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 ƒ be 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 Saddle Saddle Maximum
Index Index Index 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 | x є l, dest(l) = p}
Ascending Manifolds
A(p) = {p} { x | x є l, 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
Cells of dimension i connect critical points with index that differ by i.
Crystal
Quad
Arc
Node

15. Topology based simplification
A lemma from morse theory states that we can remove critical points in pairs whose indices differ by one, and are connected by the complex.
We use the notion of persistence to order critical point pairs, where persistence is the absolute difference in function value of the pair that is removed.
Generally we consider low persistence pairs to be small features in the data. Going back to the construction, we insterted all the extra nodes so they have low persistence, and hence are removed first.
Next: Simplification algorithm
We formalize our simplification algorithm as

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
including reeb graphs, contour trees, and other hierarchical representations.
The advantage of using a Morse-Smale complex is that it is a complete characterization of the gradient behavior, and therefore is a more complete representation.
We extend work from 2D to idendentify and simplify the Morse-Smale complex for volumetric domains. We use the basic definitions for three-dimensional Morse Smale complexes, and the notion of persistence based simplification in our approach.
Next: New approach

33. Motivation Previous Work Examples from applications
Persistent extrema are the features
Morse-Smale comple in 2D
(Multi-Scale Analysis)
There are loads of scalar functions used in scientific computing, including physical simulations, mri and CT scans, and x-ray crystallography
and we want to understand the structure of these functions. We use a global analysis technique to identify this structure, in particular, we find the three dimensional morse-smale complex of the function.
Next: motivation pic
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.)
including reeb graphs, contour trees, and other hierarchical representations.
The advantage of using a Morse-Smale complex is that it is a complete characterization of the gradient behavior, and therefore is a more complete representation.
We extend work from 2D to idendentify and simplify the Morse-Smale complex for volumetric domains. We use the basic definitions for three-dimensional Morse Smale complexes, and the notion of persistence based simplification in our approach.
Next: New approach

36. Examples from applications
Persistent arcs are the features
There are loads of scalar functions used in scientific computing, including physical simulations, mri and CT scans, and x-ray crystallography
and we want to understand the structure of these functions. We use a global analysis technique to identify this structure, in particular, we find the three dimensional morse-smale complex of the function.
Next: motivation pic
Surface Quadrangulation (Dong et al.)

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

38. Time comparison of the reconstructions
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
including reeb graphs, contour trees, and other hierarchical representations.
The advantage of using a Morse-Smale complex is that it is a complete characterization of the gradient behavior, and therefore is a more complete representation.
We extend work from 2D to idendentify and simplify the Morse-Smale complex for volumetric domains. We use the basic definitions for three-dimensional Morse Smale complexes, and the notion of persistence based simplification in our approach.
Next: New approach

40. Examples from applications
Persistent cells are the features
including reeb graphs, contour trees, and other hierarchical representations.
The advantage of using a Morse-Smale complex is that it is a complete characterization of the gradient behavior, and therefore is a more complete representation.
We extend work from 2D to idendentify and simplify the Morse-Smale complex for volumetric domains. We use the basic definitions for three-dimensional Morse Smale complexes, and the notion of persistence based simplification in our approach.
Next: New approach
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
including reeb graphs, contour trees, and other hierarchical representations.
The advantage of using a Morse-Smale complex is that it is a complete characterization of the gradient behavior, and therefore is a more complete representation.
We extend work from 2D to idendentify and simplify the Morse-Smale complex for volumetric domains. We use the basic definitions for three-dimensional Morse Smale complexes, and the notion of persistence based simplification in our approach.
Next: New approach

42. A Simple Algorithm For Constructing the Morse-Smale Complex
Contributions
A Simple Algorithm For Constructing the Morse-Smale Complex
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.
Therefore we present a simple construction for a similar function. Here we see a simple 1D PL function defined at the yellow data points. To compute the complex directly, we would have to look at local neighborhoods of each points, and trace ascending and descending paths. In 3D this is much worse, since we need to find maximally ascending and descending surfaces.
Instead, we modify the function so that the entire complex for that function is known. Here we insert bumps into the function to make every point critical, therefore we can easily give the complex for the function.
next: in 3D,....

43. Topology based simplification
Contributions
A Simple Algorithm For Constructing the Morse-Smale Complex
Topology based simplification
We go back to the 1-D example to see how this is done. The complex tells us nothing about the critical point behavior of the original function,
next: simplified

44. Topology based simplification
Contributions
A Simple Algorithm For Constructing the Morse-Smale Complex
Topology based simplification
Until we remove the extra critical points.
If we simplify in a controlled manner, we can remove exactly those critical points that do not correspond to critical points in the original function.
So the question remains what ordering do we use in our simplification to remove the right critical points?
Next: the index lemma
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?

48. Structural Analysis of a Simulated Porous Material
25% dense copper foam, 70nm  70nm  80nm
Filament structure? Length? Number of cycles? How many cuts needed?
Further experiments on the structure...

49. Time comparison of the reconstructions
We Obtain a Consistent Reconstruction of the Filament Structures in the Material
Time comparison of the reconstructions

50. Efficient Computation of Morse-Smale Complexes for 3D Scalar Fields
Attila Gyulassy, Vijay Natarajan, Valerio Pascucci, Bernd Hamann
Center for Applied Scientific Computing
Lawrence Livermore National Laboratory
Department of Computer Science and Automation,
Indian Institute of Science
Bangalore, India
Institute for Data Analysis and Visualization
Computer Science Department
University of California, Davis

51. Contributions Filtering Filtering the Morse-Smale complex
- Certain conditions can be set to determine whether or not it is
possible to cancel an arc:
Therefore we present a simple construction for a similar function. Here we see a simple 1D PL function defined at the yellow data points. To compute the complex directly, we would have to look at local neighborhoods of each points, and trace ascending and descending paths. In 3D this is much worse, since we need to find maximally ascending and descending surfaces.
Instead, we modify the function so that the entire complex for that function is known. Here we insert bumps into the function to make every point critical, therefore we can easily give the complex for the function.
next: in 3D,....
- These can be conditions for inclusion/exclusion
- still maintain a “valid” complex in terms of gradient behavior
- restrict certain cancellations