scholarly journals Finite-dimensional attractors for the Bertozzi--Esedoglu--Gillette--Cahn--Hilliard equation in image inpainting

2015 ◽  
Vol 9 (1) ◽  
pp. 105-125 ◽  
Author(s):  
Laurence Cherfils ◽  
◽  
Hussein Fakih ◽  
Alain Miranville ◽  
◽  
...  
Keyword(s):  
Image Inpainting ◽  
Hilliard Equation ◽  
Finite Dimensional ◽  
Cahn Hilliard Equation

scholarly journals An adaptive Cahn-Hilliard equation for enhanced edges in binary image inpainting

2020 ◽  
Vol 14 ◽  
pp. 174830262094143
Author(s):  
Anis Theljani ◽  
Hamdi Houichet ◽  
Anis Mohamed
Keyword(s):  
Image Inpainting ◽  
Binary Image ◽  
Mesh Adaptation ◽  
Singular Set ◽  
Discrete Level ◽  
Diffusion Operator ◽  
Hilliard Equation ◽  
Proposed Model ◽  
Order Sets ◽  
Cahn Hilliard Equation

We consider the Cahn-Hilliard equation for solving the binary image inpainting problem with emphasis on the recovery of low-order sets (edges, corners) and enhanced edges. The model consists in solving a modified Cahn-Hilliard equation by weighting the diffusion operator with a function which will be selected locally and adaptively. The diffusivity selection is dynamically adopted at the discrete level using the residual error indicator. We combine the adaptive approach with a standard mesh adaptation technique in order to well approximate and recover the singular set of the solution. We give some numerical examples and comparisons with the classical Cahn-Hillard equation for different scenarios. The numerical results illustrate the effectiveness of the proposed model.

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

2017 ◽  
Vol 15 (1) ◽  
pp. 575-605 ◽  
Author(s):  
Laurence Cherfils ◽  
Hussein Fakih ◽  
Alain Miranville
Keyword(s):  
Image Inpainting ◽  
Grayscale Image ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation

scholarly journals Enhancement of damaged-image prediction through Cahn–Hilliard image inpainting

2021 ◽  
Vol 8 (5) ◽  
pp. 201294
Author(s):  
José A. Carrillo ◽  
Serafim Kalliadasis ◽  
Fuyue Liang ◽  
Sergio P. Perez
Keyword(s):  
Neural Network ◽  
Computational Cost ◽  
Image Inpainting ◽  
Finite Volume Scheme ◽  
Binary Images ◽  
Hilliard Equation ◽  
The Neural Network ◽  
Image Prediction ◽  
Different Types ◽  
Cahn Hilliard Equation

We assess the benefit of including an image inpainting filter before passing damaged images into a classification neural network. We employ an appropriately modified Cahn–Hilliard equation as an image inpainting filter which is solved numerically with a finite-volume scheme exhibiting reduced computational cost and the properties of energy stability and boundedness. The benchmark dataset employed is Modified National Institute of Standards and Technology (MNIST) dataset, which consists of binary images of handwritten digits and is a standard dataset to validate image-processing methodologies. We train a neural network based on dense layers with MNIST, and subsequently we contaminate the test set with damages of different types and intensities. We then compare the prediction accuracy of the neural network with and without applying the Cahn–Hilliard filter to the damaged images test. Our results quantify the significant improvement of damaged-image prediction by applying the Cahn–Hilliard filter, which for specific damages can increase up to 50% and is advantageous for low to moderate damage.

scholarly journals A fourth-order parabolic equation with a logarithmic nonlinearlity

2004 ◽  
Vol 69 (1) ◽  
pp. 35-48 ◽  
Author(s):  
Ahmed Bonfoh
Keyword(s):  
Free Energy ◽  
Parabolic Equation ◽  
Fourth Order ◽  
Constitutive Equations ◽  
Existence And Uniqueness ◽  
Uniqueness Of Solutions ◽  
Hilliard Equation ◽  
Finite Dimensional ◽  
Fourth Order Parabolic Equation ◽  
Cahn Hilliard Equation

We consider some generalisations of the Cahn—Hilliard equation based on constitutive equations derived by M. Gurtin in (1996) with a logarithmic free energy. Compared to the classical Cahn—Hilliard equation (see [4, 5]), these models take into account the work of internal microforces and the anisotropy of the material. We obtain the existence and uniqueness of solutions results and then prove the existence of finite dimensional attractors.

scholarly journals On the sharp interface limit of a model for phase separation on biological membranes

Analysis ◽  
2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Helmut Abels ◽  
Johannes Kampmann
Keyword(s):  
Lipid Raft ◽  
Cell Membranes ◽  
System Modeling ◽  
Bulk Diffusion ◽  
Type Equation ◽  
Sharp Interface ◽  
Interface Problem ◽  
Hilliard Equation ◽  
System A ◽  
Cahn Hilliard Equation

AbstractWe rigorously prove the convergence of weak solutions to a model for lipid raft formation in cell membranes which was recently proposed in [H. Garcke, J. Kampmann, A. Rätz and M. Röger, A coupled surface-Cahn–Hilliard bulk-diffusion system modeling lipid raft formation in cell membranes, Math. Models Methods Appl. Sci. 26 2016, 6, 1149–1189] to weak (varifold) solutions of the corresponding sharp-interface problem for a suitable subsequence. In the system a Cahn–Hilliard type equation on the boundary of a domain is coupled to a diffusion equation inside the domain. The proof builds on techniques developed in [X. Chen, Global asymptotic limit of solutions of the Cahn–Hilliard equation, J. Differential Geom. 44 1996, 2, 262–311] for the corresponding result for the Cahn–Hilliard equation.

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