Efficient derivation of the optimal mean-square binary morphological filter from the conditional expectation via a switching algorithm for discrete power-set lattice

Once designed, implementation of an optimal mean-square binary morphological filter is extremely fast, especially when the erosions are implemented on a suitable parallel processor. On the other hand, optimal filter design involes a computationally burdensome search procedure that can, in practice, be intractable. The present paper provides an algorithm for filter design that is based on the relationship between the optimal morphological filter and the conditional expectation. The algorithm proceeds by changing the conditional expectation into a morphological filter while at the same time increasing the mean-square error by a minimal amount. It does so by switching observations between the 1-set and the 0-set of the conditional expectation. The switching algorithm is extremely efficient in many noise environments, and therefore provides a filter design that can be useful for online structuring-element updating. Owing to the relationship between stack and morphological filters, the algorithm is at once useful for finding optimal binary stack filters.