Banach Morphology: An Alternative To Impenetrability

Frederick M. Waltz

doi:10.1117/12.969942

1 February 1990 Banach Morphology: An Alternative To Impenetrability

Frederick M. Waltz

Proceedings Volume 1197, Automated Inspection and High-Speed Vision Architectures III; (1990) https://doi.org/10.1117/12.969942
Event: 1989 Symposium on Visual Communications, Image Processing, and Intelligent Robotics Systems, 1989, Philadelphia, PA, United States

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.

Citation Download Citation

Frederick M. Waltz "Banach Morphology: An Alternative To Impenetrability", Proc. SPIE 1197, Automated Inspection and High-Speed Vision Architectures III, (1 February 1990); https://doi.org/10.1117/12.969942

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