Introducing the p-Laplacian spectra

Cohen, I. Bernard; Gilboa, Guy

doi:10.1016/j.sigpro.2019.107281

Cited by 16 publications

(7 citation statements)

References 36 publications

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“…For the case of onehomogeneous functionals, it was shown how gradient descent flows can be used for decomposition (see [20,12,8]). A similar phenomenon was observed for the p-Laplacian case in [14]. Can this be extended to gradient descent of general convex functionals?…”

Section: Discussion and Open Problemssupporting

confidence: 67%

Iterative Methods for Computing Eigenvectors of Nonlinear Operators

Gilboa¹

2020

Preprint

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In this chapter we are examining several iterative methods for solving nonlinear eigenvalue problems. These arise in variational image-processing, graph partition and classification, nonlinear physics and 39 more. The canonical eigenproblem we solve is T (u) = λu, where T : R n → R n is some bounded nonlinear operator. Other variations of eigenvalue problems are also discussed. We present a progression of 5 algorithms, coauthored in recent years by the author and colleagues. Each algorithm attempts to solve a unique problem or to improve the theoretical foundations. The algorithms can be understood as nonlinear PDE's which converge to an eigenfunction in the continuous time domain. This allows a unique view and understanding of the discrete iterative process. Finally, it is shown how to evaluate numerically the results, along with some examples and insights related to priors of nonlinear denoisers, both classical algorithms and ones based on deep networks.

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Section: Discussion and Open Problemssupporting

confidence: 67%

Iterative Methods for Computing Eigenvectors of Nonlinear Operators

Gilboa¹

2020

Preprint

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“…748 [Jones et al 2003] 28.214 29.484 [Fleishman et al 2003] 31.918 [Zheng et al 2011] 35. 2682009] for a discussion on the eigenvectors of the p-Laplacian and to [Cohen and Gilboa 2019] for an approach defining a p-Laplacianbased spectral decomposition in Euclidean spaces.…”

Section: Methods Tv Energymentioning

confidence: 99%

Nonlinear spectral geometry processing via the TV transform

2020

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We introduce a novel computational framework for digital geometry processing, based upon the derivation of a nonlinear operator associated to the total variation functional. Such an operator admits a generalized notion of spectral decomposition, yielding a convenient multiscale representation akin to Laplacian-based methods, while at the same time avoiding undesirable over-smoothing effects typical of such techniques. Our approach entails accurate, detail-preserving decomposition and manipulation of 3D shape geometry while taking an especially intuitive form: non-local semantic details are well separated into different bands, which can then be filtered and re-synthesized with a straightforward linear step. Our computational framework is flexible, can be applied to a variety of signals, and is easily adapted to different geometry representations, including triangle meshes and point clouds. We showcase our method through multiple applications in graphics, ranging from surface and signal denoising to enhancement, detail transfer, and cubic stylization.CCS Concepts: • Computing methodologies → Shape analysis.

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“…In this case the solution of (1.7) has the form u(t) = a(t)f where function a(t) depends on the homogeneity of J (cf. [15,17,19,23]). If J is one-homogeneous and f is an eigenfunction, the solution of (1.7) even coincides with the solution of the variational regularization problem…”

Section: Structure Of Regularizersmentioning

confidence: 99%

Structural analysis of an $L$-infinity variational problem and relations to distance functions

Bungert,

Korolev,

Burger

2020

Preprint

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In this work we analyse the functional J (u) = ∇u ∞ defined on Lipschitz functions with homogeneous Dirichlet boundary conditions. Our analysis is performed directly on the functional without the need to approximate with smooth p-norms. We prove that its ground states coincide with multiples of the distance function to the boundary of the domain. Furthermore, we compute the L 2 -subdifferential of J and characterize the distance function as unique non-negative eigenfunction of the subdifferential operator. We also study properties of general eigenfunctions, in particular their nodal sets. Furthermore, we prove that the distance function can be computed as asymptotic profile of the gradient flow of J and construct analytic solutions of fast marching type. In addition, we give a geometric characterization of the extreme points of the unit ball of J .Finally, we transfer many of these results to a discrete version of the functional defined on a finite weighted graph. Here, we analyze properties of distance functions on graphs and their gradients. The main difference between the continuum and discrete setting is that the distance function is not the unique non-negative eigenfunction on a graph.

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Introducing the p-Laplacian spectra

Cited by 16 publications

References 36 publications

Iterative Methods for Computing Eigenvectors of Nonlinear Operators

Iterative Methods for Computing Eigenvectors of Nonlinear Operators

Nonlinear spectral geometry processing via the TV transform

Structural analysis of an $L$-infinity variational problem and relations to distance functions

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