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. Overview. Introduction Results on local minima NP-hard Two heuristics

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generating realistic terrains with higher order delaunay triangulations

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

overview
Overview
  • Introduction
  • Results on local minima
    • NP-hard
    • Two heuristics
  • Results on valley components
    • A new heuristic
motivation
Motivation
  • Terrain modeling for geomorphological applications
  • TIN as terrain representation
  • Realism necessary
  • Choice of triangulation is important
slide4
Few local minima
  • Connected valley components
  • Wrong triangulation can introduce undesirable artifacts
higher order delaunay triangulations
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)
using hodt to solve the problem
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
local minima results
Local Minima Results
  • Computing optimal HODT for minimizing local minima is NP-hard
  • Two heuristics
  • Experimental results comparing the heuristics and analysing HODT
np hardness
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
flip heuristic
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)
slide13
New edge must be “lower” than old edge
  • New triangles must be order k
hull heuristic
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)
slide15
When adding an edge, compute the hull
  • Retriangulate the hull
  • Do not add any other edges intersecting the hull
slide17
Quinn Peak
  • Elevation data grid
  • 382 x 468
  • 1 data point = 30 meter
slide18
Random sample
  • 1800 vertices
  • Delaunay triangulation
  • 53 local minima
slide19
Hull heuristic applied
  • Order 4 Delaunay triangulation
  • 25 local minima
slide20

hull heuristic

flip heuristic

drainage on tin
Drainage on TIN
  • Complex to model due to material properties
  • Water follows path of steepest descent
    • Over edge
    • Over triangle
definitions
Definitions
  • Three kinds of edges:
slide23
Valley component: maximal set of valley edges s.t. flow from these edges reaches lowest vertex of the component
drainage quality of terrain
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
isolated valley edge
Isolated valley edge
  • Try to remove it
    • No new valley edges should be created
    • New triangle order k
  • Otherwise try to extend it
extending component
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
slide27
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
results valley heuristic
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)
conclusions local minima
Conclusions Local Minima
  • Low orders already give good results
  • Hull is often better than flip
  • Hull performed almost optimal
conclusions drainage
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
future work
Future Work
  • NP-hardness for small k
  • Other properties of terrains
    • Local maxima
    • More hydrological features (watersheds)
  • Different local operators for valleyheuristic
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