Spectral Decompositions using One-Homogeneous Functionals

TL;DR

This paper introduces a nonlinear spectral decomposition framework based on absolutely one-homogeneous regularization functionals, linking it to linear filtering theory with theoretical insights and numerical validation.

Contribution

It provides a comprehensive theoretical analysis of nonlinear spectral decompositions using one-homogeneous functionals, connecting them to classical linear concepts.

Findings

Orthogonality of the nonlinear decomposition

Parseval-type identity established

Numerical results validate theoretical insights

Abstract

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.