Robust openings in the context of a prior distribution governing the parameters of the random set model

In the context of image restoration, optimal binary openings estimate an ideal random set from an observed random set. If we consider optimization relative to a homothetic scalar t that governs structuring element sizes, then opening optimization can be placed into the context of optimal granulometric bandpass filters and solution of the optimization problem for the signal-union-noise model can be given in terms of the granulometric spectral densities (GSDs) of the signal and noise. The robustness question arises if the signal and noise GSDs are parameterized, so that the model can assume a family of states (of nature): specifically, what is the cost of applying an optimal opening designed for one pair of GSDs to a model corresponding to a different pair of GSDs? This paper addresses the robustness problem in the context of a prior distribution for the parameters governing the signal and noise GSDs. It does so by considering the mean robustness, which is defined for each state of nature to be the expected increase in error resulting from using the optimal opening for that state across all states. Moreover, it considers a global filter that is defined for all states via the expected optimal homothetic scalar (relative to the prior distribution of the parameter). Finally, it compares Baysian robust openings to minimax robust openings.