Introducing the p-Laplacian Spectra

Ido Cohen (1, 2, 3) , Guy Gilboa (4)

1 IBPS - Institut de Biologie Paris Seine
2 NPS - Neuroscience Paris Seine
3 SU - Sorbonne Université
4 Technion - Israel Institute of Technology [Haifa]

In this work we develop a nonlinear transform, associated with nonlin-ear eigenfunctions of the p-Laplacian for p ∈ (1, 2). This bridges the gap between the Fourier transform (p = 2) and the recently proposed Total Variation (TV) transform (p = 1). We first analyze the behavior of nonlinear eigenfunctions of the p−Laplacian under the p−Laplacian flow. We provide an analytic solution and show finite extinction time. A main innovation of this study is concerned with operators of fractional homogeneity which require the mathematical framework of fractional calculus. The proposed p-Laplacian transform rigorously defines the notions of decomposition, reconstruction, filtering and spectrum. Although the study is focused on the p-Laplacian operator, the mathematical results are valid for any homogeneous operator of order in the range of (0, 1).

Domaines: Traitement du signal et de l'image [eess.SP]

Ido Cohen, Guy Gilboa. Introducing the p-Laplacian Spectra. 2019. ⟨hal-02085700v1⟩