<p>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.</p><p>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.</p><p>Suppose that a continuous function is defined on a subspace of an <i>n-</i>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.</p><p>We generalize and solve the extension problem for integer-valued, Khalimsky-continuous functions defined on arbitrary smallest-neighborhood spaces, also called Alexandrov spaces.</p><p>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.</p><p>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. </p><p>The last paper is about Khalimsky manifolds, spaces that are locally homeomorphic to <i>n-</i>dimensional Khalimsky space. We study different definitions and address basic questions such as uniqueness of dimension and existence of certain manifolds.</p>
| Identifer | oai:union.ndltd.org:UPSALLA/oai:DiVA.org:uu-8419 |
| Date | January 2008 |
| Creators | Melin, Erik |
| Publisher | Uppsala University, Department of Mathematics, Uppsala : Matematiska institutionen |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Doctoral thesis, comprehensive summary, text |
| Relation | Uppsala Dissertations in Mathematics, 1401-2049 ; 54 |
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