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Crisp and Fuzzy Topological Interior and Closure Operators with Inclusion Degree : Theory and Applications

Fashandi H., Peters J.F.

Fundamenta Informaticae

2013

Vol. 122, nr 3

207--225

This article introduces interior and closure operators with inclusion degree considered within a crisp or fuzzy topological framework. First, inclusion degree is introduced in an extension of the interior and closure operators in crisp topology. This idea is then introduced in fuzzy topology by incorporating a relaxed version of fuzzy subsethood. The introduction of inclusion degree leads to a means of dealing with imperfections and small errors, especially in cases such as digital images where boundaries of subsets of an image are not crisp. The properties of the new operators are presented. Applications of the proposed operators are given in terms of rough sets and mathematical morphology.

Fuzzy regions: interpretations of surface area and distance

Verstraete J.

Control and Cybernetics

2009

Vol. 38, no 2

509-526

This contribution concerns the modelling of fuzzy information in geographic databases. For this purpose, fuzzy regions and fuzzy points have been defined in the past, along with a number of suitable operations. For numerical information - such as the surface area of fuzzy regions, or the distance between fuzzy regions and/or fuzzy points - the computation depends on the interpretation given to the fuzzy regions or points. Consequently, it is important to differentiate between the interpretation in order to obtain correct results. This article explains in detail the impact of the interpretation in terms of both the surface area and the distance.