Page 147 - Textos de Matemática Vol. 46

MATHEMATICAL MORPHOLOGY WITH REGIMES IN ALGEBRAIC UNIVERSES
RENE´ GUITART
Friendly dedicated to Manuela Sobral, on the occasion of her 70th birthday
Abstract. Usual structures can be described through the elementary notion of a regime of assimilations in the category of sets. This allows a translation principle between modality in discursive analysis and mor- phology (dilatations and erosions) in visual analysis. Especially it is a natural context to analyze images (as subsets of regimes).
This notion of regime and the associated calculus of augmentation and diminution are also definable in any algebraic universe, and this allows us to reach the main purpose of this paper, which is to show that in fact Mathematical Morphology and the associated construction of hyperspaces, initially developed in a set theoretical context, can be developed in any algebraic universe.
In any algebraic universe mathematical morphology exists as essen- tially the calculus of inverse relations, and consequently we automatically obtain morphologies in any category of fuzzy sets or any topos.
Introduction
In this paper we present an extension of morphological analysis in two di- rections.
The first direction of development is about the global setting, according to which we proceed to analyze and transform images and data, which usually is the calculus of relations in the category Set of sets and maps. We show that the structural morphological operations, and basically dilatation and erosion, make sense in a very general setting, the context of an algebraic universe. This extension automatically with no new e↵orts allows to obtain toposified or fuzzified versions of morphological analysis.
Received: 12 August 2014 / Accepted: 9 January 2015.
2010 Mathematics Subject Classification. 18B, 18C, 03G30, 03E72, 06A15, 54, 54E15,
54E35, 52A.
Key words and phrases. Mathematical morphology, convexity, uniform structure, hyper-
space, regime, algebraic universe, topos, fuzzy set.
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