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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

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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, 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

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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 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 % 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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