Infimal Convolution Regularisation Functionals of BV and [Formula: see text] Spaces: Part I: The Finite [Formula: see text] Case.

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 seminorm and [Formula: see text] 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 [Formula: see text]. Furthermore, the dependency of the regularisation properties of this infimal convolution approach to the choice of is studied. It turns out that in the case [Formula: see text] this regulariser is equivalent to the 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 increases we obtain almost piecewise affine reconstructions, leading also to a better preservation of hat-like structures.