1. Lecture 6 : Level Set Method

2. Introduction Developed by Books 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 interactive surface deformations