In this paper we prove the convergence at a large scale of a non-local first order equation to an anisotropic mean curvature motion. This is an eikonal-type equation with a velocity depending in a non-local way on the solution itself, that arises in the theory of dislocations dynamics. We show that if a mean curvature motion is approximated by this type of equations then it is always of variati...

We present a new Fast Marching algorithm for an eikonal equation with a velocity changing sign. This first order equation models a front propagation in the normal direction. The algorithm is an extension of the Fast Marching Method in two respects. The first is that the new scheme can deal with a time-dependent velocity and the second is that there is no restriction on its change in sign. We an...

Shape-from-Shading (SfS) is a fundamental problem in Computer Vision. Its goal is to solve the image irradiance equation. One prominent solution is the Fast Marching Method of Kimmel & Sethian. When the light source is oblique, Kimmel & Sethian proposed to rotate the image to the light source coordinate system and then solve an ‘almost’ Eikonal equation. This paper presents a new iterative vari...

Recently, shape-based matching and retrieval of 3D polygonal models has become one of the most fundamental problems in computer vision. Dealing with families of objects instead of a single one may impose further challenges on regular geometric algorithms. In this paper we focus on the classification of 3D objects based on their geodesic distance & path calculated on a mesh using an iterative al...

In the present paper we study the geometric properties of the multivalued solutions to the eikonal equation and we give the appropriate classification theorems. Our motivation stems from geometrical optics for approximating high frequency waves in stratified media. We consider the case of a fixed Hamiltonian imposed by the medium, and we present the geometric framework that describes the geomet...

A computational study of the Fast Marching and the Fast Sweeping methods for the Eikonal equation is given. It is shown that even though Fast Sweeping requires asymptotically less operations than Fast Marching, the latter method is faster than the former for the problems under consideration and for realistic grid sizes. Fully second order generalizations of methods of this type for problems wit...

In this paper we introduce an extension of the Fast Marching Method introduced by Sethian [6] for the eikonal equation modelling front evolutions in normal direction. The new scheme can deal with a time-dependent velocity without any restriction on its sign. This scheme is then used for solving dislocation dynamics problems in which the velocity of the front depends on the position of the front...

In [4], the authors have proposed a generalization of the classical Fast Marching Method of Sethian for the eikonal equation in the case where the normal velocity depends on space and time and can change sign. The goal of this paper is to propose a modified version of the Generalized Fast Marching Method proposed in [4] for which we state a general comparison principle. We also prove the conver...

In some higher dimensional nonlinear field theories integrable subsectors with infinitely many conservation laws have been identified by imposing additional integrability conditions. Originally, the complex eikonal equation was chosen as integrability condition, but recently further generalizations have been proposed. Here we show how these new integrability conditions may be derived from the g...

We study the asymptotic limit of a family of functionals related to the theory of micromagnetism in two dimensions. We prove a compactness result for families of uniformly bounded energy. After studying the corresponding one-dimensional prooles, we exhibit the ?-limit (\wall-energy") which is a variational problem on the folding of solutions of the eikonal equation jrgj = 1. We prove that the m...