Variably Scaled Kernels for the regularized solution of the parametric Fourier imaging problem

TL;DR

This paper introduces variably scaled kernels (VSKs) for improved regularization in parametric Fourier imaging, providing theoretical error bounds and demonstrating effective interpolation/extrapolation in astronomical imaging applications.

Contribution

The study offers new theoretical insights into VSKs, including error bounds and properties of the Landweber scheme, enhancing the understanding of their role in regularized Fourier imaging.

Findings

VSKs effectively transfer information across parameters during image reconstruction.

Theoretical error bounds depend on the choice of the scaling function.

Numerical tests in astronomical imaging confirm the scheme's regularization and interpretability.

Abstract

We address the problem of approximating parametric Fourier imaging problems via interpolation/ extrapolation algorithms that impose smoothing constraints across contiguous values of the parameter. Previous works already proved that interpolating via Variably Scaled Kernels (VSKs) the scattered observations in the Fourier domain and then defining the sought approximation via the projected Landweber iterative scheme, turns out to be effective. This study provides new theoretical insights, including error bounds in the image space and properties of the projected Landweber iterative scheme, both influenced by the choice of the scaling function, which characterizes the VSK basis. Such bounds then suggest a smarter solution for the definition of the scaling functions. Indeed, by means of VSKs, the information coded in an image reconstructed for a given parameter is transferred during the reconstruction process to a contiguous parameter value. Benchmark test cases in the field of astronomical imaging, numerically show that the proposed scheme is able to regularize along the parameter direction, thus proving reliable and interpretable results.