Holes and Genus of 2D and 3D Digital Images

"Hole" has been a confusing idea in the 3D digital literature. We replace counting holes by the clear geometrical idea of counting non-separating cuts, and show that this gives the Betti number b 1, while b 0 counts components and b 2 cavities. Connected sets with equal b 1 and b 2 must ma...

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Veröffentlicht in:	CVGIP. Graphical models and image processing 1993, Vol.55 (1), p.20-47
Hauptverfasser:	Lee, C.N., Poston, T., Rosenfeld, A.
Format:	Artikel
Sprache:	eng
Schlagworte:	Applied sciences Artificial intelligence Computer science control theory systems Exact sciences and technology Pattern recognition. Digital image processing. Computational geometry
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creator	Lee, C.N. Poston, T. Rosenfeld, A.
description	"Hole" has been a confusing idea in the 3D digital literature. We replace counting holes by the clear geometrical idea of counting non-separating cuts, and show that this gives the Betti number b 1, while b 0 counts components and b 2 cavities. Connected sets with equal b 1 and b 2 must match topologically when b 1 = 0 (implying simple connectedness). When b 1 ≠ 0, contrary to digital folklore, they need not. This paper is a conceptually self-contained introduction for computer scientists to these numbers of 2D and 3D images, and to other topological features such as Euler and linking numbers.
doi_str_mv	10.1006/cgip.1993.1002
format	Article
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subjects	Applied sciences Artificial intelligence Computer science control theory systems Exact sciences and technology Pattern recognition. Digital image processing. Computational geometry
title	Holes and Genus of 2D and 3D Digital Images
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