1. Deformable Contours
Dr. E. Ribeiro

2. Classical Methods An image of blood vessel Thresholding Edge detection
4/16/2017
Classical Methods
An image of blood vessel
Thresholding
Edge detection
Slide by: Chunming Li, Vanderbilt University

3. An Advanced Method: Active Contour Model
4/16/2017
An Advanced Method: Active Contour Model
Slide by: Chunming Li, Vanderbilt University

4. Edges vs. boundaries
Edges useful signal to indicate occluding boundaries, shape.
Here the raw edge output is not so bad…
…but quite often boundaries of interest are fragmented, and we have extra “clutter” edge points.
Images from D. Jacobs

5. Deformable contours a.k.a. active contours, snakes
Given: initial contour (model) near desired object
(Single frame)
[Snakes: Active contour models, Kass, Witkin, & Terzopoulos, ICCV1987]
Fig: Y. Boykov

6. Deformable contours a.k.a. active contours, snakes
Given: initial contour (model) near desired object
Goal: evolve the contour to fit exact object boundary
(Single frame)
[Snakes: Active contour models, Kass, Witkin, & Terzopoulos, ICCV1987]
Fig: Y. Boykov

8. Why do we want to fit deformable shapes?
Non-rigid, deformable objects can change their shape over time, e.g. lips, hands.
Figure from Kass et al. 1987

9. Why do we want to fit deformable shapes?
Some objects have similar basic form but some variety in the contour shape.

10. Deformable contours: intuition
Image from
Figure from Shapiro & Stockman

11. a.k.a. active contours, snakes
Deformable contours
a.k.a. active contours, snakes
How is the current contour adjusted to find the new contour at each iteration?
Define a cost function (“energy” function) that says how good a possible configuration is.
Seek next configuration that minimizes that cost function.
initial
intermediate
final

12. Snakes energy function
The total energy (cost) of the current snake is defined as:
Internal energy: encourage prior shape preferences: e.g., smoothness, elasticity, particular known shape.
External energy (“image” energy): encourage contour to fit on places where image structures exist, e.g., edges.
A good fit between the current deformable contour and the target shape in the image will yield a low value for this cost function.

13. Parametric curve representation
(continuous case)
Fig from Y. Boykov

14. External energy: intuition
Measure how well the curve matches the image data
“Attract” the curve toward different image features
Edges, lines, etc.

15. - (Magnitude of gradient)
External image energy
How do edges affect “snap” of rubber band?
Think of external energy from image as gravitational pull towards areas of high contrast
Magnitude of gradient
- (Magnitude of gradient)

16. External image energy Image I(x,y) Gradient images and
External energy at a point v(s) on the curve is
External energy for the whole curve:

17. Internal energy: intuition
A priori, we want to favor smooth shapes, contours with low curvature, contours similar to a known shape, etc. to balance what is actually observed (i.e., in the gradient image).

18. Internal energy Internal energy for whole curve:
For a continuous curve, a common internal energy term is the “bending energy”.
At some point v(s) on the curve, this is:
The more the curve bends  the larger this energy value is.
The weights α and β dictate how much influence each component has.
Elasticity,
Tension
Stiffness,
Curvature
Internal energy for whole curve:

19. Dealing with missing data
The smoothness constraint can deal with missing data:
[Figure from Kass et al. 1987]

20. Total energy (continuous form)
// bending energy
// total edge strength
under curve

21. Parametric curve representation (discrete form)
Represent the curve with a set of n points …

22. Discrete energy function: external term
If the curve is represented by n points
Discrete image gradients

23. Summary: elastic snake
A simple elastic snake is defined by
A set of n points,
An internal elastic energy term
An external edge based energy term
To use this to locate the outline of an object
Initialize in the vicinity of the object
Modify the points to minimize the total energy

24. Level Set Representation of Curves
4/16/2017
Level Set Representation of Curves
zero level
zero level
Slide by: Chunming Li, Vanderbilt University

25. Level Set Method (Osher and Sethian, 1988)
4/16/2017
Level Set Method (Osher and Sethian, 1988)
Curve evolution:
where F is the speed function, N is normal vector to the curve C
Level set formulation: N
Slide by: Chunming Li, Vanderbilt University

26. Geodesic active contour

27. Geodesic active contour

28. Geodesic active contour

29. Segmentation using statistical models (Rousson and Deriche, 2002)
Energy functional
Probability describing the pixel values inside region i

30. Segmentation using statistical models (Rousson and Deriche, 2002)
Energy functional
Probability describing the pixel values inside region i
The energy functional can be rewritten as:

31. Two-phase case

32. Two-phase case
Energy functional
Level set function

33. Heaviside step function
Two-phase case
Energy functional
Heaviside step function
Length term
Smooth approximation

34. Indicator functions (partitions)

35. Indicator functions (partitions)

36. Distance function (level set function)

37. Estimating the Parameters of the Gaussian densities
Two-phase case
Estimating the Parameters of the Gaussian densities

38. Two-phase case

39. Experiments

40. Gray level

41. Color and Texture

42. Slide by: Chunming Li, Vanderbilt University
4/16/2017
Results
Slide by: Chunming Li, Vanderbilt University

43. 3D Segmentation of Corpus Callosum
4/16/2017
3D Segmentation of Corpus Callosum
Slide by: Chunming Li, Vanderbilt University

44. Slide by: Chunming Li, Vanderbilt University
Result
Synthetic noisy image
Slide by: Chunming Li, Vanderbilt University

45. 2D Segmentation of Real Color Images
A real image of potatoes
Slide by: Chunming Li, Vanderbilt University

46. Slide by: Chunming Li, Vanderbilt University
2D Vessel Segmentation
Slide by: Chunming Li, Vanderbilt University

47. Segmentation of White Matter in MR images
Slide by: Chunming Li, Vanderbilt University

48. Effect of the Level Set Regularization
Without level set regularization
Final zero level contour
Final level set function
Slide by: Chunming Li, Vanderbilt University

49. 3D Vessel Segmentation MRA Vessel Segmentation
Slide by: Chunming Li, Vanderbilt University

50. 3-D Ultrasound

51. 3-D Ultrasound