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 • Triangulation for terrains • Realistic terrains • Higher order Delaunay triangulations • Minimizing local minima • NP-hardness • Two heuristics: algorithms and experiments • Other realistic aspects

3. Polyhedral terrains, or TINs • Points with (x,y) and elevation as input • TIN as terrain representation • Choice of triangulation is important 25 29 25 29 24 24 21 21 19 19 78 78 73 73 15 15 10 10 12 12

4. Realistic terrains • Due to erosion, realistic terrains • have few local minima • have valley lines that continue local minimum, interrupted valley line after an edge flip

5. Terrain modeling in GIS • Terrain modeling is extensively studied in geomorphology and GIS • Need to avoid artifacts like local minima • Need correct “shape” for run-off models, hydrological models, avalanche models, ... 17 52 6 12 local minimum in a TIN 21 24

6. Delaunay triangulation • Maximizes minimum angle • Empty circle property

7. Delaunay triangulation • Does not take elevation into account • May give local minima • May give interrupted valleys

8. Triangulate to minimize local minima?

9. Triangulate to minimize local minima? Connect everything to global minimum bad triangle shape & interpolation

10. Higher order Delaunay triangulations • Compromise between good shape & interpolation, and flexibility to satisfy other constraints • k -th order: allow k points in circle 1st order 4th order 0th order

11. Higher order Delaunay triangulations • Introduced by Gudmundsson, Hammar and van Kreveld (ESA 2000) • Minimize local minima for 1st order:O(n log n) time • Minimize local minima for kth order:O(k2)-approximation algorithm inO(nk3 + nk log n) time (hull heuristic)

12. This paper, results • NP-hardness of minimizing local minima • NP-hardness for kth order, k = (n) • New flip heuristic: O(nk2 + nk log n) time • Faster hull heuristic: O(nk2 + nk log n) time • Implementation and experiments on real terrains • Heuristic to avoid interrupted valleys: valley heuristic

13. 11 11 18 18 23 23 15 15 Flip Heuristic • Start with Delaunay triangulation • Flip edges that remove, or may “help” remove a local minimum • Only flip if 2 circles have ≤k points inside • O(nk2 + nklogn) time flip

14. Hull Heuristic • Start with Delaunay triangulation • Compute all useful order k Delaunay edges that remove a local minimum useful order 4 Delaunay edge

15. Hull Heuristic • Add them incrementally, unless • it intersects a previously inserted edge • Retriangulate the polygon that appears

16. Hull Heuristic • Add them incrementally, unless • it intersects a previously inserted edge • Retriangulate the polygon that appears

17. Experiments on terrains

18. Experiments • Do higher order Delaunay triangulations help to reduce local minima? • How does this depend on the order? • Which heuristic is better: flip or hull? • Do they create any artifacts? • 5 terrains • orders 0-10 • flip and hull heuristic

19. Quinn Peak • Elevation grid of 382 x 468 • Random sample of 1800 vertices • Delaunay triangulation • 53 local minima

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

21. Hull heuristic Flip heuristic

22. Another realistic aspect • Valleys continue 26 34 26 27 32 21 20 21 17 14 19 15 normal edge ridge edge valley edge Valley edges can end in vertices that are not local minima

23. Valley Heuristic • Remove isolated valley edges by flipping them out • Extend valley edge components further down • O(nk log n) time

24. Experiments • Terrains with valley edges and local minima shown • Delaunay, Flip-8, Hull-8, Valley-8,Hull-8 + Valley-8

25. Delaunay triangulation

26. Flip-8

27. Hull-8

28. Valley-8

29. Hull-8 + valley heuristic

30. Conclusions • Hull and Flip reduce local minima by 60-70% for order 8; Hull is often better • Valley reduces the number of valley edge components by 20-40% for order 8 • Flip gives artifacts • Hull + Valley seems best

31. Future Work • NP-hardness for small k ? • Other properties of terrains • Spatial angles • Local maxima • Other hydrological features (watersheds) • Improvements valley heuristic