Adaptive reconstructive τ-openings: Convergence and the steady-state distribution

A parameterized τ-opening is a filter defined as a union of openings by a collection of compact, convex structuring elements, each scalar multiplied by the parameter. For a reconstructive τ-opening, the filter is modified by fully passing any connected component not completely eliminated. Applied to the signal-union-noise model, in which the reconstructive filter is designed to sieve out clutter while passing the signal, the optimization problem is to find a parameter value that minimizes the MAE between the filtered and ideal image processes. The present study introduces an adaptation procedure for the design of reconstructive τ-openings. The adaptive filter fits into the framework of Markov processes, the adaptive parameter being the state of the process. There exists a stationary distribution governing the parameter in the steady state and convergence is characterized via the steady-state distribution. Key filter properties such as parameter mean, parameter variance, and expected error in the steady state are characterized via the stationary distribution. The Chapman-Kolmogorov equations are developed for various scanning modes and transient behavior is examined.