A Total Variation Based Regularizer Promoting Piecewise-Lipschitz Reconstructions

  • Martin Burger
  • Yury KorolevEmail author
  • Carola-Bibiane Schönlieb
  • Christiane Stollenwerk
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11603)


We introduce a new regularizer in the total variation family that promotes reconstructions with a given Lipschitz constant (which can also vary spatially). We prove regularizing properties of this functional and investigate its connections to total variation and infimal convolution type regularizers \({{\,\mathrm{{{\,\mathrm{TVL}\,}}^p}\,}}\) and, in particular, establish topological equivalence. Our numerical experiments show that the proposed regularizer can achieve similar performance as total generalized variation while having the advantage of a very intuitive interpretation of its free parameter, which is just a local estimate of the norm of the gradient. It also provides a natural approach to spatially adaptive regularization.


Total variation Total generalized variation First order regularization Image denoising 



This work was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 777826 (NoMADS). MB acknowledges further support by ERC via Grant EU FP7 – ERC Consolidator Grant 615216 LifeInverse. YK acknowledges support of the Royal Society through a Newton International Fellowship. YK also acknowledges support of the Humbold Foundataion through a Humbold Fellowship he held at the University of Münster when this work was initiated. CBS acknowledges support from the Leverhulme Trust project on Breaking the non-convexity barrier, EPSRC grant Nr. EP/M00483X/1, the EPSRC Centre Nr. EP/N014588/1, the RISE projects CHiPS and NoMADS, the Cantab Capital Institute for the Mathematics of Information and the Alan Turing Institute. We gratefully acknowledge the support of NVIDIA Corporation with the donation of a Quadro P6000 and a Titan Xp GPUs used for this research.


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Martin Burger
    • 1
  • Yury Korolev
    • 2
    Email author
  • Carola-Bibiane Schönlieb
    • 2
  • Christiane Stollenwerk
    • 3
  1. 1.Department MathematikUniversity of Erlangen-NürnbergErlangenGermany
  2. 2.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeUK
  3. 3.Institute for Analysis and NumericsUniversity of MünsterMünsterGermany

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