### Abstract

The aim of this paper is to show that 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 us to relate these operations to alternative Bayesian or Markovian approaches to image analysis. We first show 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 become purely linear averages as β goes to zero. 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 minimum-variance estimators. Finally, we describe simulations comparing statistical morphology and Bayesian methods for image smoothing, edge detection, and noise reduction.

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

Pages (from-to) | 223-238 |

Number of pages | 16 |

Journal | Journal of Mathematical Imaging and Vision |

Volume | 1 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1992 |

### Fingerprint

### Keywords

- Bayesian models
- mean field theory
- morphology
- statistical physics

### ASJC Scopus subject areas

- Applied Mathematics
- Control and Systems Engineering
- Software
- Computer Vision and Pattern Recognition
- Computer Graphics and Computer-Aided Design
- Artificial Intelligence

### Cite this

*Journal of Mathematical Imaging and Vision*,

*1*(3), 223-238. https://doi.org/10.1007/BF00129877

**Statistical morphology and Bayesian reconstruction.** / Yuille, Alan; Vincent, Luc; Geiger, Davi.

Research output: Contribution to journal › Article

*Journal of Mathematical Imaging and Vision*, vol. 1, no. 3, pp. 223-238. https://doi.org/10.1007/BF00129877

}

TY - JOUR

T1 - Statistical morphology and Bayesian reconstruction

AU - Yuille, Alan

AU - Vincent, Luc

AU - Geiger, Davi

PY - 1992/9

Y1 - 1992/9

N2 - The aim of this paper is to show that 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 us to relate these operations to alternative Bayesian or Markovian approaches to image analysis. We first show 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 become purely linear averages as β goes to zero. 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 minimum-variance estimators. Finally, we describe simulations comparing statistical morphology and Bayesian methods for image smoothing, edge detection, and noise reduction.

AB - The aim of this paper is to show that 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 us to relate these operations to alternative Bayesian or Markovian approaches to image analysis. We first show 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 become purely linear averages as β goes to zero. 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 minimum-variance estimators. Finally, we describe simulations comparing statistical morphology and Bayesian methods for image smoothing, edge detection, and noise reduction.

KW - Bayesian models

KW - mean field theory

KW - morphology

KW - statistical physics

UR - http://www.scopus.com/inward/record.url?scp=0000381266&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000381266&partnerID=8YFLogxK

U2 - 10.1007/BF00129877

DO - 10.1007/BF00129877

M3 - Article

VL - 1

SP - 223

EP - 238

JO - Journal of Mathematical Imaging and Vision

JF - Journal of Mathematical Imaging and Vision

SN - 0924-9907

IS - 3

ER -