Streaming Construction of Delaunay Triangulations
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1 Streaming Construction of Delaunay Triangulations Jonathan Shewchuk University of California at Berkeley Joint work with Martin Isenburg, Yuanxin Liu, & Jack Snoeyink
2 Streaming Construction of Delaunay Triangulations Or, How we compute supercomputer sized Delaunay triangulations on an ordinary laptop computer.
3 The Delaunay Triangulation Every point set has a Delaunay triangulation. Think of it as a function that takes a set of points and outputs a triangulation.
4 The Delaunay Triangulation...is a triangulation whose triangles are all Delaunay. An triangle is Delaunay if it has an empty circumscribing circle one that encloses no vertex.
5 The Delaunay Triangulation The circumcircle of every Delaunay triangle is empty.
6 Applications of Triangulations Contouring Rendering Terrain Databases and Geographical Information Systems
7 Elevation Collection by LIDAR
8 Related Work Amenta, Choi, & Rote [2003] use biased randomized insertion orders to improve virtual memory performance of 3D Delaunay triangulation construction.
9 Related Work Amenta, Choi, & Rote [2003] use biased randomized insertion orders to improve virtual memory performance of 3D Delaunay triangulation construction. Agarwal, Arge, and Yi [200] implemented an external memory 2D triangulator based on randomized divide and conquer. ( billion triangles on desktop machine.)
10 Related Work Amenta, Choi, & Rote [2003] use biased randomized insertion orders to improve virtual memory performance of 3D Delaunay triangulation construction. Agarwal, Arge, and Yi [200] implemented an external memory 2D triangulator based on randomized divide and conquer. ( billion triangles on desktop machine.) Blandford, Blelloch, and Kadow [200] implemented a 3D parallel triangulator. (0 billion tetrahedra on 4 processors.)
11 Streaming Delaunay
12 Streaming Computation A restricted version of out of core computation. Algorithm makes one (or a few) pass(es) over input stream, and writes output stream.
13 Streaming Computation A restricted version of out of core computation. Algorithm makes one (or a few) pass(es) over input stream, and writes output stream. Processes data in memory buffer much smaller than the stream sizes.
14 Streaming Computation A restricted version of out of core computation. Algorithm makes one (or a few) pass(es) over input stream, and writes output stream. Processes data in memory buffer much smaller than the stream sizes. Nothing is stored temporarily to disk.
15 Advantages of Streaming Streaming tools can run concurrently in a pipeline. Often much faster than other out of core algorithms!
16 Our Accomplishments 9 billion triangles in. hours on a laptop.
17 Our Accomplishments 9 billion triangles in. hours on a laptop. 2 times faster than Agarwal Arge Yi.
18 Our Accomplishments 9 billion triangles in. hours on a laptop. 2 times faster than Agarwal Arge Yi. 800 million tetrahedra in under 3 hours.
19 Algorithm Choice The incremental insertion algorithm for constructing Delaunay triangulations.
20 Algorithm Choice The incremental insertion algorithm for constructing Delaunay triangulations. Why? We have little control over the order of points in the input stream.
21 Algorithm Choice The incremental insertion algorithm for constructing Delaunay triangulations. Why? We have little control over the order of points in the input stream. We modified existing, sequential Delaunay triangulation codes (2D & 3D) for streaming.
22 Bowyer Watson Algorithm Insert one vertex at a time.
23 Bowyer Watson Algorithm Insert one vertex at a time. Remove all triangles/tetrahedra that are no longer Delaunay.
24 Bowyer Watson Algorithm Insert one vertex at a time. Remove all triangles/tetrahedra that are no longer Delaunay. Retriangulate the cavity with a fan around the new vertex.
25 Spatial Finalization
26 Spatial Finalization Subdivide space into finalization regions.
27 Spatial Finalization Subdivide space into finalization regions. Inject into the stream spatial finalization tags that indicate there are no more points in this region. Point stream Finalized point stream
28 Spatial Finalization: Why? Triangles whose circumscribing circles lie entirely in finalized space can be written to disk immediately. Regions not yet finalized Finalized space
29 Regions not yet finalized Triangles whose circumscribing circles intersect an unfinalized region remain in memory. Finalized space
30 Streaming Delaunay Pipeline Finalizer Triangulator
31 The Finalizer
32 How to Finalize Compute bounding box.
33 How to Finalize Compute bounding box.
34 How to Finalize Compute bounding box.
35 How to Finalize Compute bounding box.
36 How to Finalize Compute bounding box.
37 How to Finalize Compute bounding box.
38 How to Finalize Compute bounding box.
39 How to Finalize Compute bounding box.
40 How to Finalize 2 Compute bounding box. Finalization grid.
41 How to Finalize Compute bounding box. Finalization grid. Count points.
42 How to Finalize Compute bounding box. Finalization grid. Count points. 3 4
43 How to Finalize 2 Compute bounding box. Finalization grid. Count points
44 How to Finalize Compute bounding box. Finalization grid. Count points.
45 How to Finalize 2 Compute bounding box. Finalization grid. Count points
46 How to Finalize Compute bounding box. Finalization grid. Count points. 2
47 How to Finalize Compute bounding box. Finalization grid. Count points. Store sprinkle points
48 How to Finalize 2 Compute bounding box. Finalization grid Count points. Store sprinkle points. 4 3 Output finalized points Release chunks. Inject finalization tags.
49 How to Finalize Compute bounding box. Finalization grid Count points. Store sprinkle points. 4 3 Output finalized points Release chunks. Inject finalization tags.
50 How to Finalize Compute bounding box. Finalization grid Count points. Store sprinkle points. 4 3 Output finalized points Release chunks. Inject finalization tags.
51 How to Finalize 2 2 Compute bounding box. Finalization grid Count points. Store sprinkle points. 3 3 Output finalized points Release chunks. Inject finalization tags.
52 How to Finalize 2 2 Compute bounding box. Finalization grid Count points. Store sprinkle points Output finalized points Release chunks. Inject finalization tags.
53 How to Finalize 2 2 Compute bounding box. Finalization grid Count points. Store sprinkle points Output finalized points Release chunks. Inject finalization tags.
54 How to Finalize 2 2 Compute bounding box. Finalization grid Count points. Store sprinkle points. 4 3 Output finalized points Release chunks. Inject finalization tags.
55 How to Finalize 2 Compute bounding box. Finalization grid Count points. Store sprinkle points. 4 3 Output finalized points Release chunks. Inject finalization tags.
56 Streaming Point Format # npoints 224 # bb_min_f # bb_max_f # datatype SP_FLOAT v v chunk v of v points v v x cell 232 v v v v v v v x cell 240 finalization tag chunk of points finalization tag
57 Spatial Coherence The correlation between the proximity of points in space and their proximity in the stream.
58 Spatial Coherence The correlation between the proximity of points in space and their proximity in the stream.
59 Spatial Coherence The correlation between the proximity of points in space and their proximity in the stream.
60 Spatial Coherence The correlation between the proximity of points in space and their proximity in the stream.
61 Danger: Potential Quadratic Time
62 Danger: Potential Quadratic Time
63 Danger: Potential Quadratic Time
64 Danger: Potential Quadratic Time
65 Danger: Potential Quadratic Time
66 Danger: Potential Quadratic Time
67 Danger: Potential Quadratic Time
68 Point Reordering by Finalizer Chunking: Buffering points in a region until all points arrive, then releasing them together. Points in a chunk are randomly reordered.
69 Point Reordering by Finalizer Chunking: Buffering points in a region until all points arrive, then releasing them together. Points in a chunk are randomly reordered. Sprinkling: Promoting a small random sample of sprinkle points to earlier in the stream. Circumvents danger of quadratic behavior.
70 The Triangulator
71 Streaming Delaunay Triangulation When point arrives: Locate enclosing triangle. Update triangulation. Regions not yet finalized Finalized space
72 Streaming Delaunay Triangulation When point arrives: Locate enclosing triangle. Update triangulation. When tag arrives: Identify final triangles. Write them out & free their memory. Regions not yet finalized
73 Streaming Delaunay Triangulation When point arrives: Locate enclosing triangle. Start at newest triangle.
74 Streaming Delaunay Triangulation When point arrives: Locate enclosing triangle. Start at newest triangle. Walk to point.
75 Streaming Delaunay Triangulation When point arrives: Locate enclosing triangle. Start at newest triangle. Walk to point.
76 Streaming Delaunay Triangulation When point arrives: Locate enclosing triangle. Start at newest triangle. Walk to point. If that fails, restart from cell triangle. If that fails, do exhaustive search.
77 Streaming Delaunay Triangulation When point arrives: Locate enclosing triangle. Update triangulation.
78 Streaming Delaunay Triangulation When tag arrives: Identify final triangles. Only these triangles need to be checked: New triangles (since last tag). Triangles waiting for this tag.
79 Streaming Delaunay Triangulation When tag arrives: Identify final triangles. Checked triangles that aren t final are assigned a tag to wait for.
80 Results
81 Results Us: 2 min finalize + 3 min triang. = 48 min. Agarwal Arge Yi: 3 hr sort +. hr triang. = 0. hr.
82 Results 0 min finalize + 28 min triang. = hr 2 min. 42 MB finalize + MB triang. = 43 MB.
83 3D
84 3D Streaming Delaunay 8 million tetrahedra in 2:4 hrs & 9 MB. (Pre finalized points.)
85 3D Streaming Delaunay Does not work for surface scans. Circumspheres are too big.
86 3D Streaming Delaunay Future work: spherical finalization regions from Delaunay triangulation of a random sample.
87 3D Streaming Delaunay Future work: spherical finalization regions from Delaunay triangulation of a random sample.
88 Conclusions
89 Conclusions Most real world points sets have lots of spatial coherence sorting doesn t help.
90 Conclusions Most real world points sets have lots of spatial coherence sorting doesn t help. Like aikido: Use your opponent s spatial coherence against him.
91 Conclusions Most real world points sets have lots of spatial coherence sorting doesn t help. Like aikido: Use your opponent s spatial coherence against him. Not an external memory algorithm! No temporary storage to disk.
92 Conclusions Most real world points sets have lots of spatial coherence sorting doesn t help. Like aikido: Use your opponent s spatial coherence against him. Not an external memory algorithm! No temporary storage to disk. 2 times faster than best previous 2D approach.
93 Conclusions Most real world points sets have lots of spatial coherence sorting doesn t help. Like aikido: Use your opponent s spatial coherence against him. Not an external memory algorithm! No temporary storage to disk. 2 times faster than best previous 2D approach. Streaming triangulation can be piped directly to another streaming application.
94 Streaming Digital Elevation Maps
95 Streaming Contour Extraction
96 Thanks Kevin Yi at Duke supplied the Neuse River Basin data. Martin Isenburg & Yuanxin Leo Liu did most of the programming.
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