Spectral decompositions using one-homogeneous functionals. (English) Zbl 1361.35123
Summary: This paper discusses the use of absolutely one-homogeneous regularization functionals in a variational, scale space, and inverse scale space setting to define a nonlinear spectral decomposition of input data. We present several theoretical results that explain the relation between the different definitions. Additionally, results on the orthogonality of the decomposition, a Parseval-type identity, and the notion of generalized (nonlinear) eigenvectors closely link our nonlinear multiscale decompositions to the well-known linear filtering theory. Numerical results are used to illustrate our findings.
MSC:
| 35P30 | Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs |
| 34L05 | General spectral theory of ordinary differential operators |
| 49R05 | Variational methods for eigenvalues of operators |
| 47A75 | Eigenvalue problems for linear operators |
Keywords:
nonlinear spectral decomposition; nonlinear eigenfunctions; total variation; convex regularizationReferences:
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