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Continuous digitization in Khalimsky spaces
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2008 (English)In: Journal of Approximation Theory, ISSN 0021-9045, Vol. 150, no 1, 96-116 p.Article in journal (Refereed) Published
Abstract [en]

A real-valued function defined on R-n can sometimes be approximated by a Khalimsky-continuous mapping defined on Z(n). We elucidate when this can be done and give a construction for the approximation. This approximation can be used to define digital Khalimsky hyperplanes that are topological embeddings of Z(n) into Z(n+1). In particular, we consider Khalimsky planes in Z(3) and show that the intersection of two non-parallel Khalimsky planes contains a curve homeomorphic to the Khalimsky line.

Place, publisher, year, edition, pages
2008. Vol. 150, no 1, 96-116 p.
Keyword [en]
digitization, digital geometry, Khalimsky topology, digital surface, digital plane
National Category
URN: urn:nbn:se:uu:diva-96677DOI: 10.1016/j.jat.2007.06.003ISI: 000253000500005OAI: oai:DiVA.org:uu-96677DiVA: diva2:171328
Available from: 2008-02-05 Created: 2008-02-05 Last updated: 2010-01-13Bibliographically approved
In thesis
1. Digital Geometry and Khalimsky Spaces
Open this publication in new window or tab >>Digital Geometry and Khalimsky Spaces
2008 (English)Doctoral thesis, comprehensive summary (Other academic)
Alternative title[sv]
Digital Geometri och Khalimskyrum
Abstract [en]

Digital geometry is the geometry of digital images. Compared to Euclid’s geometry, which has been studied for more than two thousand years, this field is very young.

Efim Khalimsky’s topology on the integers, invented in the 1970s, is a digital counterpart of the Euclidean topology on the real line. The Khalimsky topology became widely known to researchers in digital geometry and computer imagery during the early 1990s.

Suppose that a continuous function is defined on a subspace of an n-dimensional Khalimsky space. One question to ask is whether this function can be extended to a continuous function defined on the whole space. We solve this problem. A related problem is to characterize the subspaces on which every continuous function can be extended. Also this problem is solved.

We generalize and solve the extension problem for integer-valued, Khalimsky-continuous functions defined on arbitrary smallest-neighborhood spaces, also called Alexandrov spaces.

The notion of a digital straight line was clarified in 1974 by Azriel Rosenfeld. We introduce another type of digital straight line, a line that respects the Khalimsky topology in the sense that a line is a topological embedding of the Khalimsky line into the Khalimsky plane.

In higher dimensions, we generalize this construction to digital Khalimsky hyperplanes, surfaces and curves by digitization of real objects. In particular we study approximation properties and topological separation properties.

The last paper is about Khalimsky manifolds, spaces that are locally homeomorphic to n-dimensional Khalimsky space. We study different definitions and address basic questions such as uniqueness of dimension and existence of certain manifolds.

Place, publisher, year, edition, pages
Uppsala: Matematiska institutionen, 2008. vii+47 p.
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 54
Applied mathematics, Khalimsky topology, digital geometry, digital topology, Alexandrov space, digital surface, digital curve, digital manifold, continuous extension, smallest-neighborhood space, image processing, Tillämpad matematik
urn:nbn:se:uu:diva-8419 (URN)978-91-506-1983-6 (ISBN)
Public defence
2008-02-29, Häggsalen, Ångström Laboratory, 13:15
Available from: 2008-02-05 Created: 2008-02-05Bibliographically approved

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