Computing Nonlinear Eigenfunctions via Gradient Flow Extinction

Bungert, Leon; Burger, Martin; Tenbrinck, Daniel

doi:10.1007/978-3-030-22368-7_23

Cited by 11 publications

(9 citation statements)

References 17 publications

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“…For general applications a robust automatic stopping method would be helpful. Spectral analysis of nonlinear operators [10,16] may apply here. -Coherence enhancement [54] was not originally conceived for denoising.…”

Section: Resultsmentioning

confidence: 99%

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Total Variation and Mean Curvature PDEs on the Homogeneous Space of Positions and Orientations

Smets

Portegies

St-Onge³

et al. 2020

J Math Imaging Vis

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Two key ideas have greatly improved techniques for image enhancement and denoising: the lifting of image data to multi-orientation distributions and the application of nonlinear PDEs such as total variation flow (TVF) and mean curvature flow (MCF). These two ideas were recently combined by Chambolle and Pock (for TVF) and Citti et al. (for MCF) for two-dimensional images. In this work, we extend their approach to enhance and denoise images of arbitrary dimension, creating a unified geometric and algorithmic PDE framework, relying on (sub-)Riemannian geometry. In particular, we follow a different numerical approach, for which we prove convergence in the case of TVF by an application of Brezis–Komura gradient flow theory. Our framework also allows for additional data adaptation through the use of locally adaptive frames and coherence enhancement techniques. We apply TVF and MCF to the enhancement and denoising of elongated structures in 2D images via orientation scores and compare the results to Perona–Malik diffusion and BM3D. We also demonstrate our techniques in 3D in the denoising and enhancement of crossing fiber bundles in DW-MRI. In comparison with data-driven diffusions, we see a better preservation of bundle boundaries and angular sharpness in fiber orientation densities at crossings.

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Section: Resultsmentioning

confidence: 99%

“…5,Figs. 6,7,8,9,10,11,12,13,14,15,and Table 4. These experiments now include a full comparison of isotropic and anisotropic processing, and the effect of the including coherence enhancement (via locally adaptive frames) in TVF and MCF.…”

Section: Tablementioning

confidence: 99%

Total Variation and Mean Curvature PDEs on the Homogeneous Space of Positions and Orientations

Smets

Portegies

St-Onge³

et al. 2020

J Math Imaging Vis

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“…In this chapter we stick to gradient flows in Hilbert and Banach spaces which have intriguing connections to nonlinear eigenvalue problems. Such problems appear in different applications in physics, (Weinstein, 1985), mathematics (Cancès, Chakir, and Maday, 2010;Amann, 1976;Rabinowitz, 1971), and also in modern disciplines like data science (Bungert, Burger, and Tenbrinck, 2019;Gilboa, 2018;Hein and Bühler, 2010;Bühler and Hein, 2009). In the language of physics, solutions of eigenvalue problem typically describe stable energy states of a physical system, e.g., an atom.…”

Section: Introductionmentioning

confidence: 99%

“…Here, nonlinear operators like the 1-Laplacian have turned out to be suitable tools for defining nonlinear spectral decompositions and filtering of data (see Fumero, Möller, and Rodolà (2020), Gilboa (2018), Benning, Möller, et al (2017), , Gilboa, Moeller, and Burger (2016), Burger, Eckardt, Gilboa, and Moeller (2015), Gilboa (2014), and Gilboa (2013)). Another application of nonlinear eigenproblems in data science is graph clustering (Bungert, Burger, and Tenbrinck, 2019;Hein and Bühler, 2010;Bühler and Hein, 2009), where eigenfunctions of the graph p-Laplacian operators (Elmoataz, Toutain, and Tenbrinck, 2015) are used for partitioning graphs.…”

Section: Introductionmentioning

confidence: 99%

Gradient Flows and Nonlinear Power Methods for the Computation of Nonlinear Eigenfunctions

Bungert,

Burger

2021

Preprint

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This chapter describes how gradient flows and nonlinear power methods in Banach spaces can be used to solve nonlinear eigenvector-dependent eigenvalue problems, and how these can be approximated using Γ-convergence. We review several flows fromliterature, which were proposed to compute nonlinear eigenfunctions, and show that they all relate to normalized gradient flows. Furthermore, we show that the implicit Euler discretization of gradient flows gives rise to a nonlinear power method of the proximal operator and prove their convergence to nonlinear eigenfunctions. Finally, we prove that Γ-convergence of functionals implies convergence of their ground states.

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“…While this contribution is already very general and captures a wide range of homogeneity degrees p, it still lacks important regime 1 ≤ p < 2 which applies inter alia to p-Laplace equations including the total variation flow and fast diffusion equations. Furthermore, the relation between asymptotic profiles and so-called ground states, i.e., eigenfunctions with minimal eigenvalue which are especially important in applications like graph clustering (see [11,14], for instance), is not illuminated.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic profiles of nonlinear homogeneous evolution equations of gradient flow type

Bungert

Burger

2019

J. Evol. Equ.

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This work is concerned with the gradient flow of absolutely p-homogeneous convex functionals on a Hilbert space, which we show to exhibit finite (p < 2) or infinite extinction time (p ≥ 2). We give upper bounds for the finite extinction time and establish convergence rates of the flow. Moreover, we study next order asymptotics and prove that asymptotic profiles of the solution are eigenfunctions of the subdifferential operator of the functional. To this end, we compare with solutions of an ordinary differential equation which describes the evolution of eigenfunction under the flow. Our work applies, for instance, to local and nonlocal versions of PDEs like p-Laplacian evolution equations, the porous medium equation, and fast diffusion equations, herewith generalizing many results from the literature to an abstract setting.We also demonstrate how our theory extends to general homogeneous evolution equations which are not necessarily a gradient flow. Here we discover an interesting integrability condition which characterizes whether or not asymptotic profiles are eigenfunctions.These equations can also be studied as a fourth order gradient flow in H −1 , i.e.,Another class of examples are the fast diffusion equations for 1 < p < 2, the linear heat equation for p = 2, and the porous medium equation for p > 2, i.e.which, complemented with suitable boundary conditions, can also be interpreted as Hilbert space gradient flows (cf. [29] for the porous medium / fast diffusion case). Furthermore, as long as homogeneity is preserved, our general model covers non-local versions of the equations above, as well. Remarkably, we can also address an eigenvalue problem of the ∞-Laplacian operator [24,31] with our framework by regarding it from an purely energetic point of view. That means we set J(u) = ∇u ∞ for u ∈ W 1,∞ ∩ L 2 and J(u) = ∞ else, which meets all our assumptions under sufficient regularity of the domain, and interpret the ∞-Laplacian as L 2 -subdifferential operator of J.The main objective of this work is to prove that asymptotic profiles of the gradient flow (GF) are eigenfunctions of the subdifferential operator ∂J. By an asymptotic profile we refer to a suitably rescaled version of the actual solution u(t) of the gradient flow. More precisely, we look for a rescaling a(t) such that u(t)/a(t) converges to some w * as t tends to the extinction time of the flow (respectively t → ∞). Here w * is an eigenfunction of ∂J, meaning that λw * ∈ ∂J(w * ) for some λ ∈ R and by "extinction time" we refer to the (finite or infinite) time where the solution of the gradient flow stops changing, meaning ∂ t u(t) = 0 (respectively the minimal time such that that J(u(t)) = 0).The rescaling is chosen in such a way that it amplifies the shape of u(t) immediately before it reaches the state of lowest energy as described by the functional J. Furthermore, it should be noted that eigenfunctions of ∂J are self-similar in the sense that they only shrink under the gradient flow (GF) without changing their shape.If the energy is a quadratic ...

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Computing Nonlinear Eigenfunctions via Gradient Flow Extinction

Cited by 11 publications

References 17 publications

Total Variation and Mean Curvature PDEs on the Homogeneous Space of Positions and Orientations

Total Variation and Mean Curvature PDEs on the Homogeneous Space of Positions and Orientations

Gradient Flows and Nonlinear Power Methods for the Computation of Nonlinear Eigenfunctions

Asymptotic profiles of nonlinear homogeneous evolution equations of gradient flow type

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