The fast marching algorithm, and its variants, solves numerically the generalized eikonal equation associated to an underlying riemannian metric M. A major challenge for these algorithms is the non-isotropy of the riemannian metric, which magnitude is characterized by the anisotropy ratio κ(M) ∈ [1,∞]. Applications of the eikonal equation to image processing [1, 3] often involve large anisotrop...

We present and analyze a fast algorithm for directly computing the measure of a generalized Voronoi region associated with generators of arbitrary co-dimension. The algorithm is based upon solving one Eikonal equation to generate a kernel-based operator whose iteration accumulates “mass” along the closest generator. In particular, the algorithm does not require the computation of the ridge set ...

In this paper, we introduce a new deformable model for image segmentation, by reformulating a region based active contours energy into a geodesic contour energy involving a Finsler metric. As a result, we solve the region based active contours energy minimization problem without resorting to level set functions, but using a robust Eikonal equation framework. By sampling a set of control points ...

In this paper a fast sweeping method for computing the numerical solution of Eikonal equations on a rectangular grid is presented. The method is an iterative method which uses upwind difference for discretization and uses Gauss-Seidel iterations with alternating sweeping ordering to solve the discretized system. The crucial idea is that each sweeping ordering follows a family of characteristics...

The point source traveltime eld has an upwind singularity at the source point. Consequently, all formally high-order nite-diierence eikonal solvers exhibit rst-order convergence and relatively large errors. Adaptive upwind nite-diierence methods based on high-order Weighted Essentially NonOscillatory (WENO) Runge-Kutta diierence schemes for the paraxial eikonal equation overcome this diiculty. ...

Bridging the gap between clinical applications and mathematical models is one of the new challenges of medical image analysis. In this paper, we propose an efficient and accurate algorithm to solve anisotropic Eikonal equations, in order to link biological models using reaction-diffusion equations to clinical observations, such as medical images. The example application we use to demonstrate ou...

The point-source traveltime field has an upwind singularity at the source point. Consequently, all formally high-order, finite-difference eikonal solvers exhibit firstorder convergence and relatively large errors. Adaptive upwind finite-difference methods based on high-order Weighted Essentially NonOscillatory (WENO) RungeKutta difference schemes for the paraxial eikonal equation overcome this ...

I exploit the theoretical link between the eikonal equation and Fermat’s principle to derive a variational interpretation of the recently developed method for fast traveltime computations. This method, known as fast marching, possesses remarkable computational properties. Based originally on the eikonal equation, it can be derived equally well from Fermat’s principle. The new variational formul...

In this paper, we present a high-order weighted essentially non-oscillatory (WENO) scheme, coupled with a high-order fast sweeping method, for solving a dynamic continuum model for bidirectional flow. We first review the dynamic continuum model for bi-directional flow. This model is composed of a coupled system of a conservation law and an Eikonal equation. Then we present the first-order Lax-F...

We propose a non-perturbative attempt to solve the kinematic equations for spiral waves in excitable media. From the eikonal equation for the wave front we derive an implicit analytical relation between rotation frequency Ω and core radius R(0). For free, rigidly rotating spiral waves our analytical prediction is in good agreement with numerical solutions of the linear eikonal equation not only...