Spectral Decompositions using One-Homogeneous Functionals
For researchers in signal/image processing and inverse problems, this provides a theoretical foundation for nonlinear multiscale decompositions analogous to linear filtering.
The paper establishes a nonlinear spectral decomposition of data using one-homogeneous regularization functionals, proving theoretical links between variational, scale space, and inverse scale space definitions, and demonstrating orthogonality and a Parseval-type identity. Numerical examples illustrate the approach.
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.