154 research outputs found
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
Convective nonlocal Cahn-Hilliard equations with reaction terms
We introduce and analyze the nonlocal variants of two Cahn-Hilliard type
equations with reaction terms. The first one is the so-called
Cahn-Hilliard-Oono equation which models, for instance, pattern formation in
diblock-copolymers as well as in binary alloys with induced reaction and type-I
superconductors. The second one is the Cahn-Hilliard type equation introduced
by Bertozzi et al. to describe image inpainting. Here we take a free energy
functional which accounts for nonlocal interactions. Our choice is motivated by
the work of Giacomin and Lebowitz who showed that the rigorous physical
derivation of the Cahn-Hilliard equation leads to consider nonlocal
functionals. The equations also have a transport term with a given velocity
field and are subject to a homogenous Neumann boundary condition for the
chemical potential, i.e., the first variation of the free energy functional. We
first establish the well-posedness of the corresponding initial and boundary
value problems in a weak setting. Then we consider such problems as dynamical
systems and we show that they have bounded absorbing sets and global
attractors
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
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
Pde based inpainting algorithms: performance evaluation of the Cahn-Hillard model
Image inpainting consists in restoring a missing or a damaged part
of an image on the basis of the signal information in the pixels sur-
rounding the missing domain. To this aim a suitable image model is
needed to represent the signal features to be reproduced within the
inpainting domain, also depending on the size of the missing area.
With no claim of completeness, in this paper the main streamline of
the development of the PDE based models is retraced. Then, the
Cahn-Hillard model for binary images is analyzed in detail and its
performances are evaluated on some numerical experiments
Fast and Stable Schemes for Phase Fields Models
We propose and analyse new stabilized time marching schemes for Phase Fields model such as Allen-Cahn and Cahn-Hillard equations, when discretized in space with high order finite differences compact schemes. The stabilization applies to semi-implicit schemes for which the linear part is simplified using sparse pre-conditioners. The new methods allow to significant obtain a gain of CPU time. The numerical illustrations we give concern applications on pattern dynamics and on image processing (inpainting, segmentation) in two and three dimension cases
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