### Abstract

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.

Original language | English (US) |
---|---|

Title of host publication | Proceedings of SPIE - The International Society for Optical Engineering |

Publisher | Society of Photo-Optical Instrumentation Engineers |

Pages | 57-65 |

Number of pages | 9 |

Volume | 3816 |

State | Published - 1999 |

Externally published | Yes |

Event | Proceedings of the 1999 Mathematical Modeling, Bayesian Estimation, and Inverse Problems - Denver, CO, USA Duration: Jul 21 1999 → Jul 23 1999 |

### Other

Other | Proceedings of the 1999 Mathematical Modeling, Bayesian Estimation, and Inverse Problems |
---|---|

City | Denver, CO, USA |

Period | 7/21/99 → 7/23/99 |

### Fingerprint

### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Condensed Matter Physics

### Cite this

*Proceedings of SPIE - The International Society for Optical Engineering*(Vol. 3816, pp. 57-65). Society of Photo-Optical Instrumentation Engineers.

**Robust openings in the context of a prior distribution governing the parameters of the random set model.** / Dougherty, Edward R.; Chen, Yidong.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of SPIE - The International Society for Optical Engineering.*vol. 3816, Society of Photo-Optical Instrumentation Engineers, pp. 57-65, Proceedings of the 1999 Mathematical Modeling, Bayesian Estimation, and Inverse Problems, Denver, CO, USA, 7/21/99.

}

TY - GEN

T1 - Robust openings in the context of a prior distribution governing the parameters of the random set model

AU - Dougherty, Edward R.

AU - Chen, Yidong

PY - 1999

Y1 - 1999

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0033364905&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033364905&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:0033364905

VL - 3816

SP - 57

EP - 65

BT - Proceedings of SPIE - The International Society for Optical Engineering

PB - Society of Photo-Optical Instrumentation Engineers

ER -