Nonlocal Operators with Applications to Image Processing

By Guy Gilboa, Stanley J. Osher. Multiscale Modeling & simulation, 2008

1,604 citations recorded by the source corpus.

Subfields: Computer Vision and Pattern Recognition, Mathematical Physics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/d2b93dfbe50f3c642c64b8ea581cb6e449c71f82

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In Computer Vision and Pattern Recognition

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Citations in the corpus

Numerical approximation of fractional powers of elliptic operators (2013, background): “Nonlocal operators arise in a wide variety of mathematical models such as modes of long-range interaction in elastic deformations [21], nonlocal electromagnetic ﬂuid ﬂows [18], image processing [8, 20] and many more.”
Characterization of Sobolev and BV Spaces (2011, methodology): “The gradient-based regularizing functional introduced by Gilboa and Osher in [25] takes the form”
Plug-and-Play Priors for Bright Field Electron Tomography and Sparse Interpolation (2015, methodology): “local regularizers using PDE-like evolutions and total variation on the image patch graph have been proposed to solve inverse problems [27], [28].”
Traces for fractional Sobolev spaces with variable exponents (2017, background): “From an applied point of view we recall that non-local energies with constant exponents (we quote here [9, 10]) and also local equations with variable exponents (see [2]) where used in image processing.”
3D scene flow estimation with a rigid motion prior (2011, background): “1The most closely related non-local TV regularizer [8] is given as ∫ Ω ψ (∫ Ω ‖w(x)−w(y)‖ 2 2η(x,y) dy ) dx.”
The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion (2017, background): “ in continuum mechanics (elasticity, crystal dislocation, geostrophic ows,...), phase transition phenomena, population dynamics, optimal control, image processing, game theory, nance, and others. See [10, 29, 90, 136, 191, 192, 255, 259], see also Section 1.2 of [246]. After a very active period of ”
A Fractional Laplace Equation: Regularity of Solutions and Finite Element Approximations (2015, background): “Anomalous diffusion phenomena are ubiquitous in nature [29, 35]; among the several applications of these nonlocal models let us mention image processing [8, 22, 24, 32], finance [11, 13], electromagnetic fluids [34], peridynamics [42] and porous media flow [2, 14].”
On a new class of fractional partial differential equations (2014, background): “In fact, it has been mentioned without further study in some existing literature - an equivalent definition3 of the distributional Riesz fractional gradient has been given by Schikorra [27], while it can also be derived as a special case of the nonlocal gradients of Gilboa and Osher [15].”
$\Gamma$-convergence of graph Ginzburg-Landau functionals (2012, methodology): “[35] where TV is called nonlocal TV because the graph is assumed to be embedded in an Euclidean space and so what is local on the graph (neighboring vertices) might not be local in the embedding space).”
From Local Kernel to Nonlocal Multiple-Model Image Denoising (2009, background): “3): [64],[40],[39],[73],[74],[28],[106],[107].”
Non-local Regularization of Inverse Problems (2008, methodology): “To enable the recovery of sharp features such as edges, Rudin, Osher and Fatemi [50] proposed to use the total variation norm for denoising purpose, when Φ = Id Total variation regularization prior (3) has been used to solve super-resolution [38] and inpainting of small holes [13].”
Fractional p-Laplacian evolution equations (2016, background): “These operators arise in a number of applications such as: continuum mechanics, phase transition phenomena, population dynamics, image process, game theory and Lévy processes, see [8], [15], [20], [21], [25] and the references therein.”
Numerical Approximation of Fractional Powers of Regularly Accretive Operators (2015, neutral): “The mathematical study of integral or nonlocal operators has received much attention due to their wide range of applications, see for instance [16, 4, 21, 9, 24, 22, 38, 44, 18].”
Fast nonlocal filtering applied to electron cryomicroscopy (2008, methodology): “We refer the reader to [9] for a variational formulation of neighborhood filtering in the lines of nonlocal mean.”
Nonlocal Discrete Regularization on Weighted Graphs: A Framework for Image and Manifold Processing (2008, background): “u∈V | 5w f(u)|p = ∑ v∼u w(u, v) p 2 |f(v)− f(u)|, which can be considered as the discrete analogue of the recently proposed continuous nonlocal anisotropic functionals [27], [28].”
Image Recovery via Nonlocal Operators (2010, methodology): “The application of the nonlocal means filter to different image processing tasks is an active research area [4, 12, 22].”