Mean-Absolute-Error Representation and Optimization of Computational-Morphological Filters

Abstract Computational mathematical morphology provides a framework for analysis and representation of range-preserving, finite-range operators in the context of mathematical morphology. As such, it provides a framework for statistically optimal design in the framework of a Matheron-type representation; that is, each increasing, translation-invariant filter can be expressed via the erosions generated by structuring elements in a basis. The present paper develops the corresponding mean-absolute-error representation, This representation expresses the error of estimation of a filter composed of some number of erosions in terms of single-erosion filter errors. There is a recursive form of the representation that permits calculation of filter errors from errors for filters composed of fewer structuring elements, Finally, the error representation is employed in designing an optimal filter to solve an image enhancement problem in electronic printing, the transformation of a 1-bit per pixel image into a 2-bit per pixel image.