Nonlinear Spectral Analysis via One-Homogeneous Functionals: Overview and Future Prospects

Gilboa, Guy; Moeller, Michael; Burger, Martin

doi:10.1007/s10851-016-0665-5

Cited by 33 publications

(39 citation statements)

References 53 publications

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“…A very popular algorithm to solve such saddle point problems is the primal-dual hybrid gradient (PDHG) 1 algorithm [36,20,12,35,13,14]. It has been used to solve a vast amount of stateof-the-art problems-to name a few examples in imaging: image denoising with the structure tensor [21], total generalized variation denoising [10], dynamic regularization [6], multi-modal medical imaging [26], multi-spectral medical imaging [42], computation of non-linear eigenfunctions [25], regularization with directional total generalized variation [28]. Its popularity stems from two facts: First, it is very simple and therefore easy to implement.…”

mentioning

confidence: 99%

Stochastic Primal-Dual Hybrid Gradient Algorithm with Arbitrary Sampling and Imaging Applications

Chambolle¹,

Ehrhardt²,

Richtárik³

et al. 2018

SIAM J. Optim.

144

226

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We propose a stochastic extension of the primal-dual hybrid gradient algorithm studied by Chambolle and Pock in 2011 to solve saddle point problems that are separable in the dual variable. The analysis is carried out for general convex-concave saddle point problems and problems that are either partially smooth / strongly convex or fully smooth / strongly convex. We perform the analysis for arbitrary samplings of dual variables, and obtain known deterministic results as a special case. Several variants of our stochastic method significantly outperform the deterministic variant on a variety of imaging tasks.

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mentioning

confidence: 99%

Stochastic Primal-Dual Hybrid Gradient Algorithm with Arbitrary Sampling and Imaging Applications

Chambolle¹,

Ehrhardt²,

Richtárik³

et al. 2018

SIAM J. Optim.

144

226

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“…(40). For the numerical results reported in this paper, we used a fast Fourier transform and the Fourier fixed point iteration (41a) rather than inverting the operator −∂ 2 /∂x 2…”

Section: A Possible Simplifications For Nonlinear Problemsmentioning

confidence: 99%

Computing eigenfunctions and eigenvalues of boundary-value problems with the orthogonal spectral renormalization method

et al. 2018

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The spectral renormalization method was introduced in 2005 as an effective way to compute ground states of nonlinear Schrödinger and Gross-Pitaevskii type equations. In this paper, we introduce an orthogonal spectral renormalization (OSR) method to compute ground and excited states (and their respective eigenvalues) of linear and nonlinear eigenvalue problems. The implementation of the algorithm follows four simple steps: (i) reformulate the underlying eigenvalue problem as a fixed point equation, (ii) introduce a renormalization factor that controls the convergence properties of the iteration, (iii) perform a Gram-Schmidt orthogonalization process in order to prevent the iteration from converging to an unwanted mode; and (iv) compute the solution sought using a fixed-point iteration. The advantages of the OSR scheme over other known methods (such as Newton's and self-consistency) are: (i) it allows the flexibility to choose large varieties of initial guesses without diverging, (ii) easy to implement especially at higher dimensions and (iii) it can easily handle problems with complex and random potentials. The OSR method is implemented on benchmark Hermitian linear and nonlinear eigenvalue problems as well as linear and nonlinear non-Hermitian PT -symmetric models.

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“…It was shown in [5,15] that the nonlinear spectral decomposition framework actually reduces to the usual wavelet decomposition when the TV regularization is replaced by J(u) = W u 1 , where W denotes the linear operator conducting the (orthogonal) wavelet transform. We, however, are going to demonstrate that the image-adaptive nonlinear decomposition approach with TV regularization is significantly better suited for image manipulation and fusion tasks.…”

Section: Image Fusionmentioning

confidence: 99%

Nonlinear Spectral Image Fusion

Benning

Möller

Nossek

et al. 2017

Lecture Notes in Computer Science

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Abstract.In this paper we demonstrate that the framework of nonlinear spectral decompositions based on total variation (TV) regularization is very well suited for image fusion as well as more general image manipulation tasks. The well-localized and edge-preserving spectral TV decomposition allows to select frequencies of a certain image to transfer particular features, such as wrinkles in a face, from one image to another. We illustrate the effectiveness of the proposed approach in several numerical experiments, including a comparison to the competing techniques of Poisson image editing, linear osmosis, wavelet fusion and Laplacian pyramid fusion. We conclude that the proposed spectral TV image decomposition framework is a valuable tool for semi-and fullyautomatic image editing and fusion.

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Nonlinear Spectral Analysis via One-Homogeneous Functionals: Overview and Future Prospects

Cited by 33 publications

References 53 publications

Stochastic Primal-Dual Hybrid Gradient Algorithm with Arbitrary Sampling and Imaging Applications

Stochastic Primal-Dual Hybrid Gradient Algorithm with Arbitrary Sampling and Imaging Applications

Computing eigenfunctions and eigenvalues of boundary-value problems with the orthogonal spectral renormalization method

Nonlinear Spectral Image Fusion

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