms of geometry. ... Digital topology deals with properties and features of two-dimensional (2D) or three-dimensional (3D) digital images that correspond to topological properties (e. ... Tomography is imaging by sections or sectioning. ...

External links

IAPR Technical Committee on Discrete Geometry

Website on digital geometry and topology

Course on digital geometry and mathematical morphology (Ch. Kiselman)

Course on algorithms in digital geometry (R. Klette)

Books

• Rosenfeld, A. (1969). Picture Processing by Computer. Academic Press. ISBN ???.

• Rosenfeld, A. (1976). Digital Picture Analysis. Springer. ISBN ???.

• Rosenfeld, A. (1979). Picture Languages. Academic Press. ISBN 0-12-597340-3.

• Chassery, J., and A. Montanvert. (1991). Geometrie discrete en analyze d’images. Hermes. ISBN ???.

• Kong, T.Y., and A. Rosenfeld (editors) (1996). Topological Algorithms for Digital Image Processing. Elsevier. ISBN 0-44489-754-2.

• Voss, K. (1993). Discrete Images, Objects, and Functions in Zn. Springer. ISBN 0-38755-943-4.

• Gabor T. Herman (1998). Geometry of Digital Spaces. Birkhauser. ISBN 0-81-763897-0.

• Marchand-Maillet, S., and Y. M. Sharaiha (2000). Binary Digital Image Processing. Academic Press. ISBN 0-12-470505-7.

• Pierre Soille (2003). Morphological Image Analysis: Principles and Applications. Springer. ISBN 3-540-42988-3.

• Chen, L. (2004). Discrete Surfaces and Manifolds: A Theory of Digital-Discrete Geometry and Topology. SP Computing. ISBN 0-97551221-8.

• Klette, R., and A. Rosenfeld (2004). Digital Geometry. Morgan Kaufmann. ISBN 1-55860-861-3.