TY - GEN
T1 - Spectral representations of one-homogeneous functionals
AU - Burger, Martin
AU - Eckardt, Lina
AU - Gilboa, Guy
AU - Moeller, Michael
N1 - Publisher Copyright: © Springer International Publishing Switzerland 2015.
PY - 2015
Y1 - 2015
N2 - 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.
AB - 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.
KW - Convex regularization
KW - Nonlinear eigenfunctions
KW - Nonlinear spectral decomposition
KW - Total variation
UR - http://www.scopus.com/inward/record.url?scp=84931085646&partnerID=8YFLogxK
U2 - https://doi.org/10.1007/978-3-319-18461-6_2
DO - https://doi.org/10.1007/978-3-319-18461-6_2
M3 - منشور من مؤتمر
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 16
EP - 27
BT - Scale Space and Variational Methods in Computer Vision - 5th International Conference, SSVM 2015, Proceedings
A2 - Nikolova, Mila
A2 - Aujol, Jean-François
A2 - Papadakis, Nicolas
T2 - 5th International Conference on Scale Space and Variational Methods in Computer Vision, SSVM 2015
Y2 - 31 May 2015 through 4 June 2015
ER -