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

Abstract

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.