Spectral representations of one-homogeneous functionals

Conference contribution


Publication Details

Author(s): Burger M, Eckardt L, Gilboa G, Moeller M
Title edited volumes: Scale Space and Variational Methods in Computer Vision - 5th International Conference, SSVM 2015, Proceedings
Publisher: Springer Verlag
Publication year: 2015
Title of series: Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Pages range: 16-27
ISBN: 9783319184609
ISSN: 0302-9743


Abstract

This paper discusses a generalization of spectral representations related to convex one-homogeneous regularization functionals, e.g. total variation or ℓ 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 ℓ1-type functional and discuss a coupled sparsity example.


External institutions with authors

Technion - Israel Institute of Technology
Technische Universität München (TUM)
Westfälische Wilhelms-Universität (WWU) Münster


How to cite

APA:
Burger, M., Eckardt, L., Gilboa, G., & Moeller, M. (2015). Spectral representations of one-homogeneous functionals. In Proceedings of the 5th International Conference on Scale Space and Variational Methods in Computer Vision, SSVM 2015 (pp. 16-27). fra: Springer Verlag.

MLA:
Burger, Martin, et al. "Spectral representations of one-homogeneous functionals." Proceedings of the 5th International Conference on Scale Space and Variational Methods in Computer Vision, SSVM 2015, fra Springer Verlag, 2015. 16-27.

BibTeX: 

Last updated on 2019-24-08 at 07:13