介绍p-Laplacian谱
Introducing the p-Laplacian Spectra
摘要：
在这项工作中，我们开发了一个非线性分解，与非线性特征函数的p-Laplacian为p \in（1，2）。通过这种分解，我们可以处理不同平滑度的信号。我们首先分析由γ-齐次算子γ \in R生成的尺度空间的解。一个分析解制定的尺度空间初始化与相应的运营商的非线性本征函数。我们证明了对于γ \in [0，1），流在有限时间内灭绝.本研究的一个主要创新点是分数阶齐次算子，这需要分数阶微积分的数学框架。所提出的变换严格定义了分解，重建，滤波和频谱的概念。该理论适用于p-Laplacian算子，在这个框架中开发的工具进行了演示。关键词：非线性谱，滤波，保形流，p-Laplacian，非线性特征函数。
英文摘要：
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 γ-homogeneous operators, γ \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 γ \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.
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