Restoration of manifold-valued images by half-quadratic minimization

Ronny BERGMANN*, Raymond H. CHAN, Ralf HIELSCHER, Johannes PERSCH, Gabriele STEIDL

*Corresponding author for this work

Research output: Journal PublicationsJournal Article (refereed)peer-review

46 Citations (Scopus)


The paper addresses the generalization of the half-quadratic minimization method for the restoration of images having values in a complete, connected Riemannian manifold. We recall the half-quadratic minimization method using the notation of the c-transform and adapt the algorithm to our special variational setting. We prove the convergence of the method for Hadamard spaces. Extensive numerical examples for images with values on spheres, in the rotation group SO(3), and in the manifold of positive definite matrices demonstrate the excellent performance of the algorithm. In particular, the method with SO(3)-valued data shows promising results for the restoration of images obtained from Electron Backscattered Diffraction which are of interest in material science.

Original languageEnglish
Pages (from-to)281-304
Number of pages24
JournalInverse Problems and Imaging
Issue number2
Publication statusPublished - May 2016
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2016 American Institute of Mathematical Sciences.


  • DT-MRI
  • EBSD
  • Half-uadratic minimization
  • Manifold-valued data
  • Quasi-Newton method
  • Variational restoration methods


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