1. Critical points and flow structure for density data Joint work with H. Edelsbrunner, J. Harer, and V. Pascucci Vijay Natarajan Duke University

2. Problem Define Morse-Smale complexes for 3-manifolds and give an algorithm to construct them for piecewise linear data Given smooth generic function f :  3  R

3. Motivation Density function f : R 3  R X-ray crystallography electron density Medical imaging (MRI) proton density Lattice dislocation atom energy Electron microscopy atom density

4. p Critical Points Have zero gradient minimum1-saddle2-saddlemaximum regular upper link (continent) lower link (ocean) Characterized by lower link

5. Ascending 3-cell of a minimum Ascending/Descending Manifolds Ascending manifold: Points with common origin. Descending manifold: Points with common destination.

6. Desc mfldAsc mfld Min0-cell3-cell 1-saddle1-cell2-cell 2-saddle2-cell1-cell Max3-cell0-cell 1-saddle2-saddle

7. Morse-Smale Complex Overlay of Asc and Desc manifolds

8. Morse-Smale Complex A cell is a connected component of points with common origin and destination Node Arc Quadrangle Crystal

9. Continuous to Piecewise Linear Input: tetrahedral mesh, density at vertices Critical points characterized by lower link Quasi Morse-Smale complexes –same combinatorial property –cells monotonic and non-crossing 1-saddle + 2-saddle

10. Morse-Smale Complex Construction 0. Sort Vertices 1.Downward sweep descending 1- and 2-manifolds (arcs and disks) 2.Upward sweep ascending 1- and 2-manifolds (arcs and disks) Asc arc Desc arc

11. High Level Operations Starting (at 1-saddles) Extending (at all vertices) Gluing ( at minima) Desc arc construction

12. i.Construct short cycle passing thru q around annulus ii.Add triangles from p to fill disk S TART D ISK

13. Simultaneous Construction At p : 1.1. Start  1 desc disks 1.2. Extend desc disks touching p 1.3. Start (  0 –1) desc arcs 1.4. Extend desc arcs touching p p  0 = 2 oceans  1 +1 = 2 continents

14. 0. Sort Vertices 1.Downward sweep descending 1- and 2-manifolds (arcs and disks) 2.Upward sweep ascending 1- and 2-manifolds (arcs and disks) Asc arc Desc arc Morse-Smale Complex Construction

15. Substructures Descending / Ascending arcs –filter using density threshold –display corresponding isosurface Descending / Ascending disks –overlay on isosurfaces Crystals

16. Future work Case studies –dislocation study –x-ray crystallography Hierarchy –cancel pairs of critical points –order by persistence

17. References H. Edelsbrunner, J. Harer, V. Natarajan and V. Pascucci. Morse-Smale complexes for piecewise linear 3-manifolds. ACM Symposium on Computational Geometry, 2003. P. T. Bremer, H. Edelsbrunner, B. Hamann and V. Pascucci. A Multi-resolution Data Structure for Two-dimensional Morse Functions. IEEE Visualization, 2003. H. Edelsbrunner, J. Harer and A. Zomorodian. Hierarchical Morse-Smale complexes for piecewise linear 2-manifolds. Discrete and Computational Geometry, 2003.

18. No time for another slide !!

19. Simulating Disjointness Normal interval –order disks passing through a triangle Normal disk –order disks and arcs passing through an edge

20. Ascending Disks