The fuzzy transform was proposed by Perfilieva in \cite{Quoc_Perfilieva:FSS06} and became a very useful tool in many applications such as signal and image processing, compression, denoising, data analysis, etc. The fuzzy transform consists of two phases: direct and inverse. The direct phase produces a vector of fuzzy transform components by combining so-called basic functions that form a fuzzy partition of the domain space with the original function. The inverse phase is a linear-like combination of a vector of fuzzy transform components and basic functions, which leads to an approximation of the original functions. In \cite{Quoc_HolcapekViec:IEEE20}, we proposed two types of integral transforms based on a Sugeno-like integral (see \cite{Quoc_DPR16,Quoc_DH12}) and a fuzzy relation (integral kernel) for functions whose values belong to a complete residuated lattice that generalize the lower and upper fuzzy transform for lattice-valued functions also introduced in \cite{Quoc_Perfilieva:FSS06}. In the current paper \cite{Quoc_HolcapekViec:WILF21}, we demonstrate that a function can be reconstructed using integral transforms. In addition, we show that integral transforms can be used to filter biased values of ``noisy'' functions, which is a new feature that improves lattice-valued fuzzy transform and brings it closer to the fuzzy transform for real or complex-valued functions. The aim of the talk is to present an extension of integral transforms to two-dimensional lattice-valued functions. We will show an extension of Sugeno-like integrals, integral kernels, and fuzzy measures for a two-dimensional case. In addition, we will apply the extension of integral transform in image processing.

The fuzzy transform was proposed by Perfilieva in \cite{Quoc_Perfilieva:FSS06} and became a very useful tool in many applications such as signal and image processing, compression, denoising, data analysis, etc. The fuzzy transform consists of two phases: direct and inverse. The direct phase produces a vector of fuzzy transform components by combining so-called basic functions that form a fuzzy partition of the domain space with the original function. The inverse phase is a linear-like combination of a vector of fuzzy transform components and basic functions, which leads to an approximation of the original functions. In \cite{Quoc_HolcapekViec:IEEE20}, we proposed two types of integral transforms based on a Sugeno-like integral (see \cite{Quoc_DPR16,Quoc_DH12}) and a fuzzy relation (integral kernel) for functions whose values belong to a complete residuated lattice that generalize the lower and upper fuzzy transform for lattice-valued functions also introduced in \cite{Quoc_Perfilieva:FSS06}. In the current paper \cite{Quoc_HolcapekViec:WILF21}, we demonstrate that a function can be reconstructed using integral transforms. In addition, we show that integral transforms can be used to filter biased values of ``noisy'' functions, which is a new feature that improves lattice-valued fuzzy transform and brings it closer to the fuzzy transform for real or complex-valued functions. The aim of the talk is to present an extension of integral transforms to two-dimensional lattice-valued functions. We will show an extension of Sugeno-like integrals, integral kernels, and fuzzy measures for a two-dimensional case. In addition, we will apply the extension of integral transform in image processing.