Level Set Methods and Fast Marching Methods

Transcription

1 Level Set Methods and Fast Marching Methods I.Lyulina Scientific Computing Group May, 2002

2 Overview Existing Techniques for Tracking Interfaces Basic Ideas of Level Set Method and Fast Marching Method Linking moving fronts and hyperbolic conservation laws

3 Tracking a moving boundary Lagrangian approach x(s,t=0), y(s,t=0) parameterization of the curve: (x(s,t),y(s,t)) s?? How to deal with topological changes? discrete parameterization of the curve

4 Tracking a moving boundary Volume-of-fluid method: Eulerian approach ?? Drawbacks: -- approximation to the front is crude, a large number of cells -- curvature and normal is difficult to derive -- in 3D very complicated to perform

5 Level set and Fast marching methods Sethian J. A. Level Set Methods and Fast Marching Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision and Material Science. Cambridge University Press, 999

6 Level Set Method: an initial value formulation φ(x,y,t) y y x φ=0 F=F(L,G,I) original front level set function x

7 How do you move the front?

8 Why is this called an initial value formulation? Level set equation: φ x(t) : φ(x(t),t)= 0 φt + x ( t) = 0 x If front moves in normal direction: φ n = F = n x ( t) φ φ + F φ = 0 IC : φ ( x, t = 0) t If front is advected by velocity field: F = ( u, v) φ + F φ = 0 φ + u φ + v φ = 0 t t x y IC : φ ( x, t = 0)

9 Fast Marching Method: a boundary value formulation T(x) dx dt dt dx = F dt F = dx F T = T = 0 o n Γ x

10 Construction of stationary level set solution

11 Summary Boundary Value Formulation: TF= Front : Γ () t = (, x y): Txy (, ) = t { } Initial Value Formulation: φ + F φ = 0 t Front : Γ () t = ( x, y): φ( x, y,) t = 0 { } F > 0 F arbitrary

12 Advantages of these perspectives Unchanged in higher dimensions Topological changes are handled naturally Geometric properties are are easily determined φ T n = or n = normal vector φ T φ k = curvature φ Both equations can be accurately solved using numerical schemes for hyperbolic conservation laws

13 Hamilton-Jacobi equation Level set equation and stationary equation are particular cases of the more general Hamilton-Jacobi equation: α u + H ( Du, x) = 0 t H ( Du, x) = F u ( α ) Du H ( u, u, u, x, y, z) α α = = x y z 0 partial derivatives of u in each variable level set equation stationary equation Hamiltonian

14 Example: viscosity solutions Smooth front, constant speed function F= The swallowtail solution The leading wave solution Speed function in the form: ε X ( t), X ( t) curvature const lim X ( t) X ( t) ε ε 0 curvature = F = ε k ε > 0 const two solutions, then

15 Link between propagating fronts and hyperbolic conservation laws Hamilton-Jacobi equation with viscosity : α u + H ( u ) = ε u t x xx Hyperbolic conservation law: Burgers equation: Burgers equation with viscosity: u u t t [ G u ] + ( ) = 0 + uu = x x 0 u + uu = εu t x xx Conclusion: Level set and Fast marching methods rely on viscosity solutions of the associated partial differential equations in order to guarantee that unique, entropy-satisfying weak solution is obtained. Both equations can be accurately solved using numerical schemes for hyperbolic conservation laws.

16 Next lectures: Efficient numerical algorithms for the Level Set and Fast Marching methods Applications of Level Set and Fast Marching methods