1. Electron denisty, isosurfaces and contour trees Jack Snoeyink GCMB’07

2. Outline n Electron density (3D scalar fields) n The topology of isosurfaces/level sets n The Contour tree –Join tree + Split tree n Applications –Flexible contouring –Contour simplification

3. Density data f:Domain → R My route into geometric computation in molec. bio. n GIS: Geographic Information Systems –Terrain models –Watersheds n Molecular Scene Analysis –J. Glasgow & S. Fortier (Queens)

4. From crystal to structure Data from X-ray diffraction  Electron density maps  Threaded backbone ...

5. Density to surface We start with the easiest of these steps... Assume we have density, e.g., from the Uppsala Electron Density ServerUppsala Electron Density Server

6. Level sets in density data Many technologies give density samples: f(p) on a 3-D grid. Define Level Sets, a.k.a. isosurfaces, L(v)={p | f(p) = v}. n Requires interpolation.

7. Level sets in density data Many technologies give density samples: f(p) on a 3-D grid. Define Level Sets, a.k.a. isosurfaces, L(v)={p | f(p) = v}. n Define Contour as a connected component of the level set.

8. The Marching Cubes algorithm extracts an isosurface n Test values at cube corners to identify cube edges that intersect the isosurface. n 2 8 cases reduce to 14-16 cases by rotations. n Make triangles to represent portion of level set in cube. n Lorensen & Cline 87, Marching cubes n Nielsen & Hamma 91, fix ambiguities + + + ++ – -

9. Cases Ambiguity in cases w/ separate loops

10. Contour following + + + ++ – - n Can follow contour if we know a start point. n How to find start point? n Shinagawa et al. 91, extreme graph n Cignoni et al. 95, interval tree n Carr et al. 03, contour tree

11. Evolution of level sets Watch contours of L(v) as v increases

12. Evolution of level sets 7 8 9 10 5 6 4 3 1 2 Contour tree

13. Evolution of level sets 7 8 9 10 5 6 4 3 1 2 Contour tree

14. Evolution of level sets 7 8 9 10 5 6 4 3 1 2 Contour tree

15. Evolution of level sets 7 8 9 10 5 6 4 3 1 2 Contour tree

16. Evolution of level sets 7 8 9 10 5 6 4 3 1 2 Contour tree

17. Evolution of level sets Contours appear, merge, split, & vanish 7 8 9 10 5 6 4 3 1 2 Contour tree

18. Morse theory & Contour trees n Karron et al 95, digital Morse theory n Bajaj et al 97–99, seed sets n Van Kreveld et al 97, 2-D contour tree n Tarasov & Vyalyi 98, 3-D contour tree

19. How to compute contour trees without sweeping all level sets n Build join and split trees n Merge to form contour tree n Works in all dimensions...

20. The join tree Represents set {p | f(p) >= x} n Can be computed w/out the surface of the level set. 7 8 9 10 5 6 4 1

21. Join tree construction n Sort data by  value Maintain union/find for comp. of {p | f(p) >= x} 7 8 9 10 5 6 4

22. Union/Find n Data structure for integers 1..n supporting: –Initialize() each integer starts in its own group for i = 1..n, g(i) = i; –Union(i,j) union groups of i and j g(Find(i)) = Find(j); –Find(i) return name of group containing i group = i; while group != g(group), group = g(group); // find group while i != group, {nx = g(i), g(i) = group, i = nx} // compress path n Does n union/finds in O(n a(n)) steps [T72] 1 2 3 4... 42... n

23. Join tree construction n Sort data by  value Maintain union/find for comp. of {p | f(p) >= x} O(n+ t·α ( t ) ) after sort 7 8 9 10 5 6 4 1

24. The split tree 10 3 1 2 Represents the set {p | f(p) <= x} n Reverse join tree computation

25. Merge join and split trees 10 3 1 2 7 8 9 5 6 4 1 7 8 9 5 6 4 3 1 2

26. 1. Add nodes from other tree n Nodes have correct up degree in join & down degree in the split tree. 7 8 9 10 5 6 4 1 7 8 9 5 6 4 3 1 2 3 2

27. 2. Identify leaves n A leaf in one tree, with up/down degree < 1 in other tree 7 8 9 10 5 6 4 1 7 8 9 5 6 4 3 1 2 3 2

28. 3. Add to contour tree n A leaf in one tree, with up/down degree < 1 in other tree 7 8 9 10 5 6 4 1 7 8 9 5 6 4 3 1 3 3 2

29. 3. Add to contour tree 7 8 9 10 5 6 4 1 7 8 9 5 6 4 3 1 3 3 2

30. 3. Add to contour tree 7 8 9 10 5 6 4 7 8 9 5 6 4 3 1 3 3 2

31. 3. Add to contour tree n Note that 4 is not a leaf 7 8 9 10 5 6 4 7 8 9 5 6 4 1 3 2 4

32. 3. Add to contour tree n The merge time is proportional to the tree size; much smaller than the data 8 9 10 5 6 4 8 9 5 6 4 1 3 2 4 7 5

33. 4. Ends with Contour tree 7 8 9 10 5 6 4 3 1 2

34. Advantages n Doesn’t need the surface structure to build the contour tree. n The merge becomes simple, and works on a subset of the points n The tree structure can advise where to choose the threshold value n Can cut diff. branches at diff. values.

35. Flexible contouring n Threshold diff. areas at diff. values

36. Flexible contouring n Threshold diff. areas at diff. Values n Guided by the Contour Tree 7 8 9 10 5 6 4 3 1 2