37 research outputs found
A Fractional Image Inpainting Model Using a Variant of Mumford-Shah Model
In this paper, we propose a fourth order PDE model for image inpainting based
on a variant of the famous Mumford-Shah (MS) image segmentation model.
Convexity splitting is used to discrtised the time and we replace the Laplacian
by its fractional counterpart in the time discretised scheme. Fourier spectral
method is used for space discretization. Consistency, stability and convergence
of the time discretised model has been carried out. The model is tested on some
standard test images and compared them with the result of some models existed
in the literature.Comment: 19 page
Cahn--Hilliard inpainting with the double obstacle potential
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
Applications of PDEs inpainting to magnetic particle imaging and corneal topography
In this work we propose a novel application of Partial Differential Equations (PDEs) inpainting techniques to two medical contexts. The first one concerning recovering of concentration maps for superparamagnetic nanoparticles, used as tracers in the framework of Magnetic Particle Imaging. The analysis is carried out by two set of simulations, with and without adding a source of noise, to show that the inpainted images preserve the main properties of the original ones. The second medical application is related to recovering data of corneal elevation maps in ophthalmology. A new procedure consisting in applying the PDEs inpainting techniques to the radial curvature image is proposed. The images of the anterior corneal surface are properly recovered to obtain an approximation error of the required precision. We compare inpainting methods based on second, third and fourth-order PDEs with standard approximation and interpolation techniques
Convergence analysis of variable steps BDF2 method for the space fractional Cahn-Hilliard model
An implicit variable-step BDF2 scheme is established for solving the space
fractional Cahn-Hilliard equation, involving the fractional Laplacian, derived
from a gradient flow in the negative order Sobolev space ,
. The Fourier pseudo-spectral method is applied for the spatial
approximation. The proposed scheme inherits the energy dissipation law in the
form of the modified discrete energy under the sufficient restriction of the
time-step ratios. The convergence of the fully discrete scheme is rigorously
provided utilizing the newly proved discrete embedding type convolution
inequality dealing with the fractional Laplacian. Besides, the mass
conservation and the unique solvability are also theoretically guaranteed.
Numerical experiments are carried out to show the accuracy and the energy
dissipation both for various interface widths. In particular, the
multiple-time-scale evolution of the solution is captured by an adaptive
time-stepping strategy in the short-to-long time simulation
Fast iterative solvers for Cahn-Hilliard problems
Otto-von-Guericke-Universität Magdeburg, Fakultät für Mathematik, Dissertation, 2016von M. Sc. Jessica BoschLiteraturverzeichnis: Seite [247]-25
Time-fractional Cahn-Hilliard equation: Well-posedness, degeneracy, and numerical solutions
In this paper, we derive the time-fractional Cahn-Hilliard equation from
continuum mixture theory with a modification of Fick's law of diffusion. This
model describes the process of phase separation with nonlocal memory effects.
We analyze the existence, uniqueness, and regularity of weak solutions of the
time-fractional Cahn-Hilliard equation. In this regard, we consider
degenerating mobility functions and free energies of Landau, Flory--Huggins and
double-obstacle type. We apply the Faedo-Galerkin method to the system, derive
energy estimates, and use compactness theorems to pass to the limit in the
discrete form. In order to compensate for the missing chain rule of fractional
derivatives, we prove a fractional chain inequality for semiconvex functions.
The work concludes with numerical simulations and a sensitivity analysis
showing the influence of the fractional power. Here, we consider a convolution
quadrature scheme for the time-fractional component, and use a mixed finite
element method for the space discretization