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In this work, we analyze Merriman, Bence and Osher's thresholding scheme, a time discretization for mean curvature flow. We restrict to the two-phase setting and mean convex initial conditions. In the sense of the minimizing movements interpretation of Esedoğlu and Otto, we show the time-integrated energy of the approximation to converge to the time-integrated energy of the limit. As a corollary, the conditional convergence results of Otto and one of the authors become unconditional in the two-phase mean convex case. Our results are general enough to handle the extension of the scheme to anisotropic flows for which a non-negative kernel can be chosen. [ABSTRACT FROM AUTHOR]

Several recent convergence results for numerical schemes for mean-curvature flow in particular in the multi-phase case with arbitrary surface tensions are discussed. The guiding principle of all these works is the gradient-flow structure of multi-phase mean-curvature flowwhich is explained in the general framework. For simplicity, the convergence results are presented in the simpler two-phase case. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: We describe a new class of algorithms for generating a variety of geometric interfacial motions by alternating two steps: Construction of the signed distance function (i.e. redistancing) to the interface, and convolution with a suitable kernel. These algorithms can be seen as variants of Merriman, Bence, and Osher’s threshold dynamics . The new algorithms proposed here preserve the computational efficiency of the original threshold dynamics algorithm. However, unlike threshold dynamics, the new algorithms also allow attaining high accuracy on uniform grids, without adaptive refinement. [Copyright &y& Elsevier]

Abstract: We present a new method for computing the curvature-driven motion of a curve constrained to move on a given surface. It is based on the Diffusion Generated Motion algorithm, and retains both the novel simplicity of that method, as well as the natural extension to curves with junctions, to general geometric motion laws, and to higher dimensions. The result is an extremely simple algorithm for curvature-dependent motion on surfaces, wherein the only evolution operation is linear diffusion in three-dimensional Euclidean space. [Copyright &y& Elsevier]

Diffusion generated motion has been used to generate a variety of interface motions. In this paper, we present a new shape recovery model with diffusion generated motion. The approach is based on alternately diffusing and sharpening the initial region to move the sharp interface toward the boundaries of the desired objects. The shapes are recovered by an anisotropic interface motion with a local image property dependent speed. Our algorithm is simple and easy to implement. It automatically captures topological changes and works for both 2D and 3D image data. Experimental results for synthetic and real images are presented. [Copyright &y& Elsevier]

• Original numerical method for 2D and 3D high-order geometrical flows based on the diffusion of signed distance functions. • Projection method to impose the constraint of fixed area and enclosed volume of a surface as it moves. • Validation and application to the equilibrium shapes of vesicles. In this paper we present a novel algorithm for simulating geometrical flows, and in particular the Willmore flow, with conservation of volume and area. The idea is to adapt the class of diffusion-redistanciation algorithms to the Willmore flow in both two and three dimensions. These algorithms rely on alternating diffusions of the signed distance function to the interface and a redistanciation step, and with careful choice of the applied diffusions, end up moving the zero level-set of the distance function by some geometrical quantity without resorting to any explicit transport equation. The constraints are enforced between the diffusion and redistanciation steps via a simple rescaling method. The energy globally decreases at the end of each global step. The algorithms feature the computational efficiency of thresholding methods without requiring any adaptive remeshing thanks to the use of a signed distance function to describe the interface. This opens their application to dynamic fluid-structure simulations for large and realistic cases. The methodology is validated by computing the equilibrium shapes of two- and three-dimensional vesicles, as well as the Clifford torus. [ABSTRACT FROM AUTHOR]