Optimal nonlinear filter for signal-union-noise and runlength analysis in the directional one-dimensional discrete Boolean random set 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 a 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 the 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. This equivalence allows us to do runlength analysis of the one-dimensional Boolean model. Filtering for the Boolean model can also be done via the runlengths. The windowed optimal minimum-mean-absolute-error filter for union noise is computed as the binary conditional expectation. It is expressible as a function of the observed black runlengths.
