Spectral representations of one-homogeneous functionals

Martin Burger, Lina Eckardt, Guy Gilboa, Michael Moeller

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


This paper discusses a generalization of spectral representations related to convex one-homogeneous regularization functionals, e.g. total variation or ℓ1-norms. Those functionals serve as a substitute for a Hilbert space structure (and the related norm) in classical linear spectral transforms, e.g. Fourier and wavelet analysis. We discuss three meaningful definitions of spectral representations by scale space and variational methods and prove that (nonlinear) eigenfunctions of the regularization functionals are indeed atoms in the spectral representation. Moreover, we verify further useful properties related to orthogonality of the decomposition and the Parseval identity. The spectral transform is motivated by total variation and further developed to higher order variants. Moreover, we show that the approach can recover Fourier analysis as a special case using an appropriate ℓ1-type functional and discuss a coupled sparsity example.

Original languageEnglish
Title of host publicationScale Space and Variational Methods in Computer Vision - 5th International Conference, SSVM 2015, Proceedings
EditorsMila Nikolova, Jean-François Aujol, Nicolas Papadakis
Number of pages12
ISBN (Electronic)9783319184609
StatePublished - 2015
Event5th International Conference on Scale Space and Variational Methods in Computer Vision, SSVM 2015 - Lege-Cap Ferret, France
Duration: 31 May 20154 Jun 2015

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)


Conference5th International Conference on Scale Space and Variational Methods in Computer Vision, SSVM 2015
CityLege-Cap Ferret


  • Convex regularization
  • Nonlinear eigenfunctions
  • Nonlinear spectral decomposition
  • Total variation

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)


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