Optimal Conjunctive Granulometric Bandpass Filters

• Authors: Edward R. Doughert
• Venue: Journal of Mathematical Imaging and Vision
• Year: 2001

A conjunctive homothetic granulometry is an intersection of openings by independently scaled structuring elements. Like classical Euclidean granulometries, conjunctive granulometries possess size distributions and pattern spectra; however, they are based on intersections of openings by scaled structuring elements instead of unions of such openings. For a disjunctive granulometry, which is a union of openings, a grain (or part thereof) is passed if there exists a translate of at least one structuring element that is a subset of the grain; for a conjunctive granulometry, there must exist a translate of each structuring element that is a subset of the grain. Like disjunctive granulometries, they possess size distributions; however, unlike disjunctive granulometries, their pattern spectra are not probability densities. An optimal granulometric bandpass filter passes image components so as to minimize the expected area of the symmetric difference between the filtered and ideal images. This paper provides an analytic formulation of optimal conjunctive granulometric bandpass filters. The theory provides one of the few areas of nonlinear image processing in which three of the basic components of linear optimization apply: (1) there is an analytic expression determining the optimal filter; (2) there is an explicit error formula; (3) and there is a closed-form representation of the optimal filter based on a decomposition of the observed random image. These correspond to the Wiener-Hopf equation, mean-square-error formula, and filter representation via an integral canonical representation of the observed image in linear filtering.