Graph approximation and generalized Tikhonov regularization for signal deblurring

TL;DR

This paper introduces a graph-based approximation scheme for linear operators in signal deblurring, which improves the accuracy of regularized solutions by preserving spectral properties during discretization.

Contribution

It proposes a novel graph-based approximation method that ensures asymptotic spectral accuracy, enhancing generalized Tikhonov regularization for signal deblurring.

Findings

Graph approximation reduces spectral error in discretized operators.

Improved regularized solutions with spectral preservation.

Combining graph approximation with graph regularization enhances deblurring results.

Abstract

Given a compact linear operator $\K$, the (pseudo) inverse $\K^\dagger$ is usually substituted by a family of regularizing operators $\R_\alpha$ which depends on $\K$ itself. Naturally, in the actual computation we are forced to approximate the true continuous operator $\K$ with a discrete operator $\K^{(n)}$ characterized by a finesses discretization parameter $n$, and obtaining then a discretized family of regularizing operators $\R_\alpha^{(n)}$. In general, the numerical scheme applied to discretize $\K$ does not preserve, asymptotically, the full spectrum of $\K$. In the context of a generalized Tikhonov-type regularization, we show that a graph-based approximation scheme that guarantees, asymptotically, a zero maximum relative spectral error can significantly improve the approximated solutions given by $\R_\alpha^{(n)}$. This approach is combined with a graph based regularization technique with respect to the penalty term.