Optimal morphological restoration: The morphological filter mean-absolute-error theorem

Authors:
Robert P. Loge;Edward R. Dougherty
Affiliations:
Joseph C. Wilson Center for Technology, Xerox Corporation, Webster, New York, 14580, USA;Center for Imaging Science, Rochester Institute of Technology, One Lomb Memorial Drive, Rochester, New York, 14623, USA
Venue:
Journal of Visual Communication and Image Representation
Year:
1992

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Abstract

Morphological restoration is grounded on the Matheron representation for morphological filters, in the present context these being monotonically increasing, translation-invariant image-to-image operators. As conceived in its most general form, optimal-morphological-filter design involves a search over potential bases of structuring elements that can be used to form the Matheron erosion expansion. The present paper provides expressions for the mean-absolute restoration error of general morphological filters formed from erosion bases in terms of mean-absolute errors of single-erosion filters. It does so in both the binary setting, where the expansion is a union of erosions, and in the gray-scale setting, where the expansion is a maxima of erosions. Expressing the mean-absolute-error theorem in a recursive form leads to a unified methodology for the design of optimal (suboptimal) morphological restoration filters. Applications to binary-image, gray-scale signal, and order-statistic restoration on images are provided.