Lecture 6 : Level Set Method

1. Lecture 6 : Level Set Method

2. Introduction • Developed by • Stanley Osher (UCLA) • J. A. Sethian (UC Berkeley) • Books • J.A. Sethian: Level Set Methods and Fast Marching Methods, 1999 • S. Osher, R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces , 2002

3. Evolving Curves and Surfaces

4. Geometry Representation

5. Explicit Techniques for Evolution

6. Explicit Techniques - Drawbacks

7. Implicit Geometries

8. Discretized Implicit Geometries

9. Level Set Method: Overview • Generic numerical method for evolving fronts in an implicit form • Handles topological changes of the evolving interface • Define problem in 1 higher dimension • Use an implicit representation of the contour C as the zero level set of higher dimensional function - the level set function

10. Level Set Method: Overview • Move the level set function, so that it deforms in the way the user expects • contour = cross section at z=t

11. Implicit Curve Evolution

12. Level Set Evolution • Define a speed function F, that specifies how contour points move in time • Based on application-specific physics such as time, position, normal, curvature, image gradient magnitude • Build an initial level set curve • Adjust over time • Current contour is defined as

13. Equation for Level Set Evolution • Indirectly move C by manipulating where F is the speed function normal to the curve Level set equation

14. Example: an expanding circle • Level Set representation of a circle • Setting F=1 causes the circle to expand uniformly • Observe everywhere • We obtain • Explicit solution: • meaning the circle has radius r+t at time t

15. Example: an expanding circle

16. Motion under curvature • Complicated shapes? • Each piece of the curve moves perpendicular to the curve with speed proportional to the curvature • Since curvature can be either positive or negative , some parts of the curve move outwards while others move inwards • Example movie file • Setting F = curvature

17. Level Set Segmentation • We may think of as signed distance function • Negative inside the curve • Positive outside the curve • Distance function has unit gradient almost everywhere and smooth • By choosing a suitable speed function F, we may segment an object in an image

18. Level Set Segmentation • Evolving Geometry : F(X,t)=0 • Intuitively, move a lot on low intensity gradient area and move little on high intensity gradient area along normal direction • F : speed function , k : curvature , I : intensity

19. Segmentation Example • Arterial tree segmentation

20. Discretization • Use upwinded finite difference approximations (first order)

22. Acceleration Techniques • Acceleration for fast level set method • Narrow band level set method • Fast marching method

23. Narrow band level set method • The efficiency comes from updating the speed function • We do not need to update the function over the whole image or volume • Update over a narrow band (2D or 3D)

24. Fast Marching Method • Assume the front (level set) propagates always outward or always inward • Compute T(x,y)=time at which the contour crosses grid point (x,y) • At any height T, the surface gives the set of points reached at time T

25. Fast Marching Algorithm

26. Fast Marching Algorithm

27. Fast Marching Method

28. Applications • Segmentation • Level Set Surface Editing Operators • Surface Reconstruction

29. Segmetation • 2D • 3D

30. Level Set Surface Editing Operators • SIGGRAPH 2002

31. Level Set Surface Editing Operators

32. Surface Reconstruction • zhao, osher, and fedkiw 2001

33. A painting interface for interactivesurface deformations