Bilevel Parameter Optimization for Nonlocal Image Denoising Models

D’Elia, Marta; Reyes, Juan De los; Miniguano, Andres

doi:10.2172/1592945

Cited by 22 publications

(17 citation statements)

References 20 publications

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“…This implies that γ is well-defined by (11) as an equivalence class of functions on Ω × Ω. It remains to show (12). To do this, we argue as follows: If (12) does not hold, then there exists ε > 0 and A ⊂ Ω × Ω with |A| > 0 such that for all γ ∈ γ…”

Section: 2mentioning

confidence: 99%

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Learning nonlocal regularization operators

Holler¹,

Kunisch²

2022

MCRF

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A learning approach for determining which operator from a class of nonlocal operators is optimal for the regularization of an inverse problem is investigated. The considered class of nonlocal operators is motivated by the use of squared fractional order Sobolev seminorms as regularization operators. First fundamental results from the theory of regularization with local operators are extended to the nonlocal case. Then a framework based on a bilevel optimization strategy is developed which allows to choose nonlocal regularization operators from a given class which i) are optimal with respect to a suitable performance measure on a training set, and ii) enjoy particularly favorable properties. Results from numerical experiments are also provided.

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Section: 2mentioning

confidence: 99%

“…[22,18,13,2]. A learning problem for determining optimal filter parameters for nonlocal regularization operators in the context of image denoising problems was recently investigated in [12]. As a particular instance of nonlocal regularization operators, fractional-type regularization operators are considered in [3,4].…”

mentioning

confidence: 99%

Learning nonlocal regularization operators

Holler¹,

Kunisch²

2022

MCRF

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“…In fact, nonlocal operators embed length scales in their definitions and allow for irregular functions. For these reasons, nonlocal models are currently employed in several scientific and engineering applications including surface or subsurface transport [1,2,3,4,5,6], fracture mechanics [7,8,9], turbulence [10,11,12], image processing [13,14,15] and stochastic processes [16,17,18,19,20].…”

Section: Introductionmentioning

confidence: 99%

Efficient optimization-based quadrature for variational discretization of nonlocal problems

Pasetto,

Shen,

D'Elia

et al. 2022

Preprint

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Casting nonlocal problems in variational form and discretizing them with the finite element (FE) method facilitates the use of nonlocal vector calculus to prove well-posedeness, convergence, and stability of such schemes. Employing an FE method also facilitates meshing of complicated domain geometries and coupling with FE methods for local problems. However, nonlocal weak problems involve the computation of a double-integral, which is computationally expensive and presents several challenges. In particular, the inner integral of the variational form associated with the stiffness matrix is defined over the intersections of FE mesh elements with a ball of radius δ, where δ is the range of nonlocal interaction. Identifying and parameterizing these intersections is a nontrivial computational geometry problem. In this work, we propose a quadrature technique where the inner integration is performed using quadrature points distributed over the full ball, without regard for how it intersects elements, and weights are computed based on the generalized moving least squares method. Thus, as opposed to all previ-

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“…A nonlocal equation is characterized by integral operators acting on a lengthscale or "horizon", describing long-range forces and reducing the regularity requirements on the solutions. Engineering applications include surface or subsurface transport [3,4,21,30,31,47,48], fracture mechanics [29,34,52], turbulence [32,41], image processing [1,11,27,35] and stochastic processes [6,13,37,39,40].…”

Section: Introductionmentioning

confidence: 99%

Analysis of Anisotropic Nonlocal Diffusion Models: Well-posedness of Fractional Problems for Anomalous Transport

D’Elia¹,

Gulian²

2021

Preprint

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We analyze the well-posedness of an anisotropic, nonlocal diffusion equation. Establishing an equivalence between weighted and unweighted anisotropic nonlocal diffusion operators in the vein of unified nonlocal vector calculus, we apply our analysis to a class of fractional-order operators and present rigorous estimates for the solution of the corresponding anisotropic anomalous diffusion equation. Furthermore, we extend our analysis to the anisotropic diffusion-advection equation and prove well-posedness for fractional orders s ∈ [0.5, 1). We also present an application of the advection-diffusion equation to anomalous transport of solutes.

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Bilevel Parameter Optimization for Nonlocal Image Denoising Models

Cited by 22 publications

References 20 publications

Learning nonlocal regularization operators

Learning nonlocal regularization operators

Efficient optimization-based quadrature for variational discretization of nonlocal problems

Analysis of Anisotropic Nonlocal Diffusion Models: Well-posedness of Fractional Problems for Anomalous Transport

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