TY - JOUR
T1 - Introducing the p-Laplacian spectra
AU - Cohen, Ido
AU - Gilboa, Guy
N1 - Publisher Copyright:
© 2019 Elsevier B.V.
PY - 2020/2
Y1 - 2020/2
N2 - In this work we develop a nonlinear decomposition, associated with nonlinear eigenfunctions of the p-Laplacian for p ∈ (1, 2). With this decomposition we can process signals of different degrees of smoothness. We first analyze solutions of scale spaces, generated by γ-homogeneous operators, γ∈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 γ ∈ [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.
AB - In this work we develop a nonlinear decomposition, associated with nonlinear eigenfunctions of the p-Laplacian for p ∈ (1, 2). With this decomposition we can process signals of different degrees of smoothness. We first analyze solutions of scale spaces, generated by γ-homogeneous operators, γ∈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 γ ∈ [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.
KW - Filtering
KW - Nonlinear eigenfunctions
KW - Nonlinear spectra
KW - Shape preserving flows
KW - p-Laplacian
UR - http://www.scopus.com/inward/record.url?scp=85072276749&partnerID=8YFLogxK
U2 - 10.1016/j.sigpro.2019.107281
DO - 10.1016/j.sigpro.2019.107281
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AN - SCOPUS:85072276749
SN - 0165-1684
VL - 167
JO - Signal Processing
JF - Signal Processing
M1 - 107281
ER -