## 1 Introduction  
The Fuzzy Transform technique (for short F-transform) [1] is a fuzzy regression method introduced to approximate a continuous function of k variables \(f(x_{1},x_{2},\ldots,x_{k})\) defined in the domain [a\({}_{1}\),b\({}_{1}\)] \(\times\) [a\({}_{2}\),b\({}_{2}\)] \(\times\)... \(\times\) [a\({}_{k}\),b\({}_{k}\)] \(\subset\) R\({}^{k}\). If this function is known in N points P\({}_{\parallel}\) = (p\({}_{\parallel}\), p\({}_{\parallel}\), \(\ldots\) p\({}_{\parallel}\)) = 1,... N, it can be approximated by a weighted average whose weights are constants given by the components of the direct F-transform. F-transform was applied in many image and data analysis problems. An extensive description of the F-transform-based techniques used in image and data analysis is given in [2]. In [3] a generalization of the F-transform, called high degree F-transform and labelled with F-transform (s \(\geq\) 0) was proposed, where the F-transform is given by the zero-degree F\({}^{0}\)-transform. In the F\({}^{\text{s}}\)-transform, with s \(>\) 0, the constant components of the direct F-transform, were replaced by s-degree polynomial components with the aim to capture more information about the original function. The greater the polynomial degree, the finer the approximation of the original function; however, as the polynomial degree increases, the computational complexity of the algorithm that uses the F\({}^{\text{S}}\) transform increases considerably. In addition, the meeting of the constraint of sufficient data density with respect to fuzzy partitions further requires additional consumption of memory resources and CPU time [2, 4]. Some researchers apply the F\({}^{1}\)-transform in image analysis, in which only two input variables are used, and the sufficient data density constraint is always met. In [5, 6] an algorithm based on the F\({}^{1}\)-transform in two variables was applied in an image processing edge detection problem. In [7] a lossy image compression method based on the F\({}^{1}\)-transform in two variables is proposed; the authors show that this method improves the decoded image quality obtained using the F-transform image compression [8]. In [9] a hybrid deep neural network in which the F\({}^{\text{s}}\)-transform is used in image analysis to construct in a preprocessing phase the convolution kernels in the first two layers of the network. Generally, when the multidimensional F-transform is applied in classification, regression, and prediction problems, as the dimensionality of the data increases, the computational complexity greatly increases; for this reason, the application of high-order F-transforms in data regression or classification has a high computational complexity, in terms of memory and time consumption, especially in the presence of massive multidimensional data [4]. An application of the first-order unidimensional fuzzy transform was proposed in [10] where the unidimensional F\({}^{1}\)-transform is applied to seasonal time series weather datasets in a seasonal forecasting model. Comparison tests show that this model improves the forecasting time series performances obtained by applying the F\({}^{0}\)-transform model proposed in [11; 12]. In [13] the F\({}^{1}\)-transform is applied to remove seasonal components and noise in time series. Recently, in [14] a new classification algorithm based on the multidimensional F\({}^{0}\)-transform for massive datasets called Multi-dimensional F-transform Classification (for short, MFC), was proposed. In MFC the K-fold cross-validation technique is applied to avoid overfitting in the data; the multidimensional direct F\({}^{0}\)-transform is applied to each fold and a weighted mean of the multi-dimensional inverse F\({}^{0}\)-transform calculated from the direct F\({}^{0}\)-transform components obtained in each fold is used to classify data points.