1. Active Contours, Level Sets, and Image Segmentation
Chunming Li
Institute of Imaging Science
Vanderbilt University
URL:

2. Outline Curve evolution and level set methods:
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Outline
Curve evolution and level set methods:
Curve Evolution: from snake to general curve evolution
Level Set Methods: basic concepts and methods
Numerical issues:
Difference scheme: upwind scheme, ENO interpolation…
Reinitialization, velocity extension …
Introduction to calculus of variations and applications:
Segmentation: variational level set, shape prior …
Denoising: diffusion, TV, …

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2D Level Set
Click to play the movie
Click to play the movie

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3D Level Set
Click to play the movie
Click to play the movie

5. More Applications of Level Sets
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More Applications of Level Sets
Fluid Dynamics
Computer Graphics: Surface Rendering, Animation…
Material Science
Others…

6. Advantages of Active Contours and Level Sets
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Advantages of Active Contours and Level Sets
Nice representation of object boundary:
Smooth and closed, good for shape analysis and recognition and other applications.
Sub-pixel accuracy.
Can incorporate various information such as shape prior and motion.
Mature mathematical tools can be used: calculus of variations, PDE, differential geometry …

7. Snake Models

8. Image Segmentation: Classic Methods
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Image Segmentation: Classic Methods
An image of blood vessel
Thresholding
Edge detection

9. Parametric Active Contours (Kass et al 1987)
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Parametric Active Contours (Kass et al 1987)
For a contour
, define energy:

10. Evolution of Active Contours
Steepest descent:
Internal force: Regularizes the contour.
External force: Drives the contour towards the desired object boundary.
In the classic snake model, the external force field is:

11. External Force

12. Gradient Vector Flow (Xu and Prince, 1998)
Find a vector field v(x,y) that minimize the energy functional:
Solve two decoupled PDEs:

13. An Example
Gradient vector flow

14. Advantages of GVF Snake
Significantly increase capture range
Be able to detect concave object boundary

15. General Curve Evolution

16. Differential Geometry of Curves
A planar curve C is a function , with
Tangent vector
Normal vector
Arc length between and

17. Curvature Use arc length as the parameter.
The tangent vector is a unit vector
The unit vector defines the direction of the curve
Curvature describes how fast the curve changes its direction.
Take derivative again:
Note that

18. Curvature (Cont’d) Both and are orthogonal to is parallel to
For general parameterization the curvature can be expressed as:

19. Dynamic Curves A dynamic curve is a time dependent curve
The motion of the curve is governed by a curve evolution equation:
The tangent vector and normal vector forms a basis of

20. Geometric Curve Evolution
Theorem: Let be an intrinsic quantity. If evolves according to
Then, there exists another parameterization of such that
is solution of
where at the same point
General geometric curve evolution:

21. Constant Speed Motion and Mean Curvature Motion
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Constant Speed Motion and Mean Curvature Motion
Mean curvature motion (Length shortening flow)
Constant Speed Motion (Area decreasing/increasing)
Click to play the movie
Click to play the movie

22. Mean curvature motion (Length shortening flow)
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Mean curvature motion (Length shortening flow)
Constant Speed Motion (Area decreasing flow)

23. Why MCM Shortens Curve Length?
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Why MCM Shortens Curve Length?
Mean curvature motion is the steepest descent
flow (or gradient flow) that minimizes arc length of the contour:
Click to play the movie

24. Geodesic Active Contour (Caselles et al, 1997)
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Geodesic Active Contour (Caselles et al, 1997)
Euclidean metric:
Minimize a weighted length of C
where
This is a Riemannian metric.
How to find the geodesic?
Solve the gradient flow equation:

25. Geodesic Active Contour
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Geodesic Active Contour
How does the evolution stop the contour at the edges?
External force
The external force has two opposite directions on the two sides of the object boundary.
The curvature term regularizes the curve: shorten and smooth.
Problem: The contour cannot progress into concavity.
Add a balloon/pressure force:

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Example
Click to play the movie

27. Difficulties of Parameterized Curve Evolution
Re-parameterization during evolution: very difficult for 3D surface
Cannot handle topological changes

28. Solutions Some approaches within parametric active contour framework:
McInerney and Terzopoulos, 1995
Ray and Acton, 2003
Li, Liu, and Fox, 2005
More natural and better approaches:
Level set methods

29. Level Set Methods

30. Level Set Representation of Curves
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Level Set Representation of Curves
zero level
zero level

31. Advantages of Level Set Methods
All computation is carried out on a fixed grid, no need for re-parameterization.
Topological changes are handled naturally.
More …
Also have some disadvantages, will be discussed later.

32. Level Set Functions contour
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Level Set Functions
contour
Level set functions with the same zero level set

33. Signed Distance Function
Contour C

34. Useful Calculus Facts in Level Set Formulation
If a curve is a level set of a function
then, the normal vector and the curvature can be computed from the embedding function
The surface (or line) integral of a quantity p(x) over the zero level surface (or contour) of
The volume (or area) integral of p(x) over

35. From Curve Evolution to Level Set Evolution
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From Curve Evolution to Level Set Evolution
Curve evolution
where F is the speed function, N is normal vector to the curve C
Embed the dynamic curve as the zero level set of a time dependent function , i.e.
Take derivative with respect to time
Level set evolution:

36. Special Cases Mean curvature flow Constant Speed Motion
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Special Cases
Mean curvature flow
Constant Speed Motion
Geodesic active contours

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Demo of GAC
Click to play the movie

38. Unstable Evolution and Need for Reinitialization
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Unstable Evolution and Need for Reinitialization
Zero level set of degraded level set function
Degraded level set function, 50 iterations, time step=0.1

39. Procedure of Standard Level Set Methods
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Procedure of Standard Level Set Methods
Reinitialization
Initialization
Evolve level set function
converge? N Y
stop

40. More Details Difference Scheme: finite difference, upwind scheme.
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More Details
Difference Scheme: finite difference, upwind scheme.
ENO interpolation
Velocity Extension
Narrow Band Implementation
More… Next time

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Questions

42. Thank you