Optimal nonlinear filter in the random Boolean model
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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.
