Variably Scaled Kernels for the regularized solution of the parametric Fourier imaging problem
Anna Volpara, Alessandro Lupoli, Emma Perracchione
TL;DR
The paper tackles the ill-posed problem of reconstructing images from parametric Fourier data with limited measurements. It proposes a Variably Scaled Kernel (VSK) interpolation in the Fourier domain, using a parameter-dependent scaling function $\tilde{g}$ to transfer information across parameter values, followed by a projected Landweber iteration to enforce positivity and regularization. Theoretical contributions include $L^2$ image-space error bounds that depend on the scaling function and an analysis of the extrapolation/Landweber scheme, showing that a scaling function closer to the target reduces error. Numerical validation on RHESSI solar imaging data demonstrates regularization along the parameter (energy) channel, yielding more stable reconstructions and smaller uncertainties in derived spectra, with implications for sparse imaging systems and medical imaging.
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.