Bilevel Parameter Learning for Higher-Order Total Variation Regularisation Models

J. C. De los Reyes, C. -B. Schonlieb, Tuomo Valkonen

Research output: Contribution to journalArticlepeer-review

73 Scopus citations


We consider a bilevel optimisation approach for parameter learning in higher-order total variation image reconstruction models. Apart from the least squares cost functional, naturally used in bilevel learning, we propose and analyse an alternative cost based on a Huber-regularised TV seminorm. Differentiability properties of the solution operator are verified and a first-order optimality system is derived. Based on the adjoint information, a combined quasi-Newton/semismooth Newton algorithm is proposed for the numerical solution of the bilevel problems. Numerical experiments are carried out to show the suitability of our approach and the improved performance of the new cost functional. Thanks to the bilevel optimisation framework, also a detailed comparison between (Formula presented.) and (Formula presented.) is carried out, showing the advantages and shortcomings of both regularisers, depending on the structure of the processed images and their noise level.
Original languageEnglish (US)
Pages (from-to)1-25
Number of pages25
Issue number1
StatePublished - Jun 1 2016
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2022-06-03
Acknowledged KAUST grant number(s): KUK-I1-007-43
Acknowledgements: This research has been supported by King Abdullah University of Science and Technology (KAUST) Award No. KUK-I1-007-43, EPSRC grants Nr. EP/J009539/1 Sparse & Higher-order Image Restoration and Nr. EP/M00483X/1 Efficient computational tools for inverse imaging problems, Escuela Politecnica Nacional de Quito Award No. PIS 12-14. MATHAmSud project SOCDE Sparse Optimal Control of Differential Equations and the Leverhulme Trust project on Breaking the non-convexity barrier. While in Quito, I Valkonen has moreover been supported by SENESCYT (Ecuadorian Ministry of Higher Education, Science, Technology and Innovation) under a Prometeo Fellowship.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

ASJC Scopus subject areas

  • Modeling and Simulation
  • Geometry and Topology
  • Applied Mathematics
  • Computer Vision and Pattern Recognition
  • Statistics and Probability
  • Condensed Matter Physics


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