On the Bertozzi--Esedoglu--Gillette--Cahn--Hilliard Equation with Logarithmic Nonlinear Terms

Cherfils, Laurence; Fakih, Hussein; Miranville, Alain

doi:10.1137/140985627

Cited by 29 publications

(32 citation statements)

References 33 publications

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“…The analogy between the Cahn–Hilliard model in material sciences and Cahn–Hilliard model in image inpainting is that the two state phases are considered as two homogeneous regions and the interface is considered as an edge. In image inpainting, it was exploited by Bertozzi et al in , see also , by considering the following equation:

({, \begin{matrix} array \\ array \partial_{t} u + normalΔ (ε normalΔ u - \frac{1}{ε} W^{'} (u)) + λ_{D} (u - f) = 0, & array in R^{+} \times normalΩ, \\ array \frac{\partial u}{\partial n} = 0 and \frac{\partial normalΔ u}{\partial n} = 0, & array on R^{+} \times \partial normalΩ, \\ array u (0, x) = f, & array in normalΩ, \end{matrix})

which was obtained by incorporating the data fidelity term λ D ( f − u ). Note that the classical Cahn–Hilliard equation is only appropriate for two‐scale (binary) images inpainting because of the double well potential W that vanishes on only the values 0 and 1.…”

Section: Cahn–hilliard Equationmentioning

confidence: 99%

“…where .u/ The analogy between the Cahn-Hilliard model in material sciences and Cahn-Hilliard model in image inpainting is that the two state phases are considered as two homogeneous regions and the interface is considered as an edge. In image inpainting, it was exploited by Bertozzi et al in [21], see also [7,23,[43][44][45][46][47], by considering the following equation:…”

Section: Cahn-hilliard Equationmentioning

confidence: 99%

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A multiscale fourth‐order model for the image inpainting and low‐dimensional sets recovery

Theljani

Belhachmi

Kallel

et al. 2016

Math Methods in App Sciences

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We consider a fourth‐order variational model, to solve the image inpainting problem, with the emphasis on the recovery of low‐dimensional sets (edges and corners) and the curvature of the edges. The model permits also to perform simultaneously the restoration (filtering) of the initial image where this one is available. The multiscale character of the model follows from an adaptive selection of the diffusion parameters that allows us to optimize the regularization effects in the neighborhoods of the small features that we aim to preserve. In addition, because the model is based on the high‐order derivatives, it favors naturally the accurate capture of the curvature of the edges, hence to balance the tasks of obtaining long curved edges and the obtention of short edges, tip points, and corners. We analyze the method in the framework of the calculus of variations and the Γ‐convergence to show that it leads to a convergent algorithm. In particular, we obtain a simple discrete numerical method based on a standard mixed‐finite elements with well‐established approximation properties. We compare the method to the Cahn–Hilliard model for the inpainting, and we present several numerical examples to show its performances. Copyright © 2016 John Wiley & Sons, Ltd.

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({, \begin{matrix} array \\ array \partial_{t} u + normalΔ (ε normalΔ u - \frac{1}{ε} W^{'} (u)) + λ_{D} (u - f) = 0, & array in R^{+} \times normalΩ, \\ array \frac{\partial u}{\partial n} = 0 and \frac{\partial normalΔ u}{\partial n} = 0, & array on R^{+} \times \partial normalΩ, \\ array u (0, x) = f, & array in normalΩ, \end{matrix})

Section: Cahn–hilliard Equationmentioning

confidence: 99%

Section: Cahn-hilliard Equationmentioning

confidence: 99%

A multiscale fourth‐order model for the image inpainting and low‐dimensional sets recovery

Theljani

Belhachmi

Kallel

et al. 2016

Math Methods in App Sciences

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show abstract

“…Therefore, numerical methods that preserve this sort of property (for example a finite difference scheme analyzed in [11]), where the discrete solution stays in (−1, 1), circumvent any need for ad hoc processing of the recovery image. For inpainting applications, the Cahn-Hilliard inpainting model (1.1) with a logarithmic potential W log has been studied in [13] for the existence of weak solutions, and numerical simulations performed with W log reach steady states faster than simulations with W qu . Due to the technical difficulty in showing the spatial mean value of u stays strictly in the open interval (−1, 1) for arbitrary reference time intervals, the authors in [13] can only prove a local-intime existence result, and so it is not known if the solution eventually blows up after some finite time, nor if the solution converges to an equilibrium (which is the desired recovery image we seek).…”

Section: Introductionmentioning

confidence: 99%

Cahn--Hilliard Inpainting with the Double Obstacle Potential

Garcke¹,

Lam²,

Styles³

2018

SIAM J. Imaging Sci.

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The inpainting of damaged images has a wide range of applications, and many different mathematical methods have been proposed to solve this problem. Inpainting with the help of Cahn-Hilliard models has been particularly successful, and it turns out that Cahn-Hilliard inpainting with the double obstacle potential can lead to better results compared to inpainting with a smooth double well potential. However, a mathematical analysis of this approach is missing so far. In this paper we give first analytical results for a Cahn-Hilliard double obstacle inpainting model regarding existence of global solutions to the time-dependent problem and stationary solutions to the time-independent problem without constraints on the parameters involved. With the help of numerical results we show the effectiveness of the approach for binary and grayscale images.

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“…This equation was studied, endowed with Neumann boundary conditions, in [6], [7], [10], [17], and [18].…”

Section: Introductionmentioning

confidence: 99%

“…Finally, the existence of local (in time) solutions to (1.1) with logarithmic nonlinear terms has been studied in [18] (note indeed that the original Cahn-Hilliard equation was actually proposed with thermodynamically relevant logarithmic nonlinear terms which follow from a mean-field model; regular (and, in particular, cubic) nonlinear terms are approximations of such logarithmic nonlinear terms). In that case, we can obtain better results than those obtained with polynomial nonlinear terms in [17], as far as the convergence time is concerned, in particular.…”

Section: Introductionmentioning

confidence: 99%

A Complex Version of the Cahn--Hilliard Equation for Grayscale Image Inpainting

Cherfils¹,

Fakih²,

Miranville³

2017

Multiscale Model. Simul.

Self Cite

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Abstract. Our aim in this article is to propose a generalization of the BertozziEsedoglu-Gillette-Cahn-Hilliard equation, introduced for binary image inpainting, for grayscale image inpainting. In particular, we consider the solution to the corresponding Cahn-Hilliard inpainting model as a complex valued function. We are interested in the study of the well-posedness and of the asymptotic behavior, in terms of finitedimensional attractors, of the associated dynamical system. We have to face two major difficulties here. The first one comes from the fact that we no longer have the conservation of mass, i.e., of the spatial average of the order parameter u, contrary to the classical Cahn-Hilliard equation. The second one is due to the estimates on the nonlinear terms, combined with the fact that the order parameter u is complex valued. We finally give numerical simulations which confirm and extend previous ones on the efficiency of the binary model.

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On the Bertozzi--Esedoglu--Gillette--Cahn--Hilliard Equation with Logarithmic Nonlinear Terms

Cited by 29 publications

References 33 publications

A multiscale fourth‐order model for the image inpainting and low‐dimensional sets recovery

A multiscale fourth‐order model for the image inpainting and low‐dimensional sets recovery

Cahn--Hilliard Inpainting with the Double Obstacle Potential

A Complex Version of the Cahn--Hilliard Equation for Grayscale Image Inpainting

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