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Journal of Mathematical Imaging and Vision

, Volume 61, Issue 5, pp 571–601 | Cite as

Unified Models for Second-Order TV-Type Regularisation in Imaging: A New Perspective Based on Vector Operators

  • Eva-Maria BrinkmannEmail author
  • Martin Burger
  • Joana Sarah Grah
Article

Abstract

We introduce a novel regulariser based on the natural vector field operations gradient, divergence, curl and shear. For suitable choices of the weighting parameters contained in our model, it generalises well-known first- and second-order TV-type regularisation methods including TV, ICTV and TGV\(^2\) and enables interpolation between them. To better understand the influence of each parameter, we characterise the nullspaces of the respective regularisation functionals. Analysing the continuous model, we conclude that it is not sufficient to combine penalisation of the divergence and the curl to achieve high-quality results, but interestingly it seems crucial that the penalty functional includes at least one component of the shear or suitable boundary conditions. We investigate which requirements regarding the choice of weighting parameters yield a rotational invariant approach. To guarantee physically meaningful reconstructions, implying that conservation laws for vectorial differential operators remain valid, we need a careful discretisation that we therefore discuss in detail.

Keywords

Variational methods Sparse regularisation Natural differential operators Helmholtz decomposition (Higher-order) total variation (TV) regularisation Denoising 

Notes

Acknowledgements

The authors thank Kristian Bredies, Martin Holler (both University of Graz) and Christoph Schnörr (University of Heidelberg) for useful discussions and links to literature.

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Authors and Affiliations

  1. 1.Applied Mathematics: Institute for Analysis and NumericsWestfälische Wilhelms-Universität MünsterMünsterGermany
  2. 2.Department MathematikFriedrich-Alexander Universität Erlangen-NürnbergErlangenGermany
  3. 3.Institute of Computer Graphics and VisionGraz University of TechnologyGrazAustria

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