Optimal and adaptive reconstructive granulometric bandpass filters

Granulometric spectral decomposition results from partitioning an image according to the manner in which a granulometry diminishes the image. An optimal granulometric bandpass filter is one that passes spectral components in a way to minimize the expected area of the symmetric difference between the filtered and ideal images. The present paper treats bandpass optimization for reconstructive granulometries. For these, each connected grain in the input image is either fully passed or eliminated. Such filters are well-suited for elimination of clutter or, equivalently, locating grains in size-shape bands. The observed image is typically modeled as a disjoint union of signal and clutter grains and the filter is designed to best eliminate clutter while maintaining the signal. The method is very general: grains are considered to be realizations of random sets; there are no shape constraints on signal and noise grains; there are no similarity constraints between granulometric and image generators; and the method applies to overlapping grains by filtering the image model resulting from segmentation preprocessing. Three filter design paradigms are considered, one for optimal and two for adaptive filters.