Flows Generating Nonlinear Eigenfunctions

Abstract

Nonlinear variational methods have become very powerful tools for many image processing tasks. Recently a new line of research has emerged, dealing with nonlinear eigenfunctions induced by convex functionals. This has provided new insights and better theoretical understanding of convex regularization and introduced new processing methods. However, the theory of nonlinear eigenvalue problems is still at its infancy. We present a new flow that can generate nonlinear eigenfunctions of the form T (u) = λu, where T (u) is a nonlinear operator and λ ∈ R is the eigenvalue. We develop the theory where T (u) is a subgradient element of a regularizing one-homogeneous functional, such as total-variation (TV) or total-generalized-variation (TGV). We introduce two flows: a forward flow and an inverse flow; for which the steady state solution is a nonlinear eigenfunction. The forward flow monotonically smooths the solution (with respect to the regularizer) and simultaneously increases the L 2 norm. The inverse flow has the opposite characteristics. For both flows, the steady state depends on the initial condition, thus different initial conditions yield different eigenfunctions. This enables a deeper investigation into the space of nonlinear eigenfunctions, allowing to produce numerically diverse examples, which may be unknown yet. In addition we suggest an indicator to measure the affinity of a function to an eigenfunction and relate it to pseudo-eigenfunctions in the linear case.

Theoretical Analysis of Flows Estimating Eigenfunctions of One-Homogeneous Functionals

“…We first analyze the flow of [36]. Then a generalized α-flow is proposed for finding eigenfunctions.…”

“…We first introduce some basic material for one homgeneous functionals in Section 2. We then analyse the flow of [36] in Section 3. We introduce a generalized α-flow for finding eigenfunctions in Section 3.2.…”

“…In this work we present a family of new nonlinear flows, which considerably generalize the initial work of [36]. Moreover for a specific type of flow a comprehensive theoretical analysis is provided.…”

Energy dissipating flows for solving nonlinear eigenpair problems

“…It is based on the analysis of nonlinear eigenvalue problems. The work of [29] proposed a flow for finding eigenfunctions of one-homogeneous functionals, and is described in more detail below. Numerical methods for finding p-Laplacian eigenpairs were proposed in [41] and [24].…”

Rayleigh quotient minimization for absolutely one-homogeneous functionals

“…They also introduced an algorithm for solving the Rayleigh quotient for the case J = T V , H = L 1 , which approximates the Cheeger cut problem. In (Nossek & Gilboa 2018) a continuous flow for minimizing Rayleigh quotients with J being one-homogeneous and H the square L 2 norm was proposed. A comprehensive theoretical analysis was recently performed for that flow in (Aujol, Gilboa & Papadakis 2018) and an analog flow with full convergence proof was proposed.…”