Preserving Topology by a Digitization Process

  • Authors:

  • Longin Jan Latecki;Christopher Conrad;Ari Gross

  • Affiliations:

  • Department of Computer Science, University of Hamburg, Vogt-Kölln-Str. 30, 22527 Hamburg, Germany. E-mail: latecki@informatik.uni-hamburg.de;Department of Applied Mathematics, University of Hamburg, Bundesstr. 55, 20146 Hamburg, Germany. E-mail: conrad@math.uni-hamburg.de;Department of Computer Science, Graduate Center and Queens College, CUNY, Flushing, New York 11367, USA. E-mail: ari@vision.cs.qc.edu

  • Venue:
  • Journal of Mathematical Imaging and Vision
  • Year:
  • 1998

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Abstract

The main task of digital image processing is to recognizeproperties of real objects based on their digital images. These images areobtained by some sampling device, like a CCD camera, and represented asfinite sets of points that are assigned some value in a gray-level or colorscale. Based on technical properties of sampling devices, these points areusually assumed to form a square grid and are modeled as finite subsets of Z^2. Therefore, a fundamental question in digital imageprocessing is which features in the digital image correspond, under certainconditions, to properties of the underlying objects. In practicalapplications this question is mostly answered by visually judging theobtained digital images. In this paper we present a comprehensive answer tothis question with respect to topological properties. In particular, wederive conditions relating properties of real objects to the grid size ofthe sampling device which guarantee that a real object and its digitalimage are topologically equivalent. These conditions also imply that twodigital images of a given object are topologically equivalent. This means,for example, that shifting or rotating an object or the camera cannot leadto topologically different images, i.e., topological properties of obtaineddigital images are invariant under shifting and rotation.