Bilevel parameter optimization for learning nonlocal image denoising models

D’Elia, Marta; Reyes, Juan De los; Trujillo, A.

doi:10.2172/1617438

Cited by 12 publications

(5 citation statements)

References 20 publications

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“…Note that the constraint in (21.6) is equivalent to the diffusion–reaction equation in (21.5). The well-posedness of the bilevel optimization problem has been proved by D’Elia et al (2019 c ). They also introduce a second-order optimization algorithm for its solution, and give insights into implementation aspects and numerical performance.…”

Section: Image Denoisingmentioning

confidence: 93%

“…However, model parameters are often unknown (see Section 16) and the selection of w is not a trivial task. D’Elia, De los Reyes and Trujillo (2019 c ) considered a bilevel optimization approach for the identification of kernel parameters (for integrable kernels, including NL-means) and of the weight function . Because the estimation of kernel parameters is discussed in Section 16, here we only consider the formulation for the identification of w , in its simplest setting.…”

Section: Image Denoisingmentioning

confidence: 99%

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Numerical methods for nonlocal and fractional models

et al. 2020

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Partial differential equations (PDEs) are used with huge success to model phenomena across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDEs fail to adequately model observed phenomena, or are not the best available model for that purpose. On the other hand, in many situations, nonlocal models that account for interaction occurring at a distance have been shown to more faithfully and effectively model observed phenomena that involve possible singularities and other anomalies. In this article we consider a generic nonlocal model, beginning with a short review of its definition, the properties of its solution, its mathematical analysis and of specific concrete examples. We then provide extensive discussions about numerical methods, including finite element, finite difference and spectral methods, for determining approximate solutions of the nonlocal models considered. In that discussion, we pay particular attention to a special class of nonlocal models that are the most widely studied in the literature, namely those involving fractional derivatives. The article ends with brief considerations of several modelling and algorithmic extensions, which serve to show the wide applicability of nonlocal modelling.

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Section: Image Denoisingmentioning

confidence: 93%

Section: Image Denoisingmentioning

confidence: 99%

Numerical methods for nonlocal and fractional models

et al. 2020

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“…Over the last years, the study of variational models with nonlocal features has attracted increased interest in the community, motivated by the desire to develop a solid understanding of global effects, long-range interactions, and singular behavior in physical phenomena and technical applications, which standard local modeling approaches cannot capture. To mention but a few selected examples from the recent literature, functionals with a nonlocal character appear in the theory of phase transitions [22,49], in peridynamics [13,36], in new models of hyperelasticity [11,12], in image processing [15,23] or in machine learning applications [4,30]. From the mathematical perspective, the presence of nonlocality in variational problems requires substantially different techniques from standard ones, which often rest on localization arguments and are therefore not applicable.…”

Section: Introductionmentioning

confidence: 99%

Cartesian convexity as the key notion in the variational existence theory for nonlocal supremal functionals

2022

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Section: Introductionmentioning

confidence: 99%

Cartesian convexity as the key notion in the variational existence theory for nonlocal supremal functionals

Kreisbeck¹,

Ritorto²,

Zappale³

2022

Preprint

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Motivated by the direct method in the calculus of variations in L ∞ , our main result identifies the notion of convexity characterizing the weakly * lower semicontinuity of nonlocal supremal functionals: Cartesian level convexity. This new concept coincides with separate level convexity in the one-dimensional setting and is strictly weaker for higher dimensions. We discuss relaxation in the vectorial case, showing that the relaxed functional will not generally maintain the supremal form. Apart from illustrating this fact with examples of multi-well type, we present precise criteria for structure-preservation. When the structure is preserved, a representation formula is given in terms of the Cartesian level convex envelope of the (diagonalized) original supremand. This work does not only complete the picture of the analysis initiated in [Kreisbeck & Zappale, Calc. Var. PDE, 2020], but also establishes a connection with double integrals. We relate the two classes of functionals via an L p -approximation in the sense of Γ-convergence for diverging integrability exponents. The proofs exploit recent results on nonlocal inclusions and their asymptotic behavior, and use tools from Young measure theory and convex analysis.

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Bilevel parameter optimization for learning nonlocal image denoising models

Cited by 12 publications

References 20 publications

Numerical methods for nonlocal and fractional models

Numerical methods for nonlocal and fractional models

Cartesian convexity as the key notion in the variational existence theory for nonlocal supremal functionals

Cartesian convexity as the key notion in the variational existence theory for nonlocal supremal functionals

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