Asymptotic behaviour of total generalised variation

TL;DR

This paper investigates the asymptotic behavior of the second order total generalised variation (TGV) functional in image processing, especially as its parameters become very large or small, revealing when it approximates total variation (TV).

Contribution

It provides a detailed analysis of TGV's behavior in extreme parameter regimes, including conditions under which it converges to TV regularisation.

Findings

TGV coincides with TV for large ratio of parameters in symmetric 2D data.

Behavior of TGV varies significantly with parameter scaling.

Asymptotic limits of TGV are characterized in different parameter regimes.

Abstract

The recently introduced second order total generalised variation functional $\mathrm{TGV}_{\beta,\alpha}^{2}$ has been a successful regulariser for image processing purposes. Its definition involves two positive parameters $\alpha$ and $\beta$ whose values determine the amount and the quality of the regularisation. In this paper we report on the behaviour of $\mathrm{TGV}_{\beta,\alpha}^{2}$ in the cases where the parameters $\alpha, \beta$ as well as their ratio $\beta/\alpha$ becomes very large or very small. Among others, we prove that for sufficiently symmetric two dimensional data and large ratio $\beta/\alpha$, $\mathrm{TGV}_{\beta,\alpha}^{2}$ regularisation coincides with total variation ($\mathrm{TV}$) regularisation.