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.
Categories: math.OC