Nearly continuous functions in digital images

Longin Jan Latecki; Frank Prokop

doi:10.1117/12.198618

4 January 1995 Nearly continuous functions in digital images

Longin Jan Latecki, Frank Prokop

Proceedings Volume 2356, Vision Geometry III; (1995) https://doi.org/10.1117/12.198618
Event: Photonics for Industrial Applications, 1994, Boston, MA, United States

Abstract

Starting with the intuitive concept of `nearness' as a binary relation, semi-proximity spaces (sp-spaces) are defined. The restrictions on semi-proximity spaces are weaker than the restrictions on topological proximity spaces. Nevertheless, semi-proximity spaces generalize classical topological spaces. Moreover, it is possible to describe all digital pictures used in computer vision and computer graphics as non-trivial semi-proximity spaces, which is not possible in classical topology. Therefore, we use semi-proximity spaces to establish a formal relationship between the `topological' concepts of digital image processing and their continuous counterparts in Rⁿ. Especially interesting are continuous functions in semi- proximity spaces. The definition of a `nearly' bicontinuous function is given which does not require the function to be one-to-one. A nearly bicontinuous function preserves connectedness in both directions. Therefore, nearly bicontinuous functions can be used for characterizing well-behaved operations on digital images such as thinning. Further, it is shown that the deletion of a simple point can be treated as a nearly bicontinuous function. These properties and the fact that a variety of nearness relations can be defined on digital pictures indicate that nearly continuous functions are a useful tool in the difficult task of shape description.

Citation Download Citation

Longin Jan Latecki and Frank Prokop "Nearly continuous functions in digital images", Proc. SPIE 2356, Vision Geometry III, (4 January 1995); https://doi.org/10.1117/12.198618

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