Characterizing the maximum parameter of the total-variation denoising through the pseudo-inverse of the divergence

TL;DR

This paper investigates the maximum regularization parameter in anisotropic total-variation denoising, providing explicit formulas in 1D and bounds in 2D, linked to the pseudo-inverse of divergence for efficient computation.

Contribution

It introduces a closed-form expression for the maximum parameter in 1D and a practical upper-bound in 2D, advancing understanding of parameter tuning in TV denoising.

Findings

Closed-form expression for 1D case

Upper-bound estimate for 2D case

Efficient computation via Fourier domain convolutions

Abstract

We focus on the maximum regularization parameter for anisotropic total-variation denoising. It corresponds to the minimum value of the regularization parameter above which the solution remains constant. While this value is well know for the Lasso, such a critical value has not been investigated in details for the total-variation. Though, it is of importance when tuning the regularization parameter as it allows fixing an upper-bound on the grid for which the optimal parameter is sought. We establish a closed form expression for the one-dimensional case, as well as an upper-bound for the two-dimensional case, that appears reasonably tight in practice. This problem is directly linked to the computation of the pseudo-inverse of the divergence, which can be quickly obtained by performing convolutions in the Fourier domain.