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<p>In this paper, we derive the optimal structuring elements of morphological filters in image restoration. The expected pattern transformation of random sets is presented. An estimation theory framework for random sets is subsequently proposed. This framework is based on the least mean difference (LMD) estimator. The LMD estimator is defined to minimize the cardinality of the expected pattern transformation of the set-difference of the parameter and the estimate. Several important results for the determination of the LMD estimator are derived. The LMD structuring elements of morphological filters in image restoration are finally derived.</p>
image reconstruction; estimation theory; mathematical morphology; filtering and prediction theory; random processes; set theory; optimal structuring elements; morphological pattern restoration; binary images; morphological filters; image restoration; random sets; estimation theory; least mean difference estimator; cardinality

D. Schonfeld, "Optimal Structuring Elements for the Morphological Pattern Restoration of Binary Images," in IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 16, no. , pp. 589-601, 1994.
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