Introducing the p-Laplacian Spectra

TL;DR

This paper introduces a nonlinear spectral decomposition framework based on the p-Laplacian for p in (1, 2), enabling signal processing of various smoothness levels through novel mathematical tools including fractional calculus.

Contribution

It develops a new nonlinear spectral decomposition method for the p-Laplacian, incorporating fractional calculus, and provides a rigorous framework for filtering and shape-preserving flows.

Findings

Defined a nonlinear spectrum for the p-Laplacian.

Formulated analytic solutions for scale spaces with nonlinear eigenfunctions.

Demonstrated the framework's application to signal processing tasks.

Abstract

In this work we develop a nonlinear decomposition, associated with nonlinear eigenfunctions of the p-Laplacian for p \in (1, 2). With this decomposition we can process signals of different degrees of smoothness. We first analyze solutions of scale spaces, generated by {\gamma}-homogeneous operators, {\gamma} \in R. An analytic solution is formulated when the scale space is initialized with a nonlinear eigenfunction of the respective operator. We show that the flow is extinct in finite time for {\gamma} \in [0, 1). A main innovation in this study is concerned with operators of fractional homogeneity, which require the mathematical framework of fractional calculus. The proposed transform rigorously defines the notions of decomposition, reconstruction, filtering and spectrum. The theory is applied to the p-Laplacian operator, where the tools developed in this framework are demonstrated. Keywords: Nonlinear spectra, filtering, shape preserving flows, p-Laplacian, nonlinear eigenfunctions.