We present and study a new algorithm for simulating the N‐phase mean curvature motion for an arbitrary set of (isotropic) N(N−1)2 surface tensions. The departure point is the threshold dynamics algorithm of Merriman, Bence, and Osher for the two‐phase case. A new energetic interpretation of this algorithm allows us to extend it in a natural way to the case of N phases, for arbitrary surface tensions and arbitrary (isotropic) mobilities. For a large class of surface tensions, the algorithm is shown to be consistent in the sense that at every time step, it decreases an energy functional that is an approximation (in the sense of Gamma convergence) of the interfacial energy. A broad range of numerical tests shows good convergence properties. An important application is the motion of grain boundaries in polycrystalline materials: It is also established that the above‐mentioned large class of surface tensions contains the Read‐Shockley surface tensions, both in the two‐dimensional and three‐dimensional settings.© 2015 Wiley Periodicals, Inc.
Abstract. Image inpainting is the process of filling in missing parts of damaged images based on information gleaned from surrounding areas. We consider a model for inpainting binary images using a modified Cahn-Hilliard equation. We prove for the steady state problem that the isophote directions are matched at the boundary of inpainting regions. Our model has two scales, the diffuse interface scale, ε, on which it can accomplish topological transitions, and the feature scale of the image. We show via simulations that a dynamic two step method involving the diffuse interface scale allows us to connect regions across larger inpainting domains. For the model problem of stripe inpainting, we show that this issue is related to a bifurcation structure with respect to the scale ε.
An accurate and efficient algorithm, closely related to the level set method, is presented for the simulation of Mullins' model of grain growth with arbitrarily prescribed surface energies. The implicit representation of interfaces allows for seamless transitions through topological changes. Well-resolved large-scale simulations are presented, beginning with over 650,000 grains in two dimensions and 64,000 grains in three dimensions. The evolution of the misorientation distribution function (MDF) is computed, starting from random and fiber crystallographic textures with Read-Shockley surface energies. Prior work had established that with random texture the MDF shows little change as the grain network coarsened whereas with fiber texture the MDF concentrates near zero misorientation. The lack of concentration about zero of the MDF in the random texture case has not been satisfactorily explained previously since this concentration would decrease the energy of the system. In this study, very large-scale simulations confirm these previous studies. However, computations with a larger cutoff for the Read-Shockley energies and an affine surface energy show a greater tendency for the MDF to concentrate near small misorientations. This suggests that the reason the previous studies had observed little change in the MDF is kinetic in nature.In addition, patterns of similarly oriented grains are observed to form as the MDF concentrates.
Segmentation with depth is the challenging problem of obtaining three dimensional information from a single two dimensional image. Unlike the standard segmentation problem, the goal of segmentation with depth is to determine not only the boundaries of objects that appear in the image, but also their relative distances to the observer by making use of occlusion relations. ] proposed a variational formulation of this problem; according to their model, called the 2.1D sketch model, the regions that the objects occupy and their relative depth are to be extracted from the two dimensional image by minimizing a curvature based functional. Numerically, this is a highly nontrivial problem as the functional involves curvatures of the unknown contours. In this paper, we develop a level set based procedure for minimizing the Nitzberg-Mumford-Shiota energy. Unlike the minimization technique of Nitzberg-MumfordShiota [Filtering, Segmentation, and Depth, Lecture Notes in Comput. Sci. 662, Springer-Verlag, Berlin, 1993], our technique represents contours implicitly and thus allows for connections between T-junctions to take place automatically.
We propose efficient and accurate algorithms for computing certain area preserving geometric motions of curves in the plane, such as area preserving motion by curvature. These schemes are based on a new class of diffusion generated motion algorithms using signed distance functions. In particular, they alternate two very simple and fast operations, namely convolution with the Gaussian kernel and construction of the distance function, to generate the desired geometric flow in an unconditionally stable manner. We present applications of these area preserving flows to large scale simulations of coarsening.
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