Interpolating between Tikhonov regularization and spectral cutoff

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.