Mean Curvature Motion Research Articles

Counter-Example in Three Dimension and Homogenization of Geometric Motions in Two Dimension

In this paper we give a counter-example to the homogenization of the forced mean curvature motion in a periodic setting in dimension \({N \geqq 3}\) when the forcing is positive. We also prove a general homogenization result for geometric motions in dimension N = 2 under the assumption that there exists a constant δ > 0 such that every straight line moving with a normal velocity equal to δ is a subsolution for the motion. We also present a generalization in dimension 2, where we allow sign changing normal velocity and still construct bounded correctors, when there exists a subsolution with compact support expanding in all directions.

Multiscale Analysis for Images on Riemannian Manifolds

In this paper we study multiscale analyses for images defined on Riemannian manifolds and extend the axiomatic approach proposed by Álvarez, Guichard, Lions, and Morel to this general case. This covers the case of two- and three-dimensional images and video sequences. After obtaining the general classification, we consider the case of morphological scale spaces, which are given in terms of geometric equations, and the linear case given by the Laplace--Beltrami flow. We consider in some detail the case of image metrics given in terms of the structure tensor and compute some cases of such a tensor for video. Then we comment on the connections with variational formulations of image diffusion comparing the anisotropies that appear. Finally, we include numerical experiments illustrating some of the models. Namely, we compare some examples for still images using the Laplace--Beltrami flow and some variational models. We also consider several examples in video: the mean curvature motion and the extension of the morphological and Galilean invariant scale spaces to the video manifold case, and the Laplace--Beltrami flow. We point out that the number of models that appear is huge, and we have restricted ourselves to such cases for the sake of brevity and illustration.

The Influence of the Exterior Surface on Grain Boundary Mobility Measurements

Polycrystalline materials are materials made up of a typically large number of crystals (``grains''), whose boundaries (``grain boundaries'') define an interior network which effectively determines the microstructure of the specimen. As the network is not at equilibrium, it evolves in time to reduce the overall energy of the specimen. Since the grain boundaries couple to the exterior surface of the specimen, the evolution of the microstructure is influenced by the motion of the exterior surface. In thin specimens, this effect can be appreciable. Within the framework of the classical Mullins' model, grain boundaries move by mean curvature motion, $V_n=A\kappa$, and the exterior surface evolves by surface diffusion, $V_n=-B\Delta_s\kappa$, where $V_n$ and $\kappa$ denote the normal velocity and the mean curvature of the respective evolving surfaces, and $\Delta_s$ is the surface Laplacian. Accurately estimating the kinetic parameters, $A$ and $B$, constitutes an important characterization of material properties. A classical way to determine the “reduced mobility,” $A$, is to make measurements based on a specially constructed “half-loop bicrystalline geometry.” In this geometry there are two grains; one grain is embedded within the other and recedes at a roughly constant rate. This rate may be used to estimate $A$ if the effects of the exterior surface may be neglected. Here we focus on the effects of the exterior surface on grain boundary mobility measurements in the context of the half-loop bicrystalline geometry. Asymptotics, based on a small grain boundary surface energy to exterior surface energy ratio and a thin specimen aspect ratio, are used to predict the coupled grain boundary and exterior surface motion, as well as a leading order correction to mobility measurements.

An anisotropic diffusion filtering method for speckle reduction of synthetic aperture radar images

An anisotropic diffusion filtering method is proposed to reduce multiplicative speckle noise in synthetic aperture radar images. A new anisotropic diffusion equation, which can effectively smooth speckles near the edges and reduce blocking artifacts in homogeneous regions, is established by embedding the mean curvature motion controlled by an adaptive coupling function into a traditional speckle reducing anisotropic diffusion equation. Moreover, a new direction-constrained diffusion function formed by combining the local directional ratios with the improved Frost filtering is introduced into the new anisotropic diffusion equation, so that it can further reduce blocking artifacts and obviously improve the edge distortion problem in the despeckled synthetic aperture radar images. Experimental results show that the proposed method can obviously reduce blocking artifacts, sufficiently smooth speckle noise in homogeneous regions and near the edges while preserving edges, and effectively improve the edge distortion problem. Visual quality and performance indexes of the despeckled images show that the proposed method outperforms several despeckling methods.

Grain boundary migration and grooving in thin 3-D systems

We employ numerical simulations and experimental methods to study grain boundary migration in idealized 3-D systems of three grains in a thin film, focusing on hole formation and annihilation of small grains. The initial structure is taken to be columnar, and isotropy is assumed for simplicity. The grain boundaries and the external surfaces of the three-grain system are assumed to be governed by mean curvature motion and surface diffusion, respectively. Along the thermal grooves, where the grain boundaries and the exterior surfaces couple, balance of mechanical forces, continuity of the surface chemical potential, and balance of mass flux dictate the boundary conditions. Using a parametric description for the evolving surfaces yields partial differential algebraic equations which are solved with finite-difference schemes on staggered grids. The evolution of the three-grain system results in hole formation or annihilation of the smallest grain, depending on the relative size of the smallest grain. Shortly prior to annihilation, the exterior surface of the smallest grains may invert, partially losing its convexity, with annihilation being accompanied by accelerated pitting rates. Similar features were observed experimentally.

A variational method for multiphase volume-preserving interface motions

We develop a numerical method for realizing mean curvature motion of interfaces separating multiple phases, whose volumes are preserved throughout time. The foundation of the method is a thresholding algorithm of the Bence–Merriman–Osher type. The original algorithm is reformulated in a vector setting, which allows for a natural inclusion of constraints, even in the multiphase case. Moreover, a new method for overcoming the inaccuracy of thresholding methods on non-adaptive grids is designed, since this inaccuracy becomes especially prominent in volume-preserving motions. Formal analysis of the method and numerical tests are presented.

Nonlinear diffusions and structure tensor in modelling to speckle noise reduction and edge enhancement in ultrasound images.

Speckle noise generally affects medical ultrasound images quality, and tends to reduce the image resolution and contrast. Speckle noise is an inherent property in which the images are formed under random interference between the coherent natures returns of a transmitted waveform that cause from scattering phenomenon. Many solutions have been proposed so far to remove speckle noise without loosing the edge information in images. This paper proposes two diffusion processes, in which isotropic diffusion solves the problem of speckle noise reduction and anisotropic diffusion enhances edges and local details in ultrasound images. Both the cases are controlled by the nonlinear diffusion and structure tensor model. The proposed model combines between the regularized nonlinear diffusion process with the mean curvature motion equation and structure tensor. The model performs simultaneous speckle noise reduction in homogeneous region, structure enhancement in inhomogeneous region using nonlinear diffusion based on local variations of the gradient orientation of an image. The paper also presents experimental results carried out on ultrasound images affected by speckle noise for illustrating the effectiveness of the proposed model.

Transition region-based active contour model for image segmentation

Based on the curve evolution and the level set method, we proposed a geometric active contour model in a partial differential equation formulation. The evolution PDE is directly derived from the mean curvature motion with a transition region-based external force, and then is further simplified to a nonhomogeneous linear diffusion equation. Due to the desired sign property of the external force, the level set function starting with an arbitrary bounded function can adaptively move up or down in accordance with image information and has finally the opposite sign inside and outside the objects. The experiments show that the proposed model is very appropriate for a wider range of images, especially for real images with high noise and blurred boundaries.

A semi-Lagrangian scheme for mean curvature motion with nonlinear Neumann conditions

A numerical method for mean curvature motion in bounded domains with nonlinear Neumann boundary conditions is proposed and analyzed. It consists of a semi-Lagrangian scheme in the main part of the domain as proposed by Carlini, Falcone and Ferretti, combined with a finite difference scheme in small layers near the boundary to cope with the boundary condition. The consistency of the new scheme is proved for nonstructured triangular meshes in dimension two. The monotonicity of a regularized version of the scheme with some additional vanishing artificial viscosity is studied. Details on the implementation are given. Numerical tests are presented.

A nested variational time discretization for parametric Willmore flow

A novel variational time discretization of isotropic and anisotropic Willmore flow combined with a spatial parametric finite element discretization is applied to the evolution of polygonal curves and triangulated surfaces. In the underlying natural approach for the discretization of gradient flows a nested optimization problem has to be solved at each time step. Thereby, an outer variational problem reflects the time discretization of the actual Willmore flow and involves an approximate L^2 -distance between two consecutive time steps and a fully implicit approximation of the Willmore energy. The mean curvature needed to evaluate the integrant of the latter energy is replaced by the time discrete, approximate speed from an inner, fully implicit variational scheme for mean curvature motion. To solve the resulting PDE constrained optimization problem at every time step duality techniques from PDE optimization are applied. Computational results underline the robustness of the new scheme, in particular with respect to large time steps, and show applications to surface restoration and blending.

Travelling graphs for the forced mean curvature motion in an arbitrary space dimension

We construct travelling wave graphs of the form z = -ct phi(x), phi : x is an element of RN-1 -> phi(x) is an element of R, N >= 2, solutions to the N-dimensional forced mean curvature motion V-n = -co + k (c >= co) with prescribed asymptotics. For any 1-homogeneous function phi(infinity) , viscosity solution to the eikonal equation vertical bar D phi(infinity)vertical bar =root(c/c0)(2) - 1, we exhibit a smooth concave solution to the forced mean curvature motion whose asymptotics is driven by phi(infinity). We also describe phi(infinity) in terms of a probability measure on SN-2.

A Proof of Equivalence between Level Lines Shortening and Curvature Motion in Image Processing

In this paper we define the continuous Level Lines Shortening evolution of a two-dimensional image as the curve shortening operator acting simultaneously and independently on all the level lines of the initial data, and we show that it computes a viscosity solution for the mean curvature motion. This provides an exact analytical framework for its numerical implementation, which runs on line on any image at http://www.ipol.im. Analogous results hold for its affine variant version, the Level Lines Affine Shortening.

Curvature-driven diffusion-based mathematical image registration models

Abstract This paper introduces several mathematical image registration models. Image registration, an ill-posed optimization problem, is formulated as the minimization of the sum of an image similarity metric and a regularization term. Curvature-driven diffusion-based techniques, in particular Perona-Malik, anisotropic diffusion, mean curvature motion (MCM), affine invariant MCM (AIMCM), are employed as a regularization term in this optimal control formulation. Adopting the steepest-descent marching with an artificial time t, Euler-Lagrange (EL) equations with homogeneous Neumann boundary conditions are obtained. These EL equations are approximately solved by the explicit Petrov-Galerkin scheme. The method is applied to the registration of brain MR images of size 257 × 257 . Computational results indicate that all these regularization terms produce similarly good registration quality but that the cost associated with the AIMCM approach is, on average, less than that for the others. MSC:68U10, 65D18, 65J05, 97N40.

Paradoxical grain boundary motion in a wedge-shaped aluminum bicrystal

We provide an explanation, within the framework of a model based on standard mean curvature grain-boundary motion coupled with somewhat unusual boundary conditions, for the paradoxical experimental results reported by Rath and Hu (Trans Am Inst Min Eng, 245:1577, 1969) regarding grain boundary migration in aluminum bicrystals. In their experiments, they measured the velocity of the grain boundary in a wedge-shaped bicrystal of aluminum, and assuming that rt ∝ − r−n, where r is the radius of the grain boundary as measured on the external surface, they found that 3.3 < n < 4, despite the fact that according to much well-established theory and extensive experimental data to be found elsewhere, n should be equal to one. We undertake a theoretical study of the dynamics of an initially cylindrical grain embedded in a 3D-single crystal film. The grain boundary is assumed to maintain axisymmetry and to move by mean curvature motion. Boundary conditions for the grain boundary are taken to be determined by the experimental results of Rath and Hu. We find a solution using a combination of analytical and numerical methods. Our analytical model predicts that rt ∝ fn, where f, the force applied by the grain boundary on the triple junction, is given by \(f=\gamma_{\rm gb} \sin \theta\), where γgb is the grain boundary energy and θ is the tilt angle of the grain boundary at the triple junction, and n satisfies 3.3 < n < 4 as earlier. This power law prediction is not in accordance with classical Mullins’ theory; however, it should be possible to test experimentally, thus allowing our approach to be checked for consistency with experiment. We suggest that the unusual kinetics observed in the experiment are due to grain boundary interaction at the surface triple junction with an external surface covered with aluminum oxide.

Mixing Times of Monotone Surfaces and SOS Interfaces: A Mean Curvature Approach

We consider stochastic spin-flip dynamics for: (i) monotone discrete surfaces in Z^3 with planar boundary height and (ii) the one-dimensional discrete Solid-on-Solid (SOS) model confined to a box. In both cases we show almost optimal bounds O(L^2polylog(L)) for the mixing time of the chain, where L is the natural size of the system. The dynamics at a macroscopic scale should be described by a deterministic mean curvature motion such that each point of the surface feels a drift which tends to minimize the local surface tension. Inspired by this heuristics, our approach consists in bounding the dynamics with an auxiliary one which, with very high probability, follows quite closely the deterministic mean curvature evolution. Key technical ingredients are monotonicity, coupling and an argument due to D.B. Wilson in the framework of lozenge tiling Markov Chains. Our approach works equally well for both models despite the fact that their equilibrium maximal height fluctuations occur on very different scales (logarithmic for monotone surfaces and L^{1/2} for the SOS model). Finally, combining techniques from kinetically constrained spin systems together with the above mixing time result, we prove an almost diffusive lower bound of order 1/L^2 up to logarithmic corrections for the spectral gap of the SOS model with horizontal size L and unbounded heights.

Uniqueness and existence of spirals moving by forced mean curvature motion

In this paper, we study the motion of spirals by mean curvature type motion in 1 the (two dimensional) plane. Our motivation comes from dislocation dynamics; in this context, spirals appear when a screw dislocation line reaches the surface of a crystal. The first main result of this paper is a comparison principle for the corresponding parabolic quasi-linear equation. As far as motion of spirals are concerned, the novelty and originality of our setting and results come from the fact that, first, the singularity generated by the attached end point of spirals is taken into account for the first time, and second, spirals are studied in the whole space. Our second main result states that the Cauchy problem is well-posed in the class of sub-linear weak (viscosity) solutions. We also explain how to get the existence of smooth solutions when initial data satisfy an additional compatibility condition.

Image Visualization and Restoration by Curvature Motions

This paper presents a review, analysis, and comparison of numerical methods implementing the curvature motion and the affine curvature motion for two-dimensional (2D) images, shapes, and curves. These curvature scale spaces allow, in principle, one to compute an accurate multiscale curvature in digital images. The fastest and most invariant of them can be used in a complete image processing chain. This numerical chain simulates the accurate subpixel evolution of an image by mean curvature motion or by affine invariant curvature motion. To do so, it lets all the level lines of the image evolve by curvature shortening (of affine shortening), computes the image curvature directly on the smoothed level lines, and reconstructs the evolved image and its curvatures in an intrinsic, grid independent representation. The paper describes a careful implementation of this chain and analyzes its effects on many examples. The microscopic visualization of an image curvature map reveals after processing many image details. This image process improves graphic images and gets rid of compression and aliasing effects. It also gives an accurate tool to explore the validity of Attneave’s and Julesz’s theories on shape perception and texture discrimination. The “curvature microscope’’ runs online for any image at http://www.ipol.im/pub/algo/cmmm_image_curvature_microscope/.

Scale Space Meshing of Raw Data Point Sets

AbstractThis paper develops a scale space strategy for orienting and meshing exactly and completely a raw point set. The scale space is based on the intrinsic heat equation, also called mean curvature motion (MCM). A simple iterative scheme implementing MCM directly on the raw point set is described, and a mathematical proof of its consistency with MCM is given. Points evolved by this MCM implementation can be trivially backtracked to their initial raw position. Therefore, both the orientation and mesh of the data point set obtained at a smooth scale can be transported back on the original. The gain in visual accuracy is demonstrated on archaeological objects by comparison with several state of the art meshing methods.

“Zero” temperature stochastic 3D ising model and dimer covering fluctuations: A first step towards interface mean curvature motion

AbstractWe consider the Glauber dynamics for the Ising model with “+” boundary conditions, at zero temperature or at a temperature that goes to zero with the system size (hence the quotation marks in the title). In dimension d = 3 we prove that an initial domain of linear size L of “−” spins disappears within a time τ+, which is at most L2(log L)c and at least L2/(c log L) for some c &gt; 0. The proof of the upper bound proceeds via comparison with an auxiliary dynamics which mimics the motion by mean curvature that is expected to describe, on large time scales, the evolution of the interface between “+” and “−” domains. The analysis of the auxiliary dynamics requires recent results on the fluctuations of the height function associated to dimmer coverings of the infinite honeycomb lattice. Our result, apart from the spurious logarithmic factors, is the first rigorous confirmation of the Lifshitz law τ+ ≃ const × L2, conjectured on heuristic grounds [8, 13]. In dimension d = 2, τ+ can be shown to be of order L2 without logarithmic corrections: the upper bound was proven in [6], and here we provide the lower bound. For d = 2, we also prove that the spectral gap of the generator behaves like ${ c \over L}$ for L large, as conjectured in [2]. © 2011 Wiley Periodicals, Inc.

Convergence of a large time-step scheme for mean curvature motion

We analyse the properties of a semi-Lagrangian scheme for the approximation of the Mean Curvature Motion (MCM). This approximation is obtained by coupling a stochastic method for the approximation of characteristics (to be understood in a generalized sense) with a local interpolation. The main features of the scheme are that it can handle degeneracies, it is explicit and it allows for large time steps. We also propose a modified version of this scheme, for which monotonicity and consistency can be proved. Then convergence to the viscosity solution of the MCM equation follows by an extension of the Barles–Souganidis theorem. The scheme is also compared with similar existing schemes proposed by Crandall and Lions and, more recently, by Kohn and Serfaty. Finally, several numerical test problems in 2D and 3D are presented.