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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 10 12 15 73 78 24 21 25 29 19 10 12 15 73 78 24 21 25 29 19

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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,... 6 12 17 52 21 24 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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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 1 st order 0 th order 4 th order

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Higher order Delaunay triangulations Introduced by Gudmundsson, Hammar and van Kreveld (ESA 2000) Minimize local minima for 1 st order: O(n log n) time Minimize local minima for k th order: O(k 2 )-approximation algorithm in O(nk 3 + nk log n) time (hull heuristic)

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This paper, results NP-hardness of minimizing local minima NP-hardness for k th order, k = (n ) New flip heuristic: O(nk 2 + nk log n) time Faster hull heuristic: O(nk 2 + 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(nk 2 + nk log n) time flip 18 15 23 11 18 15 23 11

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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 26 21 20 15 34 32 14 17 21 19 26 27 normal edgeridge edgevalley 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

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Higher-Order Delaunay Triangulations

Higher-Order Delaunay Triangulations

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