## Abstract

This chapter proposes a general basis algorithm to compute bases for a range of τ-mappings and have subsequently extended these results to translation-invariant (TI) mappings and to gray-scale τ-mappings. The general basis algorithm proposed is composed of three tools, based on the algebra of union, intersection, and translation. All of these tools involve the combination of just two bases but, when combined with one another and used repeatedly, can be used to compute bases for τ-mappings—such as opening, closing, open-close, close-open, and all other cascaded τ-mappings. The tools used by the general basis algorithm and the reversal of these tools constitutes a complete description of all possible transformations involving the basis representation. The basis representation is a common platform from which the study, design, and implementation of all τ-mappings can be approached. The parallel representation of the basis uses explicit rather than contingent logic. The examination of the basis for any τ -mapping exposes the detailed workings of that τ-mapping, and finally, this may be used to select optimal structuring elements for τ-mappings or to create new and useful τ-mappings.

Original language | English |
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Title of host publication | Advances in Electronics and Electron Physics |

Place of Publication | United Kingdom |

Publisher | Academic Press |

Pages | 325-390 |

Number of pages | 66 |

Volume | 89 |

ISBN (Print) | 0-12-014731-9 |

DOIs | |

Publication status | Published - 1 Jan 1994 |