Introducing the p-Laplacian spectra

In this work we develop a nonlinear decomposition, associated with nonlinear eigenfunctions of the p-Laplacian for p is an element of (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 is an element of 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 is an element of [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. (C) 2019 Elsevier B.V. All rights reserved.