Jordan Surfaces in Discrete Antimatroid Topologies

• Published in: Combinatorial Image Analysis pp. 334 - 350
• Main Authors: Kopperman, Ralph, Pfaltz, John L.
• Format: Book Chapter Conference Proceeding
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg 2004; Springer
• Series: Lecture Notes in Computer Science
• Subjects: Applied sciences; Artificial intelligence; Closure Operator; Computer science; control theory; systems; Digital Imagery; Discrete Topology; Exact sciences and technology; Geometric Space; Geometric Topology; Pattern recognition. Digital image processing. Computational geometry; Computational geometry; Discrete geometry; Image analysis; Closure; Image processing; Abstract interpretation; Numerical method; Topology; Combinatorial analysis; Boundary element method
• ISBN: 9783540239420; 3540239421
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-540-30503-3_25

In this paper we develop a discrete, T0 topology in which (1) closed sets play a more prominent role than open sets, (2) atoms comprising the space have discrete dimension, which (3) is used to define boundary elements, and (4) configurations within the topology can have connectivity (or separation) of different degrees. To justify this discrete, closure based topological approach we use it to establish an n-dimensional Jordan surface theorem of some interest. As surfaces in digital imagery are increasingly rendered by triangulated decompositions, this kind of discrete topology can replace the highly regular pixel approach as an abstract model of n-dimensional computational geometry.