F(φ(t)) = Φ(f) = ∫ ∞ −∞ f(t)e−2πiftdt F−1(Φ(f)) = φ(t) = ∫ ∞ −∞ Φ(f)edf The notation φ(t) ⊃ Φ(f) is also used to indicate that Φ(f) is the Fourier transform of φ(t). Another common notational convention is that capital letters are used for the names of functions in the frequency domain, and corresponding lower case letters are used for functions in the time domain. In this notation, s is often ...

The theory of fuzzy transform (F-transform, for short) has been developed extensively in recent years. Based on the current successes of F-transform, we expect that in the future, F-transform will become as successful and as widely used as the well-known transforms such as Laplace, Fourier, and wavelet transforms. The most successful applications of F-transform are in the fields of image proces...

The aim of this contribution is to show how the Ftransform technique can be generalized from the case of constant components to the case of polynomial components. For this purpose, we choose complete functional spaces with inner products. After a general presentation of Fm-transform where m ≥ 0, a detailed characterization of the F -transform is given. We applied a technique of numeric integrat...

In this paper, the application of the fuzzy transforms of the zero degree (F -transform) and of the first degree (F -transform) to the image registration is demonstrated. The main idea is to use only one technique (F-transform generally) to solve various tasks of the image registration. The F -transform is used for an extraction of feature points in edge detection step. The correspondence betwe...

A relationship between the discrete F-transform and aggregation functions is analyzed. We show that the discrete F-transform (direct or inverse) can be associated with a set of linear aggregation functions that respect a fuzzy partition of a universe. On the other side, we discover conditions that should be added to a set of linear aggregation functions in order to obtain the discrete F-transfo...

The article summarizes current approaches to image fusion problem using fuzzy transformation (F-transform) with their weak points and proposes improved version of the algorithm which suppress them. The first part of this contribution brings brief theoretical introduction into problem domain. Next part analyses weak points of current implementations. Last part introduces improved algorithm and c...

This article proposes a new interval-valued fuzzy transform. Its construction is based on a possibilistic interpretation of the partition on which the fuzzy transform is built. The main advantage of this approach is that it provides specific interval valued functions whose interpretation is straightforward. This interpretation relates to a traditional sampling/reconstruction framework where lit...

We present a quantum algorithm for the f -conditioned phase transform which does not require any initialization of ancillary register. We also develop a quantum algorithm that can solve the generalized Deutsch-Jozsa problem by a single evaluation of a function.