Asymptotic behaviour of total generalised variation

The recently introduced second order total generalised variation functional TGVβ,α2 has been a successful regulariser for image processing purposes. Its definition involves two positive parameters α and β whose values determine the amount and the quality of the regularisation. In this paper we report on the behaviour of TGVβ,α2 in the cases where the parameters α, β as well as their ratio β/α becomes very large or very small. Among others, we prove that for sufficiently symmetric two dimensional data and large ratio β/α, TGVβ,α2 regularisation coincides with total variation (TV) regularisation.