Level Set Methods and Fast Marching Methods I.Lyulina Scientific Computing Group May, 2002
Overview Existing Techniques for Tracking Interfaces Basic Ideas of Level Set Method and Fast Marching Method Linking moving fronts and hyperbolic conservation laws
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
Tracking a moving boundary Volume-of-fluid method: Eulerian approach.5.9.7.3.2.3.2.6.7.8.9.9.8 0.2.7 0..5?? 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
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 http:/math.berkeley.edu/~sethian/level_set.html
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
How do you move the front?
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)
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
Construction of stationary level set solution
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
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
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
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
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.
Next lectures: Efficient numerical algorithms for the Level Set and Fast Marching methods Applications of Level Set and Fast Marching methods