Digital topology
Digital topology deals with properties and features of
Concepts and results of digital topology are used to specify and justify important (low-level) image analysis algorithms, including algorithms for 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
A related work called the
In the early 1980s,
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"
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,[clarification needed] and (2) digital topology also deals with non-Jordan manifolds.
A combinatorial manifold is a kind of manifold which is a discretization of a manifold. It usually means a
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
- ,
where 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., , then . (See also Euler characteristic.)
See also
- Digital geometry
- Combinatorial topology
- Computational geometry
- Computational topology
- Topological data analysis
- Topology
- Discrete mathematics
- Geospatial topology
References
- Herman, Gabor T. (1998). Geometry of Digital Spaces. Applied and Numerical Harmonic Analysis. Boston, MA: Birkhäuser Boston, Inc. MR 1711168.
- Kong, Tat Yung; Rosenfeld, Azriel, eds. (1996). Topological Algorithms for Digital Image Processing. Elsevier. ISBN 0-444-89754-2.
- Voss, Klaus (1993). Discrete Images, Objects, and Functions in . Algorithms and Combinatorics. Vol. 11. Berlin: Springer-Verlag. MR 1224678.
- Chen, L. (2004). Discrete Surfaces and Manifolds: A Theory of Digital-Discrete Geometry and Topology. SP Computing. ISBN 0-9755122-1-8.
- Klette, R.; Rosenfeld, Azriel (2004). Digital Geometry. Morgan Kaufmann. ISBN 1-55860-861-3.
- Morgenthaler, David G.; Rosenfeld, Azriel (1981). "Surfaces in three-dimensional digital images". MR 0686842.
- Pavlidis, Theo (1982). Algorithms for graphics and image processing. Lecture Notes in Mathematics. Vol. 877. Rockville, MD: Computer Science Press. MR 0643798.
- ISBN 978-3-9812252-0-4.