Optimal mean-absolute-error hit-or-miss filters: morphological representation and estimation of the binary conditional expectation

The hit-or-miss operator is used as the building block of optimal binary restoration filters. Filter design methodologies are given for general-, maximum-, and minimum-noise environments, the latter two producing optimal thinning and thickening filters, respectively, and for iterative filters. The approach is based on the expression of translation-invariant filters as unions of hit-or-miss transforms. Unions of hit-or-miss transforms are expressed as canonical logical sums of products, and the final hit-or-miss templates are obtained by logic reduction. The net effect is a morphological representation and estimation of the conditional expectation, which is the overall optimal mean-absolute-error filter.