generating realistic terrains with higher-order delaunay triangulations thierry de kok marc van...
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Generating Realistic Terrains with Higher-Order Delaunay
Triangulations
Thierry de Kok
Marc van Kreveld
Maarten Löffler
Center for Geometry, Imagingand Virtual Environments
Utrecht University
Overview
• Introduction– Triangulation for terrains– Realistic terrains– Higher order Delaunay triangulations
• Minimizing local minima– NP-hardness– Two heuristics: algorithms and experiments
• Other realistic aspects
Polyhedral terrains, or TINs
• Points with (x,y) and elevation as input
• TIN as terrain representation
• Choice of triangulation is important
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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
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, ...
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local minimum in a TIN
Delaunay triangulation
• Maximizes minimum angle
• Empty circle property
Delaunay triangulation
• Does not take elevation into account
• May give local minima
• May give interrupted valleys
Triangulate to minimize local minima?
Triangulate to minimize local minima?
Connect everything to global minimum bad triangle shape & interpolation
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
0th order
4th order
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)
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
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 + nk log n) time
flip18
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Hull Heuristic
• Start with Delaunay triangulation
• Compute all useful order k Delaunay edges that remove a local minimum
useful order 4 Delaunay edge
Hull Heuristic
• Add them incrementally, unless– it intersects a previously inserted edge
• Retriangulate the polygon that appears
Hull Heuristic
• Add them incrementally, unless– it intersects a previously inserted edge
• Retriangulate the polygon that appears
Experiments on terrains
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
• Quinn Peak• Elevation grid of
382 x 468
• Random sample of 1800 vertices
• Delaunay triangulation
• 53 local minima
• Hull heuristic applied
• Order 4 Delaunay triangulation
• 25 local minima
0
10
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60
0 1 2 3 4 5 6 7 8 9 10
order
loca
l m
inim
aHull heuristic
Flip heuristic
Another realistic aspect
• Valleys continue
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normal edge ridge edge valley edge
Valley edges can end in vertices that are not local minima
Valley Heuristic
• Remove isolated valley edges by flipping them out
• Extend valley edge components further down
• O(nk log n) time
Experiments
• Terrains with valley edges and local minima shown
• Delaunay, Flip-8, Hull-8, Valley-8,Hull-8 + Valley-8
Delaunay triangulation
Flip-8
Hull-8
Valley-8
Hull-8 + valley heuristic
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
Future Work
• NP-hardness for small k ?
• Other properties of terrains– Spatial angles– Local maxima– Other hydrological features (watersheds)
• Improvements valley heuristic