Dimension for Alexandrov spaces

Petra Wiederhold; Richard Wilson

doi:10.1117/12.142181

9 April 1993 Dimension for Alexandrov spaces

Petra Wiederhold, Richard Wilson

Proceedings Volume 1832, Vision Geometry; (1993) https://doi.org/10.1117/12.142181
Event: Applications in Optical Science and Engineering, 1992, Boston, MA, United States

Abstract

This paper continues the study of the topological model of the support of a digital image published by Kronheimer in 1992. There, he interpreted the generation of the support D of the image from a topological space S by means of some 'discretization' as the construction of a quotient space (Delta) of S, which represents the set D an d has a reasonable (non-discrete) topology. Under some conditions the space (Delta) is an Alexandrov space. Having in mind the practical example S equals Rⁿ and D equals Zⁿ we speak of 'n-dimensional images', although there is no dimension on the space (Delta) . We define in this paper a so-called Alexandrov dimension for arbitrary Alexandrov spaces. Under this definition an image which was sampled from a function defined on Rⁿ has dimension n. If the Alexandrov space (Delta) is T₀, then it corresponds to a canonical partially ordered set ((Delta) , ≤). We prove, that in this case the Alexandrov dimension coincides with the height of ((Delta) , ≤).

Citation Download Citation

Petra Wiederhold and Richard Wilson "Dimension for Alexandrov spaces", Proc. SPIE 1832, Vision Geometry, (9 April 1993); https://doi.org/10.1117/12.142181

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