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 Triangulations

  • Introduction

  • Results on local minima

    • NP-hard

    • Two heuristics

  • Results on valley components

    • A new heuristic


Motivation
Motivation Triangulations

  • Terrain modeling for geomorphological applications

  • TIN as terrain representation

  • Realism necessary

  • Choice of triangulation is important


Generating realistic terrains with higher order delaunay triangulations

  • Few local minima Triangulations

  • Connected valley components

  • Wrong triangulation can introduce undesirable artifacts


Triangulations
Triangulations Triangulations


Higher order delaunay triangulations
Higher-Order Delaunay Triangulations 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 Triangulations

  • 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 Triangulations

  • Computing optimal HODT for minimizing local minima is NP-hard

  • Two heuristics

  • Experimental results comparing the heuristics and analysing HODT


Np hardness
NP-hardness Triangulations

  • 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 Triangulations

  • Start with Delaunay triangulation

  • Flip edges that might potentially remove a local minimum

  • Preserve order k property

  • O (n.k2 + n.k.logn)


Generating realistic terrains with higher order delaunay triangulations


Hull heuristic
Hull Heuristic Triangulations

  • 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)


Generating realistic terrains with higher order delaunay triangulations



Generating realistic terrains with higher order delaunay triangulations

  • Quinn Peak Triangulations

  • Elevation data grid

  • 382 x 468

  • 1 data point = 30 meter


Generating realistic terrains with higher order delaunay triangulations

  • Random sample Triangulations

  • 1800 vertices

  • Delaunay triangulation

  • 53 local minima


Generating realistic terrains with higher order delaunay triangulations


Generating realistic terrains with higher order delaunay triangulations

hull heuristic Triangulations

flip heuristic


Drainage on tin
Drainage on TIN Triangulations

  • Complex to model due to material properties

  • Water follows path of steepest descent

    • Over edge

    • Over triangle


Definitions
Definitions Triangulations

  • Three kinds of edges:


Generating realistic terrains with higher order delaunay triangulations


Drainage quality of terrain
Drainage quality of terrain these edges reaches lowest vertex of the component

  • 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 these edges reaches lowest vertex of the component

  • Try to remove it

    • No new valley edges should be created

    • New triangle order k

  • Otherwise try to extend it


Extending component
Extending component these edges reaches lowest vertex of the 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


Generating realistic terrains with higher order delaunay triangulations

  • Extend if: these edges reaches lowest vertex of the component

    • 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 these edges reaches lowest vertex of the component

  • 25-40% decrease in number of valley components

  • +/- 30 % decrease in number of local minima (far less than flip and hull heuristic)


Results on a terrain
Results on a terrain these edges reaches lowest vertex of the component


Results compared to flip and hull
Results compared to flip and hull these edges reaches lowest vertex of the component


Delaunay triangulation
Delaunay triangulation these edges reaches lowest vertex of the component


Flip 8
Flip-8 these edges reaches lowest vertex of the component


Hull 8
Hull-8 these edges reaches lowest vertex of the component


Valley 8
Valley-8 these edges reaches lowest vertex of the component


Flip 8 valley heuristic
Flip-8 + valley heuristic these edges reaches lowest vertex of the component


Hull 8 valley heuristic
Hull-8 + valley heuristic these edges reaches lowest vertex of the component


Conclusions local minima
Conclusions Local Minima these edges reaches lowest vertex of the component

  • Low orders already give good results

  • Hull is often better than flip

  • Hull performed almost optimal


Conclusions drainage
Conclusions Drainage these edges reaches lowest vertex of the component

  • 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 these edges reaches lowest vertex of the component

  • NP-hardness for small k

  • Other properties of terrains

    • Local maxima

    • More hydrological features (watersheds)

  • Different local operators for valleyheuristic