Markovian analysis of adaptive reconstructive multiparameter τ-openings

A classical single-parameter τ-opening is a union of openings in which each structuring element is scaled by the same parameter. Multiparameter binary τ-openings generalize the model in two ways: first, parameters for each opening are individually defined; second, a structuring element can be parameterized relative to its overall shape, not merely sized. The reconstructive filter corresponding to an opening is defined by fully passing any grain (connected component) that is not fully eliminated by the opening and deleting all other grains. Adaptive design results from treating the parameter vector of a reconstructive multiparameter τ-opening as the state space of a Markov chain. Signal and noise are modeled as unions of randomly parameterized and randomly translated primary grains, and the parameter vector is transitioned depending on whether an observed grain is correctly or incorrectly passed. Various adaptive models are considered, transition probabilities are discussed, the state-probability increment equations are deduced from the appropriate Chapman-Kolmogorov equations, and convergence of the adaptation is characterized by the steady-state distribution relating to the Markov chain.