Adaptive Membership Functions and F-transform

The definition of the F-transform has been limited mainly to 1D signals and 2D data due to the difficulty of defining membership functions, their centres, and support on a domain with arbitrary dimensionality and topology. We propose a novel method for the adaptive selection of the optimal centres and supports of a class of radial membership functions based on minimising the reconstruction error of the input signal as the Ftransform and its inverse, or as a weighted linear combination of the membership functions. Replacing uniformly sampled centres of the membership functions with adaptive centres and fixed supports with adaptive supports allows us to preserve the input signal's local and global features and achieve a good approximation accuracy with fewer membership functions. We compare our method with uniform sampling and previous work. As a result, we improve the image reconstruction with respect to compared methods and we reduce the underlying computational cost and storage overhead. Finally, our approach applies to any class of continuous membership functions.