Spectral representations of one-homogeneous functionals. (English) Zbl 1444.94007
Aujol, Jean-François (ed.) et al., Scale space and variational methods in computer vision. 5th international conference, SSVM 2015, Lège-Cap Ferret, France, May 31 – June 4, 2015. Proceedings. Cham: Springer. Lect. Notes Comput. Sci. 9087, 16-27 (2015).
Summary: 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.
For the entire collection see [Zbl 1362.68008].
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.
For the entire collection see [Zbl 1362.68008].
MSC:
| 94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |
| 49N90 | Applications of optimal control and differential games |
Keywords:
nonlinear spectral decomposition; nonlinear eigenfunctions; total variation; convex regularizationReferences:
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