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

Infimal convolution regularisation functionals of BV and $\mathrm{L}^{p}$ spaces. Part I: The finite $p$ case