1. Martin Burger Institut für Numerische und Angewandte Mathematik CeNoS Level set methods for imaging and application to MRI segmentation

2. Martin Burger MRI Segmentation 2 8.5.2008 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. Martin Burger MRI Segmentation 3 8.5.2008 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. Martin Burger MRI Segmentation 4 8.5.2008 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. Martin Burger MRI Segmentation 5 8.5.2008 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. Martin Burger MRI Segmentation 6 8.5.2008 Active Contours - Snakes Image-driven force related to gradient of the image (local gray-value difference) Regularization force is (mean) curvature

7. Martin Burger MRI Segmentation 7 8.5.2008 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. Martin Burger MRI Segmentation 8 8.5.2008 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. Martin Burger MRI Segmentation 9 8.5.2008 Curve / Surface Representation Level Set Methods yield boundary representation with appropriate curvatures and subpixel resolution

10. Martin Burger MRI Segmentation 10 8.5.2008 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. Martin Burger MRI Segmentation 11 8.5.2008 Level Set Methods Change of front translated to change of function

12. Martin Burger MRI Segmentation 12 8.5.2008 Level Set Methods Implicit representation of dynamic shapes with time-dependent level set function

13. Martin Burger MRI Segmentation 13 8.5.2008 Level Set Methods Evolution of the shape corresponds to evolution of the level set function (and vice versa) Movie by J.Sethian

14. Martin Burger MRI Segmentation 14 8.5.2008 Level Set Methods Topology change is automatic Movie by J.Sethian

15. Martin Burger MRI Segmentation 15 8.5.2008 Geometric Motion Start for simplicity with the evolution of a curve Evolution in a velocity field, each point evolves via ODE

16. Martin Burger MRI Segmentation 16 8.5.2008 Geometric Motion Use any parametric representation Due to definition of the level set function Consequently

17. Martin Burger MRI Segmentation 17 8.5.2008 Geometric Motion By the chain rule Insert ODE for moving points:

18. Martin Burger MRI Segmentation 18 8.5.2008 Geometric Motion For level set function being a solution of each level set of  is moving with velocity V

19. Martin Burger MRI Segmentation 19 8.5.2008 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. Martin Burger MRI Segmentation 20 8.5.2008 Geometric Motion Normal can be computed from level set function: By the chain rule

21. Martin Burger MRI Segmentation 21 8.5.2008 Geometric Motion Note that is a tangential direction Hence, is a normal direction, unit normal is given by

22. Martin Burger MRI Segmentation 22 8.5.2008 Geometric Motion Evolution becomes nonlinear Hamilton-Jacobi equation: „Level set equation“

23. Martin Burger MRI Segmentation 23 8.5.2008 Geometric Motion Evolution could be anisotropic, i.e. normal velocity depends on the orientation with one-homogeneous extension H, yields Hamilton- Jacobi equation

24. Martin Burger MRI Segmentation 24 8.5.2008 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. Martin Burger MRI Segmentation 25 8.5.2008 Examples Eikonal equation Positive velocity field yield monotone advancement of fronts Arrival time Solves

26. Martin Burger MRI Segmentation 26 8.5.2008 Example: Eikonal Equation

27. Martin Burger MRI Segmentation 27 8.5.2008 Examples Mean curvature flow Classical example of higher-order geometric motion Normal velocity equal to curvature of curve (or mean curvature of surface)

28. Martin Burger MRI Segmentation 28 8.5.2008 Mean Curvature Flow

29. Martin Burger MRI Segmentation 29 8.5.2008 Optimal Geometries Classical problem for optimal geometry: Plateau Problem (Minimal Surface Problem) Minimize area of surface between fixed boundary curves.

30. Martin Burger MRI Segmentation 30 8.5.2008 Optimal Geometries Minimal surface (L.T.Cheng, PhD 2002)

31. Martin Burger MRI Segmentation 31 8.5.2008 Optimal Geometries Wulff-Shapes: Pb[111] in Cu[111] Surnev et al, J.Vacuum Sci. Tech. A, 1998

32. Martin Burger MRI Segmentation 32 8.5.2008 Mumford-Shah Free discontinuity problems: find the set of discontinuity from a noisy observation of a function. Mumford-Shah functional

33. Martin Burger MRI Segmentation 33 8.5.2008 Mumford-Shah Image decomposition

34. Martin Burger MRI Segmentation 34 8.5.2008 Mumford-Shah Limitations

35. Martin Burger MRI Segmentation 35 8.5.2008 Improved Model Decomposition in 3 parts: smooth, oscillating, edges

36. Martin Burger MRI Segmentation 36 8.5.2008 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. Martin Burger MRI Segmentation 37 8.5.2008 Level Set Formulation Level set function  and Heaviside function H (= 0 negative, = 1 positive)

38. Martin Burger MRI Segmentation 38 8.5.2008 Reduced Problem: fixed mean value

39. Martin Burger MRI Segmentation 39 8.5.2008 Image Segmentation Noisy Image

40. Martin Burger MRI Segmentation 40 8.5.2008 Image Segmentation Noise level 10%,  =10 3

41. Martin Burger MRI Segmentation 41 8.5.2008 Regularization For skull segmentation (smooth) regularization based on length minimization is perfect For brain structure (sulci) similar issues as for active contours

42. Martin Burger MRI Segmentation 42 8.5.2008 MR Results

43. Martin Burger MRI Segmentation 43 8.5.2008 MR Results

44. Martin Burger MRI Segmentation 44 8.5.2008 Skull Segmentation from MR-PD

45. Martin Burger MRI Segmentation 45 8.5.2008 Head Segmentation

46. Martin Burger MRI Segmentation 46 8.5.2008 Bias of one functional often too strong Better: use a family of functionals parametrized by Example: adaptive anisotropy Adaptive Bias / Parametrization J ( u;® ) ® 2 A

47. Martin Burger MRI Segmentation 47 8.5.2008 In aerial images the typical anisotropy is rectangular, houses have 90° angles But not all of them have the same orientation Adaptive Anisotropy

48. Martin Burger MRI Segmentation 48 8.5.2008 Bias for edges with 90° angles from functional of the form R  is rotation matrix for angle  to capture the orientation Since orientation is not constant over the image,  has to vary and to be found adaptively by minimization Adaptive Anisotropy J ( u;® ) = Z (j v 1 j + j v 2 j) d x ; v = R ® r u

49. Martin Burger MRI Segmentation 49 8.5.2008 To avoid microstructure, variation of  has to be regularized, too Possible regularization functional Adaptive Anisotropy

50. Martin Burger MRI Segmentation 50 8.5.2008 Improves angles, still loses contrast Adaptive Anisotropy

51. Martin Burger MRI Segmentation 51 8.5.2008 Contrast correction by iterative refinement Angle parameter provides classification of orientations in the image Adaptive Anisotropy

52. Martin Burger MRI Segmentation 52 8.5.2008 Cartoon reconstruction and orientational classification of aerial images Berkels, mb, Droske, Nemitz, Rumpf 06 Adaptive Anisotropy

53. Martin Burger MRI Segmentation 53 8.5.2008 Analogous problem in segmentation of MRI brain images for EEG/MEG Adapt anisotropy (locally like sharp ellipse) to find sulci accurately and provide classification of normals (for dipole fitting, source reconstruction) Adaptive Anisotropy

54. Martin Burger MRI Segmentation 54 8.5.2008 Fixed Anisotropy, 45° orientation

55. Martin Burger MRI Segmentation 55 8.5.2008 Adaptive Anisotropy

56. Martin Burger MRI Segmentation 56 8.5.2008 Adaptive Anisotropy

57. Martin Burger MRI Segmentation 57 8.5.2008 Adaptive Anisotropy, 3D