Asymptotic Behaviour of Total Generalised Variation

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9087)

Abstract

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

Keywords

Total variation Total generalised variation Regularisation parameters Asymptotic behaviour of regularisers 

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeUK

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