A Review on Integral, Discrete, and Lattice-Valued Fuzzy Transforms and Their Applications

Perfilieva first used the term “Fuzzy Transform” ([Formula: see text]-Transform) in the literature in 2006, the majority of study has focused on both the potential uses of the concept as well as the development of new ideas related to it. In this work, the integral [Formula: see text]-transform, discrete [Formula: see text]-transform, and lattice-valued [Formula: see text]-transform are presented chronologically. The theoretical developments extend the [Formula: see text]-transform from real-valued functions to those defined on complete lattices, enabling the formulation of direct and inverse transforms in this generalized setting. These developments also incorporate fuzzy partitions, revealing important links between [Formula: see text]-transforms and fuzzy topological structures. Recent research has focused on generalizing the [Formula: see text]-transform using arbitrary fuzzy partitions and on unifying multiple fuzzy topologies within the framework of fuzzy relational structures. On the application side, the [Formula: see text]-transform has proven effective in diverse fields such as image compression and processing, time series analysis, neural networks, and solving differential equations. By transforming functions into a new domain and reconstructing them through an inverse operation, the [Formula: see text]-transform maintains interpretability while offering advantageous filtering capabilities, making it a versatile tool for data approximation and noise reduction.