### Abstract

A classical morphological technique to restore binary images degraded by union noise is to perform an opening to remove the noise. The more general approach is to employ a union of openings, the resulting filter being known as a τ-opening. Assuming the structuring elements to be parameterized in terms of a single parameter, a fundamental problem is to determine the optimal parameter, namely the one that produces the filter having minimum error according to some error measure. Relative to symmetric-difference error and a certain random-grain model, the present paper develops an optimization procedure based upon the individual pattern spectra of the signal and noise. If the image and noise grains are disjoint, then the pattern-spectra parametric estimation procedure yields exactly the optimal value of the parameter. Of special concern in the present paper is the robustness of the method with respect to the disjointness criterion. It is analytically demonstrated for a particular model that the estimation procedure produces close to the optimal value when image and noise are not disjoint. Robustness is also experimentally demonstrated for a large number of more complex image-noise models.

Original language | English (US) |
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Pages (from-to) | 265-281 |

Number of pages | 17 |

Journal | Signal Processing |

Volume | 29 |

Issue number | 3 |

DOIs | |

State | Published - Dec 1992 |

Externally published | Yes |

### Keywords

- Mathematical morphology
- granulometry
- optimal filtering
- pattern spectrum
- union noise
- τ-opening

### ASJC Scopus subject areas

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

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## Cite this

*Signal Processing*,

*29*(3), 265-281. https://doi.org/10.1016/0165-1684(92)90086-C