Optimal mean-square N-observation digital morphological filters I. Optimal binary filters
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The present paper places binary morphological filtering into the framework of statistical estimation, the intent being to develop the theory of mean-square (MS) optimization. Classical binary morphological operations are interpreted as numerical functionals on binary N-vectors, so that in the random setting they can be treated as estimators dependent on N binary observation random variables. For single-erosion filters, optimization is achieved by finding the structuring element that minimizes MS error. Using the Matheron representation as a guide, we generalize the analysis to morphological filters given by unions of multiple erosions and optimize by minimizing MS error over all collections of erosions, or over a prefixed number of erosions. In all cases, MS error is relative to the estimation of an unobserved variable by a morphological function of observed variables. A key element in the method is use of the basis form of the Matheron expansion to reduce significantly the structuring-element search. The technique is adapted to special morphological filters by constraining the basis representation in accordance with the class of interest. It is demonstrated that optimization in terms of erosions is equivalent to optimization in terms of dilations. 1992.
