Nonlocal Bounded Variations with Applications

Antil, Harbir; Díaz, Hugo; Jing, Tian; Schikorra, Armin

doi:10.1137/22m1520876

SIAM J. Math. Anal.

2024

DOI: 10.1137/22m1520876

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Nonlocal Bounded Variations with Applications

Harbir Antil,

Hugo Díaz,

Tian Jing

et al.

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2024

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Cited by 2 publications

References 38 publications

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Segmentation in Measure Spaces

Moll,

Pallardó-Julià,

Solera

2024

Appl Math Optim

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We consider an abstract concept of perimeter measure space as a very general framework in which one can properly consider two of the most well-studied variational models in image processing: the Rudin–Osher–Fatemi model for image denoising (ROF) and the Mumford–Shah model for image segmentation (MS). We show the linkage between the ROF model and the two phases piecewise constant case of MS in perimeter measure spaces. We show applications of our results to nonlocal image segmentation, via discrete weighted graphs, and to multiclass classification on high dimensional spaces.

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Segmentation in Measure Spaces

Moll,

Pallardó-Julià,

Solera

2024

Appl Math Optim

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Integer Optimal Control with Fractional Perimeter Regularization

Antil,

Manns

2024

Appl Math Optim

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Motivated by many applications, optimal control problems with integer controls have recently received a significant attention. Some state-of-the-art work uses perimeter-regularization to derive stationarity conditions and trust-region algorithms. However, the discretization is difficult in this case because the perimeter is concentrated on a set of dimension $$d - 1$$ d - 1 for a domain of dimension d. This article proposes a potential way to overcome this challenge by using the fractional nonlocal perimeter with fractional exponent $$0<\alpha <1$$ 0 < α < 1 . In this way, the boundary integrals in the perimeter regularization are replaced by volume integrals. Besides establishing some non-trivial properties associated with this perimeter, a $$\Gamma $$ Γ -convergence result is derived. This result establishes convergence of minimizers of fractional perimeter-regularized problem, to the standard one, as the exponent $$\alpha $$ α tends to 1. In addition, the stationarity results are derived and algorithmic convergence analysis is carried out for $$\alpha \in (0.5,1)$$ α ∈ ( 0.5 , 1 ) under an additional assumption on the gradient of the reduced objective. The theoretical results are supplemented by a preliminary computational experiment. We observe that the isotropy of the total variation may be approximated by means of the fractional perimeter functional.

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Nonlocal Bounded Variations with Applications

Cited by 2 publications

References 38 publications

Segmentation in Measure Spaces

Segmentation in Measure Spaces

Integer Optimal Control with Fractional Perimeter Regularization

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