Optimal Binary Morphological Bandpass Filters Induced by Granulometric Spectral Representation

Euclidean granulometries are used to decompose a binary image into a disjoint union based on interaction between image shape and the structuring elements generating the granulometry. Each subset of the resulting granulometric spectral bands composing the union defines a filter by passing precisely the bands in the subset. Given an observed image and an ideal image to be estimated, an optimal filter must minimize the expected symmetric-difference error between the ideal image and filtered observed image. For the signal-union-noise model, and for both discrete and Euclidean images, given a granulometry, a procedure is developed for finding a filter that optimally passes bands of the observed noisy image. The key is characterization of an optimal filter in the Euclidean case. Optimization is achieved by decomposing the mean functions of the signal and noise size distributions into singular and differentiable parts, deriving an error representation based on the decomposition, and describingoptimality in terms of generalized derivatives for the singular parts and ordinary derivatives for the differentiable parts. Owing to the way in which spectral bands are optimally passed, there are strong analogies with the Wiener filter.