Robust optimal granulometric bandpass filters

A granulometric bandpass filter (GBF) passes certain size bands in a binary image. There exists an analytic formulation for the optimization of GBFs i n terms of the granulometric size density of a random set. This size densit y plays the role of the power spectral density for optimization of linear f ilters. An optimal filter depends on the parameters governing the distribut ion of the random set. In practice, the filter will be applied in statistic al conditions that do not exactly match those for which it has been designe d. Hence the robustness question: to what extent does filter performance de grade as conditions vary from those under which it has been designed? This paper considers GBF robustness. It examines three issues: minimax robust fi lters, Bayesian robust filters, and global filters. A minimax robust filter is one whose worst performance over all states of nature is best among the optimal filters over all states. Minimax robustness does not take into acc ount the probability mass of the states of nature. Bayesian robustness anal ysis takes state mass into account and is focused on mean robustness, which is the expected error increase owing to applying a filter designed for a s pecific state over all possible states. We would like to find maximally rob ust states. Finally, we consider global filters. A global filter is designe d according to some mean condition of the states and is applied across all states. Here, we hope for a uniformly most robust global filter, which mean s that its expected error increase across all states is less than the mean robustness for any state-specific filter. At least we would like a global f ilter whose expected increase is close to that for a maximally robust state . (C) 2001 Elsevier Science B.V. All rights reserved.