Random sets provide a powerful class of models for images containing randomly placed objects of random shapes and orientation. Those pixels within the foreground are members of a random set realization. The discrete Boolean model is the simplest general random set model in which a Bernoulli point process (called a germ process) is coupled with an independent shape or grain process. A typical realization consists of many overlapping shapes. Estimation in these models is difficult owing to the fact that many outcomes of the process obscure other outcomes. The directional one-dimensional (ID) model, in which random- length line segments emanate to the right from germs on the line, is analyzed via recursive expressions to provide a complete characterization of these discrete models in terms of the distributions of their black and white runlengths. An analytic representation is given for the optimal windowed filter for the signalunion- noise process, where both signal and noise are Boolean models. Several of these results are extended to the nondirectional case where segments can emanate to the left and right. Sufficient conditions are presented for a two-dimensional (2D) discrete Boolean model to induce a one dimensional Boolean model on an intersecting line. When inducement holds, the likelihood of runlength observations of the two-dimensional model is used to provide maximum-likelihood estimation of parameters of the 2D model. The ID directional discrete Boolean model is equivalent to the discrete-time infinite-server queue. Analysis for the Boolean model is extended to provide densities for many random variables of interest in queueing theory.
Library of Congress Subject Headings
Image processing--Mathematics; Set theory; Algebra, Boolean; Model theory; Stochastic processes; Recursion theory
Department, Program, or Center
Chester F. Carlson Center for Imaging Science (COS)
Handley, John, "Recursive analysis and estimation for the discrete Boolean random set model" (1996). Thesis. Rochester Institute of Technology. Accessed from
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