Logical Granulometric Filtering in the Signal—Union—Clutter Model

Abstract

A basic problem of binary morphological image filtering is to remove background clutter (noise) in order to reveal a desired target (signal). The present paper discusses the manner in which filtering can be achieved using morphological granulometric filters. Logical granulometries are unions of intersections of reconstructive openings and these use shape elements to identify image components to be passed (in full), whereas others are deleted. Assuming opening structuring elements are parameterized, the task is to find parameters that result in optimal filtering. Optimization is achieved via the notion of granulometric sizing. For situations where optimization is impractical or intractable, filter design can be achieved via adaptation. Based upon correct or incorrect decisions as to whether or not to pass a component, the filter parameter vector is adapted during training in accordance with a protocol that adapts towards correct decisions. The adaptation scheme yields a Markov chain in which the parameter space is the state space of the chain. Convergence of the adaptation procedure is characterized by the stationary distribution of the parameter vector. State-probability equations are derived via the Chapman-Kolmogorov equations and these are used to describe the steady-state distribution.