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Generating Realistic Terrains with Higher-Order Delaunay TriangulationsPowerPoint Presentation

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 Triangulations

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

- Results on valley components
- A new heuristic

Motivation Triangulations

- Terrain modeling for geomorphological applications
- TIN as terrain representation
- Realism necessary
- Choice of triangulation is important

- Few local minima Triangulations
- Connected valley components
- Wrong triangulation can introduce undesirable artifacts

Triangulations 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 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 Triangulations

- Computing optimal HODT for minimizing local minima is NP-hard
- Two heuristics
- Experimental results comparing the heuristics and analysing HODT

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 Triangulations

- Start with Delaunay triangulation
- Flip edges that might potentially remove a local minimum
- Preserve order k property
- O (n.k2 + n.k.logn)

- New edge must be “lower” than old edge Triangulations
- New triangles must be order k

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)

- When adding an edge, compute the hull Triangulations
- Retriangulate the hull
- Do not add any other edges intersecting the hull

Experiments on real Terrains Triangulations

- Quinn Peak Triangulations
- Elevation data grid
- 382 x 468
- 1 data point = 30 meter

- Random sample Triangulations
- 1800 vertices
- Delaunay triangulation
- 53 local minima

- Hull heuristic applied Triangulations
- Order 4 Delaunay triangulation
- 25 local minima

hull heuristic Triangulations

flip heuristic

Drainage on TIN Triangulations

- Complex to model due to material properties
- Water follows path of steepest descent
- Over edge
- Over triangle

Definitions Triangulations

- Three kinds of edges:

- Valley component: maximal set of valley edges s.t. flow from these edges reaches lowest vertex of the component

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 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 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

- 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 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 these edges reaches lowest vertex of the component

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

Delaunay triangulation these edges reaches lowest vertex of the component

Flip-8 these edges reaches lowest vertex of the component

Hull-8 these edges reaches lowest vertex of the component

Valley-8 these edges reaches lowest vertex of the component

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

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

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 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 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

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