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In this paper we analyse an infimal convolution type regularisationfunctional called $\mathrm{TVL}^{\infty}$, based on the total variation($\mathrm{TV}$) and the $\mathrm{L}^{\infty}$ norm of the gradient. Thefunctional belongs to a more general family of $\mathrm{TVL}^{p}$ functionals($1<p\le \infty$). We show via analytical and numerical results that theminimisation of the $\mathrm{TVL}^{\infty}$ functional promotes piecewiseaffine structures in the reconstructed images similar to the state of the arttotal generalised variation ($\mathrm{TGV}$) but improving upon preservation ofhat--like structures. We also propose a spatially adapted version of our modelthat produces results comparable to $\mathrm{TGV}$ and allows space for furtherimprovement.

Infimal Convolution Regularisation Functionals of BV and $\mathrm{L}^{p}$ Spaces. The Case p$=\infty$