Markovian analysis of adaptive reconstructive multiparameter tau-openings

A classical single-parameter tau-opening is a union of openings in which ea ch structuring element is scaled by the same parameter. Multiparameter bina ry tau-openings generalize the model in two ways: first, parameters for eac h opening are individually defined; second, a structuring element can be pa rameterized relative to its overall shape, not merely sized. The reconstruc tive filter corresponding to an opening is defined by fully passing any gra in (connected component) that is not fully eliminated by the opening and de leting all other grains. Adaptive design results from treating the paramete r vector of a reconstructive multiparameter tau-opening as the state space of a Markov chain. Signal and noise are modeled as unions of randomly param eterized and randomly translated primary grains, and the parameter vector i s transitioned depending on whether an observed grain is correctly or incor rectly passed. Various adaptive models are considered, transition probabili ties are discussed, the state-probability increment equations are deduced f rom the appropriate Chapman-Kolmogorov equations, and convergence of the ad aptation is characterized by the steady-state distribution relating to the Markov chain.