A Natural Extension of F-Transform to Triangular and Triangulated Domains Necessitates the Use of Triangular Membership Functions

Authors

Hana Zámečiková, University of OstravaFollow
Irina Perfilieva, University of OstravaFollow
Olga Kosheleva, The University of Texas at El PasoFollow
Vladik Kreinovich, The University of Texas at El PasoFollow

Publication Date

2-1-2025

Comments

Technical Report: UTEP-CS-25-5

Abstract

In many practical situations when we process 1-D data, the method of F-transform turned out to be very useful. In this method, we can use either triangular membership functions or more complex ones. Because this method has been so successful in 1-D applications, a natural idea is to extend it to functions defined on 2-D and higher-dimensional domains -- e.g., to images. This method allows natural generalization to rectangular domains, where it indeed turned out to be very effective. A recent paper showed that it can extended to more general domains -- e.g., to triangular domains and to more general domains that are divided into triangular domains by triangulation. Interestingly, while all 1-D membership functions can be extended to the rectangular domains, the current extension to triangular and more general domains was produced only for triangular membership functions. In this paper, we show that this restriction is not accidental: a natural extension of F-transform to triangular domains is only possible for triangular membership functions. This may explain why such membership functions are often very effective.