Level set methods for imaging and application to MRI segmentation

1. Level set methods for imaging and application to MRI segmentation

2. MRI Segmentation Acknowledgements • Based on results by Denis Neiter (Ecole Polytechnique) during internship 2007, partly using results by Simon Huffeteau (Ecole Polytechnique), internship 2006 • Using Carsten Wolters‘ MR Data

3. MRI Segmentation Problem Setting • Given MR Image(s), find (in an automated way): • the borders between different head compartments (segmentation) • an appropriate map of the normal directions, in particular of the brain surface (classification)- a representation useful for further finite element modelling

4. MRI Segmentation Mathematical Issues • Segmentation needs • to discriminate noise and textures (small scale structures) • to incorporate prior knowledge • - to be flexible with respect to complicated shapes (or even topology) • First two issues treated via regularization, third via level set methods

5. MRI Segmentation Object-Based Segmentation • Classical object based segmentation computes curves (2D) or surfaces (3D) marking the object boundary (contour) • Traditional approach: start with curve and let it evolve towards the contour by some criteria • Velocity of evolving curve determined by two counteracting parts: image-driven part and regularization

6. MRI Segmentation Active Contours - Snakes • Image-driven force related to gradient of the image (local gray-value difference) • Regularization force is (mean) curvature

7. MRI Segmentation Active Contours - Snakes • Popular, but various shortcomings: • needs preprocessing of the image (noise removal, intensity map so that edges are in valleys) • local minima • issues with narrow structures: big trouble in brain images

8. MRI Segmentation Statistical Models • K-Means / C-Means: • Based on optimization, find a 0-1 function (0 in pixels outside, 1 inside) • Optimization goal consists of same parts: image-driven and regularization • No useful boundary representation

9. MRI Segmentation Curve / Surface Representation • Level Set Methods yield boundary representation with appropriate curvatures and subpixel resolution

10. MRI Segmentation Level Set Methods • Osher & Sethian, JCP 1987, Sethian, Cambridge Univ. Press 1999, Osher & Fedkiw, Springer, 2002 • Basic idea: implicit shape representation • with continuous level-set function

11. MRI Segmentation Level Set Methods • Change of front translated to change of function

12. MRI Segmentation Level Set Methods • Implicit representation of dynamic shapes • with time-dependent level set function

13. MRI Segmentation Level Set Methods • Evolution of the shape corresponds to evolution of the level set function (and vice versa) • Movie by • J.Sethian

14. MRI Segmentation Level Set Methods • Topology change is automatic • Movie by • J.Sethian

15. MRI Segmentation Geometric Motion • Start for simplicity with the evolution of a curve • Evolution in a velocity field , each point evolves via ODE

16. MRI Segmentation Geometric Motion • Use any parametric representation • Due to definition of the level set function • Consequently

17. MRI Segmentation Geometric Motion • By the chain rule • Insert ODE for moving points:

18. MRI Segmentation Geometric Motion • For level set function being a solution of • each level set of f is moving with velocity V

19. MRI Segmentation Geometric Motion • In most cases, the full velocity field V is unknown, only normal velocity component v known • Tangential component of the velocity field is not important anyway, it does not change the motion (only change of parametrization)

20. MRI Segmentation Geometric Motion • Normal can be computed from level set function: • By the chain rule

21. MRI Segmentation Geometric Motion • Note that is a tangential direction • Hence, is a normal direction, unit normal is given by

22. MRI Segmentation Geometric Motion • Evolution becomes nonlinear Hamilton-Jacobi equation: • „Level set equation“

23. MRI Segmentation Geometric Motion • Evolution could be anisotropic, i.e. normal velocity depends on the orientation • with one-homogeneous extension H , yields Hamilton-Jacobi equation

24. MRI Segmentation Geometric Motion • Evolution could be of higher order, e.g. normal velocity depends on the mean curvature • Level set equation becomes fully nonlinear second-order parabolic PDE

25. MRI Segmentation Examples • Eikonal equation • Positive velocity field yield monotone advancement of fronts • Arrival time • Solves

26. MRI Segmentation Example: Eikonal Equation

27. MRI Segmentation Examples • Mean curvature flow • Classical example of higher-order geometric motion • Normal velocity equal to curvature of curve (or mean curvature of surface)

28. MRI Segmentation Mean Curvature Flow

29. MRI Segmentation Optimal Geometries • Classical problem for optimal geometry: • Plateau Problem (Minimal Surface Problem) • Minimize area of surface between fixed boundary curves.

30. MRI Segmentation Optimal Geometries • Minimal surface (L.T.Cheng, PhD 2002)

31. MRI Segmentation Optimal Geometries • Wulff-Shapes: Pb[111] in Cu[111] • Surnev et al, J.Vacuum Sci. Tech. A, 1998

32. MRI Segmentation Mumford-Shah • Free discontinuity problems: • find the set of discontinuity from a noisy observation of a function. • Mumford-Shah functional

33. MRI Segmentation Mumford-Shah • Image decomposition

34. MRI Segmentation Mumford-Shah • Limitations

35. MRI Segmentation Improved Model • Decomposition in • 3 parts: smooth, oscillating, edges

36. MRI Segmentation Object-based Mumford-Shah • Chan-Vese: • Approximate smooth component by its mean value inside and outside object • Curve / Surface can be evolved via simple criterion, in each time step mean-value inside and outside need to be computed • Via convex relaxation techniques convergence to global minimum can be ensured

37. MRI Segmentation Level Set Formulation • Level set function y and • Heaviside function H (= 0 negative, = 1 positive)

38. MRI Segmentation Reduced Problem: fixed mean value

39. MRI Segmentation Image Segmentation • Noisy Image

40. MRI Segmentation Image Segmentation • Noise level 10%, l=103

41. MRI Segmentation Regularization • For skull segmentation (smooth) regularization based on length minimization is perfect • For brain structure (sulci) similar issues as for active contours

42. MRI Segmentation MR Results

43. MRI Segmentation MR Results

44. MRI Segmentation Skull Segmentation from MR-PD

45. MRI Segmentation Head Segmentation

46. ( ) J A 2 ® u ; ® MRI Segmentation Adaptive Bias / Parametrization Bias of one functional often too strong Better: use a family of functionals parametrized by Example: adaptive anisotropy

47. MRI Segmentation Adaptive Anisotropy In aerial images the typical anisotropy is rectangular, houses have 90° angles But not all of them have the same orientation

48. Z ( ) ( j j j j ) d J R r + u ; ® v v x v u = = 1 2 ® ; MRI Segmentation Adaptive Anisotropy Bias for edges with 90° angles from functional of the form Ra is rotation matrix for angle a to capture the orientation Since orientation is not constant over the image, a has to vary and to be found adaptively by minimization

49. MRI Segmentation Adaptive Anisotropy To avoid microstructure, variation of a has to be regularized, too Possible regularization functional

50. MRI Segmentation Adaptive Anisotropy Improves angles, still loses contrast