Streaming Computation of Delaunay Triangulations

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Streaming Computation of Delaunay
Triangulations
Jonathan ShewchukUC Berkeley
Martin IsenburgUC Berkeley
Jack SnoeyinkUNC Chapel Hill
Yuanxin LiuUNC Chapel Hill
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The 2D Delaunay Triangulation

• important tool for interpolating points
• graphics, GIS, finite elements,

• a triangulation in which every triangle has an
  empty circumscribing circle

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Billions of Points
courtesy of www.ncfloodmaps.com
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Streaming Computation
output to disk
input from disk
small bufferin main memory

• restrict computations to data in buffer
• process data in the order it arrives
• output results early and reuse the memory
• do not use external memory

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Spatial Finalization

• inject tags intothe point streamthat inform
  whengrid cells are freeof future points

finalized region
future pointsonly arrive inthese cells

• use tags to certify triangles as beingpart of
  the finaltriangulation andoutput them

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Streaming Delaunay
Spatial Finalization
Incremental Triangulation
0.5 Billion Points 11 GB
1.0 Billion Triangles 17 GB
?
on a laptop ? in 48 minutes ? using 60 10 MB
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Immediate Processing of Output
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Features of Streaming Delaunay

• efficient
• 10 faster than previous method

• extremely scalable
• billions of terrain points on a laptop

• streaming output
• process triangulation without storing it

• only uniform points distributions
• can tetrahedralize 3D points
• but not when point are along a surface

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Related Work
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Streaming Geometry Processing

• Streaming Meshes
• topological finalization of vertices

Isenburg Lindstrom 05
Streaming Compression
Streaming Simplification
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Delaunay Triangulators

• Sweep line
• pretty complex implementation
• points along sweep line ? global sort

• Divide and Conquer
• fastest in practice (in main memory)
• partition points recursively ? global sort

• Incremental Point Insertion
• usually fast (except for bad orders)
• can process points in any order

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Incremental Point Insertion
Lawson 77 Bowyer 81 Watson 81

• locate triangle enclosing the point

• find and remove all triangles with non-empty
  circumcircles

• triangulate by connecting new point

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Triangulating Lots of Points

• buy more main memory
• need 36 GB for 0.5 billion points

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External Memory Methods
sorted points
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Our Streaming Approach
finalized points
output mesh
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Streaming Delaunay Pipeline
spfinalize -i points.raw -ospb spdelaunay ispb
o tin.smb
enhances points withspatial finalization
uses spatial finalizationto certify triangles as
final
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The Finalizer
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Spatial Finalization of Points
? compute bounding box
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Spatial Finalization of Points

• ? compute bounding box
• ? create finalization grid
• count number of points per cell

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Spatial Finalization of Points

• ? compute bounding box
• ? create finalization grid
• count number of points per cell
• ? output finalized points
• buffer per grid cell
• if full, output points in one randomized
  chunk followed by finalization tag

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Spatial Finalization of Points

• ? compute bounding box
• ? create finalization grid
• count number of points per cell
• ? output finalized points
• buffer per grid cell
• if full, output points in one randomized
  chunk followed by finalization tag

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Spatial Finalization of Points

• ? compute bounding box
• ? create finalization grid
• count number of points per cell
• ? output finalized points
• buffer per grid cell
• if full, output points in one randomized
  chunk followed by finalization tag

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Spatial Finalization of Points

• ? compute bounding box
• ? create finalization grid
• count number of points per cell
• ? output finalized points
• buffer per grid cell
• if full, output points in one randomized
  chunk followed by finalization tag

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Spatial Finalization of Points

• ? compute bounding box
• ? create finalization grid
• count number of points per cell
• ? output finalized points
• buffer per grid cell
• if full, output points in one randomized
  chunk followed by finalization tag

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Spatial Finalization of Points

• ? compute bounding box
• ? create finalization grid
• count number of points per cell
• ? output finalized points
• buffer per grid cell
• if full, output points in one randomized
  chunk followed by finalization tag

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Spatial Finalization of Points

• ? compute bounding box
• ? create finalization grid
• count number of points per cell
• ? output finalized points
• buffer per grid cell
• if full, output points in one randomized
  chunk followed by finalization tag

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Spatial Finalization of Points

• ? compute bounding box
• ? create finalization grid
• count number of points per cell
• ? output finalized points
• buffer per grid cell
• if full, output points in one randomized
  chunk followed by finalization tag

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Spatial Coherence
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Spatial Coherence
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Exploit the Existing Coherence

• document with finalization tags
• enhance by releasing chunks of points
• ? benefits of global sort, but cheaper

raw point stream
finalized point stream
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Exploit the Existing Coherence

• document with finalization tags
• enhance by releasing chunks of points
• ? benefits of global sort, but cheaper

raw point stream
finalized point stream
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Unfinalized Region as Quadtree

• as tags finalize cell after cell, maintain
  remaining cells in adaptive quadtree
• ? used for efficient overlap tests by the
  triangulator

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The Triangulator
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Modified Incremental Delaunay

• insert points

• when reading afinalization tag
• find certified triangles
• output them to disk or pipe
• deallocate data structure

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Modified Incremental Delaunay

• insert points

• when reading afinalization tag
• find certified triangles
• output them to disk or pipe
• deallocate data structure

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Finding Certified Triangles

• do circumcircleslie completely infinalized
  region?

? overlap test

• efficient testing
• avoid tests
• try cheaper test

• link overlappingcircles to cells

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Modified Point Location

• usual strategy
• walk current triangulation starting at the
  newest triangle

• if we fail
• restart from a triangle linked with the cell
• brute force

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Results
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The Neuse-River Basin
6 min
10 MB60,388 triangles
8 MB512 x 512 grid
6 min
52 MB 2,249,268 points
36 min
1.0 Billion Triangles 17 GB
0.5 Billion Points 11 GB
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on a laptop ? in 48 minutes ? using 60 10 MB
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Scalability 3 x 3 Tiling of Neuse
9.0 Billion Triangles 151 GB
4.5 Billion Points 101 GB
?
on a laptop ? under 7 hours ? using 152 11
MB
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Comparisons

• to best out-of-core method
• over 10 times faster on 0.5 billion points
• laptop 2 disks desktop 4 disks
• 0 scratch space 50 GB scratch space

Arge et al. 05
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Discussion
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Streaming Delaunay
spfinalize -i points.raw -ospb spdelaunay ispb
o tin.smb
enhances points withspatial finalization
uses spatial finalizationto certify triangles as
final
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Geometry Processing Pipelines

• isoline extraction

• raster DEM generation

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Spatial Finalization Tags

• order of magnitude faster than methods that use
  external memory

• no temporary storage to disk
• output results early reuse memory

• interleave input computation output
• 100 CPU usage while triangulating

• no global sort
• dont fight the data, use existing coherence

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Streaming Delaunay in 3D
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Streaming Delaunay in 3D

• uniform distributed points
• works, but bigger footprint
• example result
• 14 GB mesh, 240 hrs, 795 MB

• surface scans
• does not work
• big circumspheres intersect many cells
• most tetrahedra cannot be certified early

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Acknowledgements

• data
• Kevin Yi, Duke
• support
• NSF grants 0429901 0430065
• "Collaborative Research Fundamentalsand
  Algorithms for Streaming Meshes."
• NGA award HM1582-05-2-0003
• Alfred P. Sloan Research Fellowship

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Thank You
executables, video, slides, paper, and soon also
source code http//www.cs.unc.edu/isenburg/sd/
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Thank You
executables, video, slides, paper, and soon also
source code http//www.cs.unc.edu/isenburg/sd/
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Future / Current Work

• Finalize without Bounding Box
• eliminate 1st pass of finalizer

• Streaming Constrained Delaunay
• crucial in GIS applications

• Triangulate 3D surface points
• data-dependent spatial finalization

• Spatial Finalization for other algorithm
• surface reconstruction,

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Sprinkle Points

• add global randomness to the point insertion order

without sprinkle points
with sprinkle points