Out-of-Sample Extension of the Fuzzy Transform

This article addresses the definition and computation of the out-of-sample membership functions and the resulting out-of-sample fuzzy transform (FT), which extend their discrete counterparts to the continuous case. Through the out-of-sample FT, we introduce a coherent analysis of the discrete and continuous FTs, which is applied to extrapolate the behavior of the FT on new data and to achieve an accurate approximation of the continuous FT of signals on arbitrary data. To this end, we apply either an approximated approach, which considers the link between integral kernels and the spectrum of the corresponding Gram matrix, or an interpolation of the discrete kernel eigenfunctions with radial basis functions. In this setting, we show the generality of the proposed approach to the input data (e.g., graphs, 3-D domains) and signal reconstruction.