Feature augmentation for the inversion of the Fourier transform with limited data

TL;DR

This paper introduces a shape-driven interpolation method using Variably Scaled Kernels to improve the inversion of Fourier transform data with limited observations, demonstrated on astrophysical imaging benchmarks.

Contribution

It develops a novel VSK-based interpolation approach tailored for inverse Fourier problems, enhancing inversion accuracy with limited data.

Findings

Theoretical analysis of the VSK collocation matrix spectrum.

Successful application to astrophysical imaging benchmarks.

Improved inversion results with limited Fourier data.

Abstract

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.