Logical structural filters

Binary granulometric filters are formed from unions of parameterized openings, a point passing the filter if and only if a translate of at least one structuring element fits in the image and contains the point. A granulometry induces a reconstructive granulometry by passing any image component not eliminated by the granulometry. As historically studied in the context of Matheron's granulometric theory, reconstructive granulometries appear as unions of reconstructive parameterized openings. The theory is extended to a much wider class of filters: a logical structural filter (LSF) is formed as a union of intersections of both reconstructive and complementary reconstructive openings. A reconstructive opening passes a component if and only if at least one translate of the structuring element fits inside; a complementary reconstructive opening passes a component if and only if no translate of the structuring element fits inside. The original reconstructive granulometries form the special class of disjunctive LSFs. Complement-free LSFs form granulometries in a slightly more general sense; LSFs containing complements are not increasing and therefore not openings. Along with the relevant algebraic representations for LSFs, the theories of optimal and adaptive granulometric filters are extended to LSFs, a systematic formulation of adaptive transitions is given, transition probabilities for adaptation are found, and two applications to biological imaging are presented.