Generating Realistic Terrains with Higher-Order Delaunay Triangulations

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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
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Overview

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

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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
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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
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Delaunay triangulation

• Maximizes minimum angle
• Empty circle property

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Delaunay triangulation

• Does not take elevation into account
• May give local minima
• May give interrupted valleys

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Triangulate to minimize local minima?
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Triangulate to minimize local minima?
Connect everything to global minimum? bad
triangle shape interpolation
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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
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Higher order Delaunay triangulations

• Introduced by Gudmundsson, Hammar and van Kreveld
  (ESA 2000)
• Minimize local minima for 1st orderO(n log n)
  time
• Minimize local minima for kth orderO(k2)-approxi
  mation algorithm inO(nk3 nk log n) time (hull
  heuristic)

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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

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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

flip
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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
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Hull Heuristic

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

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Hull Heuristic

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

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Experiments on terrains
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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

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

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

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Hull heuristic Flip heuristic
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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
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Valley Heuristic

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

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Experiments

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

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Delaunay triangulation
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Flip-8
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Hull-8
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Valley-8
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Hull-8 valley heuristic
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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

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Future Work

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