1. Introduction to Level Set Methods: Part II
Chunming Li
Institute of Imaging Science
Vanderbilt University
URL:

2. Outline Numerical issues: Difference scheme: upwind scheme
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Outline
Numerical issues:
Difference scheme: upwind scheme
Velocity extension
Reinitialization
Variational level set method:
Level set evolution without reinitialization.
Active contour without edges
Multiphase level set methods

3. Numerical Implementation of Level Set Evolution
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Numerical Implementation of Level Set Evolution

4. Explicit Euler Scheme Consider general evolution equation:
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Explicit Euler Scheme
Consider general evolution equation:
Approximate spatial derivatives by certain difference scheme
Approximate temporal derivatives by forward difference
Update equation at each iteration:

5. Basic Finite Difference Scheme for Spatial Variable
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Basic Finite Difference Scheme for Spatial Variable
Backward difference
Forward difference
Central difference

6. Mean Curvature Motion Mean curvature motion Update equation: where
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Mean Curvature Motion
Mean curvature motion
Update equation:
where
Mean curvature motion is stable.

7. Motion in Normal Direction
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Motion in Normal Direction
Motion in normal direction:
Right hand side is approximated by:
where
Update equation:

8. Advection Pure advection equation: , with
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Advection
Pure advection equation: , with
Right hand side is approximated by:
where
Update equation:

9. Geodesic Active Contour
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Geodesic Active Contour
Geodesic active contour:
Update equation:

10. General Evolution Equation
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General Evolution Equation
Level set evolution:

11. Time Step Level set evolution:
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Time Step
Level set evolution:
For stable evolution, the time step and spatial step must satisfy the CFL condition:

12. Unstable Evolution in Standard Level Set Methods
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Unstable Evolution in Standard Level Set Methods
Evolution of level set function
Evolution of zero level set
Click here to see the movie
Click here to see the movie

13. Reinitialization (Redistance)
Reinitialization: periodically stop the evolution and repair the degraded level set function as a signed distance function.
Solve to steady state:
Reinitialization equation
“Steady state” means exists, denoted by
Signed distance function.
Use upwind scheme to numerically solve the above reinitialization equation.
Other reinitialization methods: direct compute SDF or fast marching.

14. Velocity Extension
In some applications, the speed function is only defined on the zero level set (interface)
To extend to the speed function to the entire domain or a narrow band of the zero level set, solve the boundary value problem:
or solve to steady state of the initial value problem:
with
Note: is the directional derivative of along the normal to the interface

15. Summary of Standard Level Set Methods
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Summary of Standard Level Set Methods
Initialization
Compute signed distance
Complex upwind scheme and small time step
Evolve level set function
Solve a PDE
Velocity extension
Solve another PDE!
Reinitialization N
converge? Y
stop

16. Variational Level Set Formulation
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Variational Level Set Formulation

17. Drawbacks of Reinitialization
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Drawbacks of Reinitialization
Drawbacks of Reinitialization:
Still a serious problem: when and how to reinitialize?
(no general answer so far)
Error in location of the zero level set
Computationally expensive

18. Level Set Evolution without Reinitialization (Li et al, 2005)
Goal: Find a level set evolution algorithm that can simultaneously move the zero level set while maintaining the signed distance profile throughout the entire evolution.
Characteristics of signed distance function:
signed distance function + constant
Deviation from a signed distance function:

19. Variational Level Set Formulation
Define an energy functional on level set function:
Internal energy:
Penalize the deviation from a signed distance function
External energy:
Drive the motion of the zero level set
Gradient flow (or steepest descent):
Gateaux derivative:

20. External Energy for Image Segmentation
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External Energy for Image Segmentation
Edge indicator function for image I
Define external Energy:
Weighted length term:
Weighted region term:

21. Energy Functional and Gradient Flow
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Energy Functional and Gradient Flow
Define energy functional:
The gradient flow of the functional is:

22. Mechanism of Maintaining Signed Distance
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Mechanism of Maintaining Signed Distance
Rewrite the gradient flow of internal energy:
Diffusion rate:
Positive diffusion rate
Decrease gradient (usual diffusion)
Negative diffusion rate
Increase gradient (reverse diffusion)

23. Implementation Use the smoothed Dirac function Discretization of PDE:
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Implementation
Use the smoothed Dirac function
Discretization of PDE:
Forward difference for temporal derivative
Can use relatively larger time step
Central difference for spatial derivative

24. Experimental Results Evolution of level set function
Evolution of zero level set
Click here to see the movie
Click here to see the movie

25. Flexible and Efficient Initialization
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Flexible and Efficient Initialization
The initial level set function is no longer required to be a signed
distance function in our method
A region-based initialization scheme:

26. Experimental Results
Click here to see the movie

27. Initial level set function
Evolution
Click here to see the movie

28. Detect Weak Object Boundaries
Microscope image of cells
Click here to see the movie

29. Experimental Results: Initialization from Thresholding
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Experimental Results: Initialization from Thresholding
MR image of corpus callosum
Initial level set function
Image courtesy of Hong Liu,
National Institute of Mental Health
Evolution
Click here to see the movie

30. Initial level set function
Evolution
Click here to see the movie

31. 3D Extension The proposed level set formulation
Click here to see the movie
The proposed level set formulation
and implementation can be easily
extended to 3D.

32. Thank you