Asymptotic behaviour of total generalised variation

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.