Deformable Contours

1. Deformable Contours Dr. E. Ribeiro

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

3. 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 http://www.healthline.com/blogs/exercise_fitness/uploaded_images/HandBand2-795868.JPG Figure from Shapiro & Stockman

11. initial intermediate final 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.

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. 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). http://www3.imperial.ac.uk/pls/portallive/docs/1/52679.JPG

18. Internal energy 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 byn 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. zero level zero level Level Set Representation of Curves Slide by: Chunming Li, Vanderbilt University

25. Level set formulation: N Level Set Method (Osher and Sethian, 1988) • Curve evolution: • where F is the speed function, N is normal vector to the curve C 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. 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. Two-phase case Estimating the Parameters of the Gaussian densities

38. Two-phase case

39. Experiments

40. Gray level

41. Color and Texture

42. Results Slide by: Chunming Li, Vanderbilt University

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

44. 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. 2D Vessel Segmentation Slide by: Chunming Li, Vanderbilt University

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

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

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

50. 3-D Ultrasound http://mrcas.mpe.ntu.edu.sg/research/imgpro/3d_level_set.htm