  We study a general class of infimal convolution type regularisation
functionals suitable for applications in image processing. These functionals
incorporate a combination of the total variation ($\mathrm{TV}$) seminorm and
$\mathrm{L}^{p}$ norms. A unified well-posedness analysis is presented and a
detailed study of the one dimensional model is performed, by computing exact
solutions for the corresponding denoising problem and the case $p=2$.
Furthermore, the dependency of the regularisation properties of this infimal
convolution approach to the choice of $p$ is studied. It turns out that in the
case $p=2$ this regulariser is equivalent to Huber-type variant of total
variation regularisation. We provide numerical examples for image decomposition
as well as for image denoising. We show that our model is capable of
eliminating the staircasing effect, a well-known disadvantage of total
variation regularisation. Moreover as $p$ increases we obtain almost piecewise
affine reconstructions, leading also to a better preservation of hat-like
structures.
