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Proceedings Paper

Banach Morphology: An Alternative To Impenetrability
Author(s): Frederick M. Waltz
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Paper Abstract

Least-squares template matching and correlation have long been used for detecting, locating, and quantifying features in images. More recently, binary and grey-level morphology have been used to accomplish similar functions. Least squares techniques involve fitting a function to the image in such a way as to minimize the norm of the error in,L2(Hilbert) space, and result in errors being distributed approximately equally on both sides (+ and -) of the fitted function. Morphological techniques involve fitting a function (i.e., a structuring element) to the image in L_space, and result in all the errors being distributed on the same side of the fitted function. Thus, the final result is completely determined by the worst case error, without regard for the other errors. As a result, morphological techniques are very sensitive to noise. There is a need for a technique falling between these two extremes of "full penetration" (12 space) and "no penetration" (L2 space). Banach spaces (Lp-spaces) provide the basis for such a technique. The Banach-space norm for discretized images has the form II f IIp = [(1/N)ΣIfjIP](1/p)where the summation is taken over all of the N pixels in the structuring element. By an appropriate choice of p, the degree of penetration appropriate to a given type of problem and image noise level can be achieved. This paper presents the mathematical basis for Banach morphology, gives some simple examples, describes possible implementations on existing hardware, and suggests an architecture suited to high-speed implementation.

Paper Details

Date Published: 1 February 1990
PDF: 6 pages
Proc. SPIE 1197, Automated Inspection and High-Speed Vision Architectures III, (1 February 1990); doi: 10.1117/12.969942
Show Author Affiliations
Frederick M. Waltz, 3M Company (United States)

Published in SPIE Proceedings Vol. 1197:
Automated Inspection and High-Speed Vision Architectures III
Michael J. W. Chen, Editor(s)

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