Spectral Representations of One-Homogeneous Functionals

  • Martin Burger
  • Lina Eckardt
  • Guy Gilboa
  • Michael Moeller
Conference paper

DOI: 10.1007/978-3-319-18461-6_2

Volume 9087 of the book series Lecture Notes in Computer Science (LNCS)
Cite this paper as:
Burger M., Eckardt L., Gilboa G., Moeller M. (2015) Spectral Representations of One-Homogeneous Functionals. In: Aujol JF., Nikolova M., Papadakis N. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2015. Lecture Notes in Computer Science, vol 9087. Springer, Cham

Abstract

This paper discusses a generalization of spectral representations related to convex one-homogeneous regularization functionals, e.g. total variation or \(\ell ^1\)-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 \(\ell ^1\)-type functional and discuss a coupled sparsity example.

Keywords

Nonlinear spectral decomposition Nonlinear eigenfunctions Total variation Convex regularization 

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Martin Burger
    • 1
  • Lina Eckardt
    • 1
  • Guy Gilboa
    • 2
  • Michael Moeller
    • 3
  1. 1.Institute for Computational and Applied MathematicsUniversity of MünsterMünsterGermany
  2. 2.Electrical Engineering DepartmentTechnion IITHaifaIsrael
  3. 3.Department of MathematicsTechnische Universität MünchenMunichGermany