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

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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


Generating realistic terrains with higher order delaunay triangulations

  • Few local minima

  • Connected valley components

  • Wrong triangulation can introduce undesirable artifacts


Triangulations

Triangulations


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)


Generating realistic terrains with higher order delaunay triangulations

  • 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)


Generating realistic terrains with higher order delaunay triangulations

  • When adding an edge, compute the hull

  • Retriangulate the hull

  • Do not add any other edges intersecting the hull


Experiments on real terrains

Experiments on real Terrains


Generating realistic terrains with higher order delaunay triangulations

  • Quinn Peak

  • Elevation data grid

  • 382 x 468

  • 1 data point = 30 meter


Generating realistic terrains with higher order delaunay triangulations

  • Random sample

  • 1800 vertices

  • Delaunay triangulation

  • 53 local minima


Generating realistic terrains with higher order delaunay triangulations

  • Hull heuristic applied

  • Order 4 Delaunay triangulation

  • 25 local minima


Generating realistic terrains with higher order delaunay triangulations

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:


Generating realistic terrains with higher order delaunay triangulations

  • 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


Generating realistic terrains with higher order delaunay triangulations

  • 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)


Results on a terrain

Results on a terrain


Results compared to flip and hull

Results compared to flip and hull


Delaunay triangulation

Delaunay triangulation


Flip 8

Flip-8


Hull 8

Hull-8


Valley 8

Valley-8


Flip 8 valley heuristic

Flip-8 + valley heuristic


Hull 8 valley heuristic

Hull-8 + valley 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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