### Abstract

A granulometric bandpass filter (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 filters. An optimal filter depends on the parameters governing the distribution of the random set. In practice, the filter 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 filter performance degrade as conditions vary from those under which it has been designed? This paper considers GBF robustness. It examines three issues: minimax robust filters, 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 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 filter designed for a specific state over all possible states. We would like to find maximally robust states. Finally, we consider global filters. A global filter 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 filter, which means 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 filter whose expected increase is close to that for a maximally robust state.

Original language | English (US) |
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Pages (from-to) | 1357-1372 |

Number of pages | 16 |

Journal | Signal Processing |

Volume | 81 |

Issue number | 7 |

DOIs | |

State | Published - Jul 1 2001 |

Externally published | Yes |

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### Keywords

- Bayesian robust filter
- Granulometry
- Minimax robust filter
- Optimal filter

### ASJC Scopus subject areas

- Control and Systems Engineering
- Software
- Signal Processing
- Computer Vision and Pattern Recognition
- Electrical and Electronic Engineering

### Cite this

*Signal Processing*,

*81*(7), 1357-1372. https://doi.org/10.1016/S0165-1684(01)00017-2