Asymptotic behaviour of total generalised variation

The recently introduced second order total generalised variation functional ${\mathrm{T}\mathrm{G}\mathrm{V}}_{\beta ,\alpha }^2$ 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 ${\mathrm{T}\mathrm{G}\mathrm{V}}_{\beta ,\alpha }^2$ in the cases where the parameters $\alpha, \beta$aswellastheirratio$\beta/\alpha$ becomes very large or very small. Among others, we prove that for sufficiently symmetric two dimensional data and large ratio $\beta \mathrm{/}\alpha$, ${\mathrm{T}\mathrm{G}\mathrm{V}}_{\beta ,\alpha }^2$ regularisation coincides with total variation ($\mathrm{T}\mathrm{V}$) regularisation.