Data Analysis and Visualization Using the Morse-Smale complex

1. Data Analysis and Visualization Using the Morse-Smale complex Attila Gyulassy Institute for Data Analysis and Visualization Computer Science Department University of California, Davis Center for Applied Scientific Computing LLNS

2. Seminar Overview • Introduce basic concepts from topology • Intuitive definition of the Morse-Smale complex • What is a feature? • Examples from various application areas • Algorithm

3. Topology Background - Critical Points Let ƒbe a scalar valued function whose critical points are not degenerate. We call ƒ a Morse function, and in the neighborhood of a critical point p, the function can be represented as ƒ(p) = 0 ƒ(x, y, z) = ƒ(p) ±x2±y2±z2 ∆ Regular Minimum 1-Saddle 2-Saddle Maximum Index 0 Index 1 Index 2 Index 3

4. Topology Background - Integral Lines An integral line is a maximal path that agrees with the gradient of at every point

5. Topology Background - Manifolds Descending Manifolds D(p) = {p} { x | xєl, dest(l) = p} Ascending Manifolds A(p) = {p} { x | xєl, orig(l) = p} ∩ ∩ 3-Manifold 1-Manifold 3-Manifold 1-Manifold 2-Manifold 0-Manifold 2-Manifold 0-Manifold Maximum 2-Saddle 1-Saddle Minimum

6. What is the Morse-Smale Complex? • The intersection of all descending and ascending manifolds D(p) ∩ A(q), for all pairs p,q of f • Any cell in the complex has the property that all integral lines in that cell share an origin and a destination The Morse-Smale complex is a segmentation of the domain that clusters integral lines that share a common origin and destination.

7. Morse-Smale Complex - 1D Example

8. Morse-Smale Complex - 1D Example

9. Morse-Smale Complex - 1D Example

10. Morse-Smale Complex - 2D Example

11. Morse-Smale Complex - 2D Example

12. Morse-Smale Complex - 2D Example

13. Morse-Smale Complex - 2D Example

14. Morse-Smale Complex - 3D Example Morse-Smale Complex - 3D Example Cells of dimension i connect critical points with index that differ by i. Crystal Quad Arc Node

15. Topology based simplification Topology based simplification

16. What is a feature?

17. What is a feature?

18. What is a feature?

19. What is a feature?

20. What is a feature?

21. What is a feature?

22. What is a feature?

23. What is a feature?

24. What is a feature?

25. What is a feature?

26. What is a feature?

27. What is a feature?

28. What is a feature?

29. What is a feature? Critical Points

30. What is a feature? Arcs

31. What is a feature? Higher Degree Cells

32. Examples from applications Persistent extrema are the features Finding atom locations in molecular simulations

33. Previous Work Examples from applications Motivation Morse-Smale comple in 2D Persistent extrema are the features (Multi-Scale Analysis) Tracking the formation of bubbles in turbulent mixing fluids (Laney et al.)

34. Examples from applications Persistent extrema are the features • Testing the “smoothness” of a generated function • How does the critical point count change as a function of persistence? • Length of the persistent arcs? • Size of the persistent cells? Critical point count Persistence

35. Examples from applications Persistent arcs are the features Terrain representation (Bremer et al.)

36. Examples from applications Persistent arcs are the features Surface Quadrangulation (Dong et al.)

37. Examples from applications Persistent arcs are the features Analysis of porous media

38. Examples from applications Persistent arcs are the features Time comparison of the reconstructions

39. Examples from applications Persistent arcs are the features Analysis of the structure of galaxies

40. Examples from applications Persistent cells are the features Analysis of a combustion simulation

41. A Simple Algorithm For Constructing the Morse-Smale Complex • Construct the known complex for a similar function called the augmented function • Simplify the artificial complex

42. A Simple Algorithm For Constructing the Morse-Smale Complex Contributions Constructing the Morse-Smale complex of an Augmented Morse Function The augmented Morse function has a very regular structure. Every vertex of S is critical, with index = dimension of its cell in K. Arcs of the complex are the edges of S.

43. A Simple Algorithm For Constructing the Morse-Smale Complex Contributions Topology based simplification

44. A Simple Algorithm For Constructing the Morse-Smale Complex Contributions Topology based simplification • Remove extra critical points • Correct Morse-Smale complex within small error bound

45. A Simple Algorithm For Constructing the Morse-Smale Complex Original data points

46. A Simple Algorithm For Constructing the Morse-Smale Complex

47. Questions? aggyulassy@ucdavis.edu