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

Fast Solvers for Cahn-Hilliard Inpainting

We consider the efficient solution of the modified Cahn-Hilliard equation for binary image inpainting using convexity splitting, which allows an unconditionally gradient stable time-discretization scheme. We look at a double-well as well as a double obstacle potential. For the latter we get a nonlinear system for which we apply a semi-smooth Newton method combined with a Moreau-Yosida regularization technique. At the heart of both methods lies the solution of large and sparse linear systems. We introduce and study block-triangular preconditioners using an efficient and easy to apply Schur complement approximation. Numerical results indicate that our preconditioners work very well for both problems and show that qualitatively better results can be obtained using the double obstacle potential

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

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. </jats:p

Primal-dual active set methods for Allen-Cahn variational inequalities

This thesis aims to introduce and analyse a primal-dual active set strategy for solving Allen-Cahn variational inequalities. We consider the standard Allen-Cahn equation with non-local constraints and a vector-valued Allen-Cahn equation with and without non-local constraints. Existence and uniqueness results are derived in a formulation involving Lagrange multipliers for local and non-local constraints. Local Convergence is shown by interpreting the primal-dual active set approach as a semi-smooth Newton method. Properties of the method are discussed and several numerical simulations in two and three space dimensions demonstrate its efficiency. In the second part of the thesis various applications of the Allen-Cahn equation are discussed. The non-local Allen-Cahn equation can be coupled with an elasticity equation to solve problems in structural topology optimisation. The model can be extended to handle multiple structures by using the vector-valued Allen-Cahn variational inequality with non-local constraints. Since many applications of the Allen-Cahn equation involve evolution of interfaces in materials an important extension of the standard Allen-Cahn model is to allow materials to exhibit anisotropic behaviour. We introduce an anisotropic version of the Allen-Cahn variational inequality and we show that it is possible to apply the primal-dual active set strategy efficiently to this model. Finally, the Allen-Cahn model is applied to problems in image processing, such as segmentation, denoising and inpainting. The primal-dual active set method proves exible and reliable for all the applications considered in this thesis

Numerical analysis of an adjusted Cahn-Hilliard equation for binary image inpainting

This thesis aims to analyse a finite element method applied to an adjusted Cahn-Hilliard equation that has been used for digital image inpainting applications. We consider both the standard model with a smooth double well potential and an alternative where an obstacle potential has been used. Existence and uniqueness results are derived for both formulations by adapting techniques existing in literature for other problems. For each formulation we then propose approximations, by discretising first in space and then in time, and we derive error bounds between the weak solution of the original formulation and the solution of the discrete approximations in terms of the discretisation parameters. We then propose and implement a practical numerical scheme for both models and investigate their use in applications, alongside some other models from literature. We investigate various real digital image examples and compare the resulting inpaintings for these competing models, considering their suitability for real-world applications

Numerical solution of phase field models for two-phase flows

Phase-field models describe the motion of multiphase flows using smooth interfaces across which the composition changes continuously. The phase-field variable represents a measure of phase as it quantifies relative differences or fractions of the fluid s components. The Cahn-Hilliard equation was originally proposed to model spinodal decomposition and coarsening in binary alloys. To this day, it has become broad ranged in its applicability. This thesis focuses on solving the Cahn-Hilliard equation numerically. A review of the mathematical modelling is made in order to develop numerical methods. Different numerical simulations in two dimensions are implemented to study the numerical and physical properties. Two realistic physical examples are also numerically solved

Fast iterative solvers for Cahn-Hilliard problems

Otto-von-Guericke-Universität Magdeburg, Fakultät für Mathematik, Dissertation, 2016von M. Sc. Jessica BoschLiteraturverzeichnis: Seite [247]-25