Asymptotic behaviour of total generalised variation

The recently introduced second order total generalised variation functional $\mathrm{TGV}_{β,α}^{2}$ 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 $\mathrm{TGV}_{β,α}^{2}$ 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 $β/α$, $\mathrm{TGV}_{β,α}^{2}$ regularisation coincides with total variation ($\mathrm{TV}$) regularisation.

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