Fuzzy metrics. Applications to colour image
Filtering


Andrés López Crevillén



One of the most important problems in Fuzzy Topology is to obtain an appropriate concept of fuzzy metric space. This problem has been approached by many authors and from different points of view. In particular, the notion of fuzzy metric on a set that, with the help of continuous t-norms was introduced by George and Veeramani is of great interest. In this dissertation, we have advanced in the study of fuzzy metric spaces, we have given new examples that will be helpful to develop theoretical issues, and we have approached other points related to convergence, to continuity and
to two particular types of fuzzy metrics called principal and strong .

In this Thesis, we provide new examples of fuzzy metrics and, in some cases, we conclude that the most common fuzzy metrics in the literature are particular cases of those provided here. We also deal with the extension of two stationary fuzzy metrics when they coincide with the intersection of two sets. In addition, we have advanced in the study of the p-convergency concept in fuzzy metric spaces introduced by D. Mihet, where we have dealt with some related aspects and we have given a characterization for those fuzzy metric spaces that we call principal, in which the family of the p-convergent sequences coincide with the family of convergent sequences. Moreover, we provide an example of a non completable and non principal fuzzy metric space.

We define the concept of t-continuous application between fuzzy metric spaces, which is stronger than the continuity concept, and we revise some aspects related to this kind of applications obtaining as a result, that, if the initial fuzzy metric space is principal, then the t-continuous applications coincide with the continuous applications.

With respect to the chapter of strong fuzzy metrics, we study a class of stationary fuzzy metrics that includes the class of stationary fuzzy ultrametrics, which admits completion.

Finally, provided that recent works have shown that fuzzy metrics are interesting for engineering problems and useful in a variety of applications, we study how fuzzy metrics can be used for colour image filtering. The image filtering process consists of the replacement of the pixels in a noisy original image with other noise-free pixels which are determined by means of a procedure that involves the usage of a metric. In this context, usually classical metrics have been employed and we study the usage of certain fuzzy metrics, instead. In a first application, we use four different fuzzy metrics
for the filtering process and we compare the results with the classical L2 and L? metrics. In a second application, we employ a novel fuzzy metric which is defined as the product of two fuzzy metrics in order to combine two different distance criteria: pixel spatial closeness and pixel RGB colour similarity. Both applications show that fuzzy metrics are a promising tool for image processing and, in general, for engineering problems.