Infimal Convolution Regularisation Functionals of BV and L Spaces


Burger, Martin
Papafitsoros, Konstantinos
Papoutsellis, Evangelos
Schönlieb, Carola-Bibiane

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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 Lp 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 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 p increases we obtain almost piecewise affine reconstructions, leading also to a better preservation of hat-like structures.

Burger M, Papafitsoros K, Papoutsellis E, Schönlieb C-B (2016) Infimal Convolution Regularisation Functionals of BV and Lp Spaces. Journal of Mathematical Imaging and Vision 55: 343–369. Available:

The authors would like to thank the anonymous reviewers for their interesting comments and suggestions which especially motivated our more detailed discussion on the generalised Huber total variation functional. The authors acknowledge support of the Royal Society International Exchange Award No. IE110314. This work is further supported by the King Abdullah University for Science and Technology (KAUST) Award No. KUK-I1-007-43, the EPSRC first Grant No. EP/J009539/1 and the EPSRC Grant No. EP/M00483X/1. MB acknowledges further support by ERC via Grant EU FP 7-ERC Consolidator Grant 615216 LifeInverse. KP acknowledges the financial support of EPSRC and the Alexander von Humboldt Foundation while in UK and Germany, respectively. EP acknowledges support by Jesus College, Cambridge and Embiricos Trust Scholarship.

Springer Nature

Journal of Mathematical Imaging and Vision


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