Fuzzy transforms: Theory and applications

  • Authors:

  • Irina Perfilieva

  • Affiliations:

  • University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 701 03 Ostrava 1, Czech Republic

  • Venue:
  • Fuzzy Sets and Systems
  • Year:
  • 2006

Quantified Score

Hi-index 0.22

Visualization

Abstract

The technique of direct and inverse fuzzy (F-)transforms of three different types is introduced and approximating properties of the inverse F-transforms are described. All three types of the direct F-transform are transformations from a function space to a finite dimensional vector space. The first (ordinary) F-transform is constructed on the basis of the ordinary algebra of reals, while the other two types of the F-transform are constructed on the basis of residuated lattice. The core idea of the technique of F-transforms is a fuzzy partition of a universe into fuzzy subsets (factors, clusters, granules etc.). We claim that for a sufficient representation of a function defined on this universe, we may consider its average values over fuzzy subsets from the partition. Thus, a function can be associated with a mapping from a set of fuzzy subsets to the set of its thus obtained average values. A number of theorems establishing best approximation properties of the inverse F-transforms are proved. In fact, three types of the inverse F-transform are the best approximations in average, from below, and from above respectively. As one of many possible applications, we present a method of image compression and reconstruction on the basis of the F-transform.