  We investigate an interpolation/extrapolation method that, given scattered
observations of the Fourier transform, approximates its inverse. The
interpolation algorithm takes advantage of modelling the available data via a
shape-driven interpolation based on Variably Scaled Kernels (VSKs), whose
implementation is here tailored for inverse problems. The so-constructed
interpolants are used as inputs for a standard iterative inversion scheme.
After providing theoretical results concerning the spectrum of the VSK
collocation matrix, we test the method on astrophysical imaging benchmarks.
