Optimal nonlinear filter in the random Boolean model

John C. Handley; Edward R. Dougherty

doi:10.1117/12.205230

28 March 1995 Optimal nonlinear filter in the random Boolean model

John C. Handley, Edward R. Dougherty

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Proceedings Volume 2424, Nonlinear Image Processing VI; (1995) https://doi.org/10.1117/12.205230
Event: IS&T/SPIE's Symposium on Electronic Imaging: Science and Technology, 1995, San Jose, CA, United States

Abstract

A one-dimensional discrete Boolean model is a random process on the discrete line where random-length line segments are positioned according to the outcomes of a Bernoulli process. Points on the discrete line are either covered or left uncovered by a realization of the process. An observation of the process consists of runs of covered and not-covered points, called black and white runlengths, respectively. The black and white runlengths form an alternating sequence of independent random variables. We show how the Boolean model is completely determined by probability distributions of these random variables by giving explicit formulas linking the marking probability of the Bernoulli process and segment length distribution with the runlength distributions. The black runlength density is expressed recursively in terms of the marking probability and segment length distribution and white runlengths are shown to have a geometric probability law. Filtering for the Boolean model can also be done via runlengths. The optimal minimum mean absolute error filter for union noise is computed as the binary conditional expectation for windowed observations, expressible as a function observed black runlengths.

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John C. Handley and Edward R. Dougherty "Optimal nonlinear filter in the random Boolean model", Proc. SPIE 2424, Nonlinear Image Processing VI, (28 March 1995); https://doi.org/10.1117/12.205230

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KEYWORDS

Statistical modeling

Signal processing

Binary data

Optimal filtering

Filtering (signal processing)

Radon

Digital filtering

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