Robust optimal granulometric bandpass filters

A granulometric bandpass "lter (GBF) passes certain size bands in a binary image. There exists an analytic formulation for the optimization of GBFs in terms of the granulometric size density of a random set. This size density plays the role of the power spectral density for optimization of linear "lters. An optimal "lter depends on the parameters governing the distribution of the random set. In practice, the "lter will be applied in statistical conditions that do not exactly match those for which it has been designed. Hence the robustness question: to what extent does "lter performance degrade as conditions vary from those under which it has been designed? This paper considers GBF robustness. It examines three issues: minimax robust "lters, Bayesian robust "lters, and global "lters. A minimax robust "lter is one whose worst performance over all states of nature is best among the optimal "lters over all states. Minimax robustness does not take into account the probability mass of the states of nature. Bayesian robustness analysis takes state mass into account and is focused on mean robustness, which is the expected error increase owing to applying a "lter designed for a speci"c state over all possible states. We would like to "nd maximally robust states. Finally, we consider global "lters. A global "lter is designed according to some mean condition of the states and is applied across all states. Here, we hope for a uniformly most robust global "lter, whichmeans that its expected error increase across all states is less than the mean robustness for any state-speci"c "lter. At least we would like a global "lter whose expected increase is close to that for a maximally robust state. 2001 Elsevier Science B.V. All rights reserved.