### Abstract

Basic morphological operations can be incorporated within a statistical physics formulation as a limit when the temperature of the system tends to zero. These operations can then be expressed in terms of finding minimum variance estimators of probability distributions. It enables one to relate these operations to alternative Bayesian or Markovian approaches to image analysis. It is shown how to derive elementary dilations (winner-take-all) and erosions (loser-take-all). These operations, referred to as statistical dilations and erosion, depend on a temperature parameter β = 1/T. They become purely morphological as β goes to infinity and purely linear averages as β goes to 0. Experimental results are given for a range of intermediate values of β. Concatenations of elementary operations can be naturally expressed by stringing together conditional probability distributions, each corresponding to the original operations, thus yielding statistical openings and closings. Techniques are given for computing the minimal variance estimators.

Original language | English (US) |
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Title of host publication | Proceedings of SPIE - The International Society for Optical Engineering |

Publisher | Publ by Int Soc for Optical Engineering |

Pages | 271-282 |

Number of pages | 12 |

Volume | 1568 |

ISBN (Print) | 0819406961 |

State | Published - 1991 |

Event | Image Algebra and Morphological Image Processing II - San Diego, CA, USA Duration: Jul 23 1991 → Jul 24 1991 |

### Other

Other | Image Algebra and Morphological Image Processing II |
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City | San Diego, CA, USA |

Period | 7/23/91 → 7/24/91 |

### Fingerprint

### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Condensed Matter Physics

### Cite this

*Proceedings of SPIE - The International Society for Optical Engineering*(Vol. 1568, pp. 271-282). Publ by Int Soc for Optical Engineering.

**Statistical morphology.** / Yuille, Alan L.; Vincent, Luc M.; Geiger, Davi.

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

*Proceedings of SPIE - The International Society for Optical Engineering.*vol. 1568, Publ by Int Soc for Optical Engineering, pp. 271-282, Image Algebra and Morphological Image Processing II, San Diego, CA, USA, 7/23/91.

}

TY - GEN

T1 - Statistical morphology

AU - Yuille, Alan L.

AU - Vincent, Luc M.

AU - Geiger, Davi

PY - 1991

Y1 - 1991

N2 - Basic morphological operations can be incorporated within a statistical physics formulation as a limit when the temperature of the system tends to zero. These operations can then be expressed in terms of finding minimum variance estimators of probability distributions. It enables one to relate these operations to alternative Bayesian or Markovian approaches to image analysis. It is shown how to derive elementary dilations (winner-take-all) and erosions (loser-take-all). These operations, referred to as statistical dilations and erosion, depend on a temperature parameter β = 1/T. They become purely morphological as β goes to infinity and purely linear averages as β goes to 0. Experimental results are given for a range of intermediate values of β. Concatenations of elementary operations can be naturally expressed by stringing together conditional probability distributions, each corresponding to the original operations, thus yielding statistical openings and closings. Techniques are given for computing the minimal variance estimators.

AB - Basic morphological operations can be incorporated within a statistical physics formulation as a limit when the temperature of the system tends to zero. These operations can then be expressed in terms of finding minimum variance estimators of probability distributions. It enables one to relate these operations to alternative Bayesian or Markovian approaches to image analysis. It is shown how to derive elementary dilations (winner-take-all) and erosions (loser-take-all). These operations, referred to as statistical dilations and erosion, depend on a temperature parameter β = 1/T. They become purely morphological as β goes to infinity and purely linear averages as β goes to 0. Experimental results are given for a range of intermediate values of β. Concatenations of elementary operations can be naturally expressed by stringing together conditional probability distributions, each corresponding to the original operations, thus yielding statistical openings and closings. Techniques are given for computing the minimal variance estimators.

UR - http://www.scopus.com/inward/record.url?scp=0025789338&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0025789338&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:0025789338

SN - 0819406961

VL - 1568

SP - 271

EP - 282

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

PB - Publ by Int Soc for Optical Engineering

ER -