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

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.