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