Generating Realistic Terrains with Higher-Order Delaunay Triangulations

1. Generating Realistic Terrains with Higher-Order Delaunay Triangulations Thierry de Kok Marc van Kreveld Maarten Löffler Center for Geometry, Imaging and Virtual Environments Utrecht University

2. Overview • Introduction • Results on local minima • NP-hard • Two heuristics • Results on valley components • A new heuristic

3. Motivation • Terrain modeling for geomorphological applications • TIN as terrain representation • Realism necessary • Choice of triangulation is important

4. Few local minima • Connected valley components • Wrong triangulation can introduce undesirable artifacts

5. Triangulations

6. Higher-Order Delaunay Triangulations • At most k points in circle • Order 0 DT is normal DT • If k > 0, order k DT is not unique • Introduced by Gudmundsson et al. (2002)

7. Using HODT to Solve the Problem • Well shaped triangles, plus room to optimize other criteria • We want to minimize local minima • For k > 1, optimal order k DT is no longer easy to compute • Heuristics are needed

8. Local Minima Results • Computing optimal HODT for minimizing local minima is NP-hard • Two heuristics • Experimental results comparing the heuristics and analysing HODT

9. NP-hardness • Minimizing local minima for degenerate pointsets is NP-hard • Minimizing local minima for non-degenerate pointsets is NP-hard too, when using order k DT • Reduction from maximum non-intersecting set of line segments

12. Flip Heuristic • Start with Delaunay triangulation • Flip edges that might potentially remove a local minimum • Preserve order k property • O (n.k2 + n.k.logn)

13. New edge must be “lower” than old edge • New triangles must be order k

14. Hull Heuristic • Compute a list of all useful order k edges that remove a local minimum • Add as many as possible • Make sure they do not interfere • O (n.k2 + n.k.logn)

15. When adding an edge, compute the hull • Retriangulate the hull • Do not add any other edges intersecting the hull

16. Experiments on real Terrains

17. Quinn Peak • Elevation data grid • 382 x 468 • 1 data point = 30 meter

18. Random sample • 1800 vertices • Delaunay triangulation • 53 local minima

19. Hull heuristic applied • Order 4 Delaunay triangulation • 25 local minima

20. hull heuristic flip heuristic

21. Drainage on TIN • Complex to model due to material properties • Water follows path of steepest descent • Over edge • Over triangle

22. Definitions • Three kinds of edges:

23. Valley component: maximal set of valley edges s.t. flow from these edges reaches lowest vertex of the component

24. Drainage quality of terrain • Quality defined by: • Number of local minima • Number of valley components not ending a local minimum • Improve quality by: • Deleting single edge networks • Extending networks downwards to local minima

25. Isolated valley edge • Try to remove it • No new valley edges should be created • New triangle order k • Otherwise try to extend it

26. Extending component • Extend: • Single edge network that cannot be removed (at this order) • Multiple edge networks that do end in a local minimum • Multiple edge networks that do not end in a local minimum

27. Extend if: • bqrp is convex • br is valley edge • brp and bqr are order k • br is steepest descent direction from b • r < b, r < q, r < p • No interrupted valley components in p or q

28. Results valley heuristic • 25-40% decrease in number of valley components • +/- 30 % decrease in number of local minima (far less than flip and hull heuristic)

29. Results on a terrain

30. Results compared to flip and hull

31. Delaunay triangulation

32. Flip-8

33. Hull-8

34. Valley-8

35. Flip-8 + valley heuristic

36. Hull-8 + valley heuristic

37. Conclusions Local Minima • Low orders already give good results • Hull is often better than flip • Hull performed almost optimal

38. Conclusions Drainage • Low order already give good results • Significant reduction in number of valley components • Drainage quality is improved the most when hullheuristic is combined with valley heuristic

39. Future Work • NP-hardness for small k • Other properties of terrains • Local maxima • More hydrological features (watersheds) • Different local operators for valleyheuristic