Cell complexes through time

Reinhard Klette

doi:10.1117/12.404813

23 October 2000 Cell complexes through time

Reinhard Klette

Proceedings Volume 4117, Vision Geometry IX; (2000) https://doi.org/10.1117/12.404813
Event: International Symposium on Optical Science and Technology, 2000, San Diego, CA, United States

Abstract

The history of cell complexes is closely related to the birth and development of topology in general. Johann Benedict Listing (1802 - 1882) introduced the term 'topology' into mathematics in a paper published in 1847, and he also defined cell complexes for the first time in a paper published in 1862. Carl Friedrich Gauss (1777 - 1855) is often cited as the one who initiated these ideas, but he did not publish either on topology or on cell complexes. The pioneering work of Leonhard Euler (1707 - 1783) on graphs is also often cited as the birth of topology, and Euler's work was cited by Listing in 1862 as a stimulus for his research on cell complexes. There are different branches in topology which have little in common: point set topology, algebraic topology, differential topology etc. Confusion may arise if just 'topology' is specified, without clarifying the used concept. Topological subjects in mathematics are often related to continuous models, and therefore quite irrelevant to computer based solutions in image analysis. Compared to this, only a minority of topology publications in mathematics addresses discrete spaces which are appropriate for computer-based image analysis. In these cases, often the notion of a cell complex plays a crucial role. This paper briefly reports on a few of these publications. This paper is not intended to cover the very lively progress in cell complex studies within the context of image analysis during the last two decades. Basically it stops its historic review at the time when this subject in image analysis research gained speed in 1980 - 1990. As a general point of view, the paper indicates that image analysis contributes to a fusion of topological concepts, the geometric and the abstract cell structure approach and point set topology, which may lead towards new problems for the study of topologies defined on geometric or abstract cell complexes.

Citation Download Citation

Reinhard Klette "Cell complexes through time", Proc. SPIE 4117, Vision Geometry IX, (23 October 2000); https://doi.org/10.1117/12.404813

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