Interpolating between Tikhonov regularization and spectral cutoff

TL;DR

This paper introduces a new regularization method that interpolates between Tikhonov and spectral cutoff techniques, allowing for tailored solutions to ill-posed linear operator equations with demonstrated benefits in signal and image processing.

Contribution

A novel one-parameter family of regularizations that bridges Tikhonov and spectral cutoff methods, enhancing flexibility and performance in solving ill-posed problems.

Findings

The interpolating regularization mitigates limitations of traditional methods.

Numerical simulations show improved results in signal processing.

The approach adapts to specific operator equations for better regularization.

Abstract

Regularizing a linear ill-posed operator equation can be achieved by manipulating the spectrum of the operator's pseudo-inverse. Tikhonov regularization and spectral cutoff are well-known techniques within this category. This paper introduces an interpolating formula that defines a one-parameter family of regularizations, where Tikhonov and spectral cutoff methods are represented as limiting cases. By adjusting the interpolating parameter taking into account the specific operator equation under consideration, it is possible to mitigate the limitations associated with both Tikhonov and spectral cutoff regularizations. The proposed approach is demonstrated through numerical simulations in the fields of signal and image processing.