63 research outputs found
Multiclass Data Segmentation using Diffuse Interface Methods on Graphs
We present two graph-based algorithms for multiclass segmentation of
high-dimensional data. The algorithms use a diffuse interface model based on
the Ginzburg-Landau functional, related to total variation compressed sensing
and image processing. A multiclass extension is introduced using the Gibbs
simplex, with the functional's double-well potential modified to handle the
multiclass case. The first algorithm minimizes the functional using a convex
splitting numerical scheme. The second algorithm is a uses a graph adaptation
of the classical numerical Merriman-Bence-Osher (MBO) scheme, which alternates
between diffusion and thresholding. We demonstrate the performance of both
algorithms experimentally on synthetic data, grayscale and color images, and
several benchmark data sets such as MNIST, COIL and WebKB. We also make use of
fast numerical solvers for finding the eigenvectors and eigenvalues of the
graph Laplacian, and take advantage of the sparsity of the matrix. Experiments
indicate that the results are competitive with or better than the current
state-of-the-art multiclass segmentation algorithms.Comment: 14 page
Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction
In this paper we study the long-time behavior of a nonlocal Cahn-Hilliard
system with singular potential, degenerate mobility, and a reaction term. In
particular, we prove the existence of a global attractor with finite fractal
dimension, the existence of an exponential attractor, and convergence to
equilibria for two physically relevant classes of reaction terms
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
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
Mini-Workshop: Analytical and Numerical Methods in Image and Surface Processing
The workshop successfully brought together researchers from mathematical analysis, numerical mathematics, computer graphics and image processing. The focus was on variational methods in image and surface processing such as active contour models, Mumford-Shah type functionals, image and surface denoising based on geometric evolution problems in image and surface fairing, physical modeling of surfaces, the restoration of images and surfaces using higher order variational formulations
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
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