OutofCore Surface Simplification

1
Out-of-Core Algorithms for Scientific
Visualization and Computer Graphics

• Out-of-Core Surface Simplification
• Peter Lindstrom
• Center for Applied Scientific Computing
• Lawrence Livermore National Laboratory
• This work was performed under the auspices of the
  U.S. Department of Energy by
• University of California, Lawrence Livermore
  National Laboratory under Contract W-7405-Eng-48.

2
Talk Overview

• introduction
• spatial clustering
• surface segmentation
• streaming
• summary

3
Introduction

• geometric datasets are growing rapidly in size
• millions to billions of geometric primitives now
  common
• sources simulations, 3D scanning, modeling
  software,
• managing large datasets is difficult
• data exceeds resources for storage and
  transmission
• gigabytes to terabytes of raw data
• conventional processing and visualization methods
  do not scale
• assume random access, in-core storage of entire
  dataset
• new approaches to large-data processing needed

4
Large-Data Example 1

• Richtmeyer-Meshkov instability simulation at LLNL
• 2K3 voxels, 27,000 time steps, 470 million
  triangle isosurface

5
Large-Data Example 2

• Digital Michelangelo project
• ¼ millimeter resolution range scan 370 million
  triangles

6
Large-Data Example 3

• Visible Human project
• ? millimeter resolution scan over 100 billion
  voxels

7
Managing Large Data

• one solution to large-data problem
  simplification
• approximate dataset with fewer primitives while
  satisfying error tolerance or size constraint
• enables levels of detail, multiresolution
  representations, progressive/adaptive/view-depende
  nt refinement

8
Levels of Detail for VisualizationPerceived
Detail Decreases with Distance
100
10
2
9
Triangle Mesh Simplification

• most surfaces are represented as triangle meshes
• common triangle mesh coarsening operations
• vertex removal (Schroeder et al. 92)
• edge collapse (Hoppe et al. 93)
• half-edge collapse (Kobbelt et al. 98)
• triangle collapse (Hamann 94)
• vertex pair contraction (Garland Heckbert 97)
• vertex clustering (Rossignac Borrel 93)
• all operations instances of vertex set
  partitioning
• (except general vertex removal)

10
Vertex Set Partitioning

• partition vertices into disjoint sets (clusters)
• V V1 È V2 È È Vn
• collapse each set into single (possibly new)
  vertex
• Vi vi
• discard degenerate triangles
• surviving triangles span three different
  clusters
• variations between simplification operations
• legality of partitions (e.g. clusters must be
  edge-connected)
• vertex positioning (e.g. subsampling vs.
  optimization)

11
Examples of Vertex Set Partitions

• uniform partitions (spatial clustering on 3D
  grid)
• adaptive partitions (error-driven edge collapse)

12
The Dilemma

• surface simplification can reduce data size,
  but...
• simplification traditionally done in-core
• load entire surface mesh into memory
• queue up sequence of fine-grained coarsening
  operations (e.g. vertex removal, edge collapse)
  in order of least error
• proceed by iteratively coarsening mesh
• memory O(n) storage, random (i.e. incoherent)
  access
• running time O(n log n)
• example in-core simplification of 1
  billion-triangle mesh
• 80 GB of RAM, 10 days to simplify using
  efficient method
• conclusion does not scale well

13
Out-of-Core Simplification

• recent research focused on external memory
  methods
• key concepts data segmentation,
  coherent/sequential access, disk-based algorithms
  (e.g. B-trees, external sorts)
• simplification of arbitrarily large surfaces now
  possible
• three distinct approaches to out-of-core
  simplification
• spatial clustering
• Lindstrom 00, Lindstrom Silva 01, Shaffer
  Garland 01,Fei et al. 02, Garland Shaffer
  02
• surface segmentation
• El-Sana Chiang 00, Prince 00, Bernardini et
  al. 02,Choudhury Watson 02, Cignoni et al.
  03
• streaming
• Wu Kobbelt 03, Isenburg et al. 03

14
Talk Overview

• introduction
• spatial clustering
• surface segmentation
• streaming
• summary

15
Spatial Clustering

• idea partition space that surface lies in into
  3D cells
• vertex set partitions given geometrically by cell
  containment
• no topological constraints or considerations
• cell geometry can be adapted to surface
• e.g. smaller cells in detailed regions, adapt
  cell shape to surface
• simplest partition uniform 3D Cartesian grid

16
OoCS Uniform Spatial ClusteringLindstrom 00

• algorithm overview
• partition space using a fixed 3D rectilinear grid
• sparse data structure (hash table) represents
  occupied cells
• position vertices by minimizing quadric error
• weighted sum of squared distances to set of
  planes
• keep triangles whose vertices fall in three
  different cells

17
OoCS Algorithm

• 1. for each triangle t in Tin
• 2. compute quadric matrix Qt ntntT
• 3. for each vertex v of t
• 4. map v to cluster cell c
• 5. add Qt to Qc
• 6. if t not degenerate, add t to Tout
• 7. for each cluster c
• 8. compute vertex xc from Qc
• 9. add xc to Xout

1. for each triangle t in Tin 2. compute quadric
matrix Qt ntntT 3. for each vertex v of
t 4. map v to cluster cell c 5. add Qt to
Qc 6. if t not degenerate, add t to Tout 7. for
each cluster c 8. compute vertex xc from
Qc 9. add xc to Xout
18
OoCS Algorithm

• algorithm characteristics
• features
• fast linear-time, single-pass algorithm
• reads arbitrarily large triangle soup stream
  sequentially
• drawbacks
• requires enough core memory to store simplified
  mesh
• connectivity oblivious ? non-manifold meshes
• non-adaptive partitions yield poor quality
• basis for many out-of-core spatial clustering
  methods

19
OoCS ResultsSt. Matthew Statue (close-up view)
simplified 3.1M triangles
original 390M triangles
dataset courtesy of the Digital Michelangelo
Project
20
OoCS ResultsMesh Quality
21
OoCS ResultsResource Usage

• simplification speed
• processes 400K triangles per second on desktop PC
• less than one hour to simplify billion-triangle
  mesh
• memory requirements
• 6372 bytes per output triangle (depends on hash
  table load)
• compare with 80 bytes per input triangle for
  typical in-core method
• allows 15 million-triangle simplified mesh on 1GB
  RAM computer

22
Output Sensitive Methods

• OoCS handles arbitrarily large inputs, but
• output size limited by available RAM
• what if simplified mesh is not accurate enough?
• cannot use finer grid because of memory
  constraints
• disk cheaper, orders of magnitude larger than RAM
• so can we use disk instead of RAM during
  simplification?
• yes, but we must avoid slow random accesses!

23
OoCSx Memory Insensitive Clustering Lindstrom
Silva 01

• achieve coherent disk access via external sort
• only random access in OoCS is to clusters
  quadric information (via in-core hash lookup)
• sort quadric information on disk, then process
  sequentially

24
OoCSx Algorithm Overview

• simplification
• scan input mesh
• read triangles, compute plane equations
• output cluster ID, plane equation for each vertex
• output non-degenerate triangles to triangle file
• sort plane equation file on cluster ID
• compute cluster quadrics, output optimal vertices
• re-indexing
• sort triangle file on given vertex field
• scan and replace cluster ID with vertex ID
• repeat for each of three vertex fields

25
OoCSx Results

• simplification speed
• 30-70K triangles per second (vs. 400K for OoCS)
• resource usage
• temporary disk 5 times size of input mesh
• depends on external sort used
• memory arbitrarily little (specified by user)
• 4 MB used in current implementation

26
Drawbacks of Uniform Spatial Clustering

• uniform grid not well adapted to surface features
• spatial proximity does not imply well-shaped
  clusters
• uneven and/or disconnected surface patches
  frequent
• fixed-resolution grid limits size of smallest
  feature preserved

27
Adaptive Spatial ClusteringShaffer Garland 01

• idea analyze mesh and repartition space
• accumulate quadrics on uniform grid in first pass
  (as in OoCS)
• use PCA of primal/dual quadrics to determine
  better partitions
• construct BSP-tree top-down for arbitrary convex
  cells
• recluster mesh on irregular grid in second pass

28
Adaptive Spatial Clustering

• algorithm characteristics
• features
• higher quality, better distribution of triangles
  than uniform spatial clustering
• drawbacks
• higher grid resolution (thus more memory) needed
  in first pass
• still output sensitive memory requirements

29
Adaptive Spatial Clustering Results
original
uniform
adaptive
uniform
adaptive
images courtesy of Eric Shaffer and Michael
Garland
30
Other Spatial Clustering Methods

• Fei et al. 02a (SIGGRAPH 2002 tech. sketch)
• simplification sensitive to translation of
  cluster grid
• degree of sensitivity (DOS) depends on amount of
  surface detail
• estimate local DOS by clustering on two
  interlaced grids
• use more detail in regions of high DOS
• Fei et al. 02b (Computer Graphics Forum 2002)
• following initial uniform clustering pass, use
  per-cell quadric error to determine which cells
  to refine/coarsen
• introduce split planes based on PCA of quadric
  matrices
• additional pass(es) used to incrementally,
  adaptively refine/coarsen mesh
• more memory efficient but slower adaptation than
  inShaffer Garland 01

31
Talk Overview

• introduction
• spatial clustering
• surface segmentation
• streaming
• summary

32
Surface Segmentation

• idea segment surface into small, disjoint
  patches
• ensure patches small enough to fit in-core
• simplify each patch using in-core method (e.g.
  edge collapse)
• stitch together simplified patches
• repeat if necessary

33
Vertex Set Partitioning usingSurface Segmentation
V
34
Uniform Spatial Surface SegmentationBernardini
et al. 02

• surface segmentation on a uniform rectilinear
  grid
• 3D grid partitions mesh into small patches
• simplify patches in-core to given error tolerance
  
• must leave patch boundaries intact to allow
  future merging
• stitch patches together along common boundaries
• displace grid by half a grid cell in each
  direction
• repeat procedure to coarsen patch seams
• features
• fairly easy to implement, higher quality than
  spatial clustering
• drawbacks
• two orders of magnitude slower than spatial
  clustering
• outputs only single level of detail

35
OEMM Adaptive Surface Segmentation Cignoni et
al. 03

• octree-based external memory mesh data structure
• rectilinear octree grid segments surface
• general data structure for out-of-core mesh
  processing
• not only for simplification

figure courtesy of Paolo Cignoni
36
OEMM Overview

• similar approach to Bernardini et al. 02, but
• octree grid adapts in resolution to better match
  surface detail
• siblings merged when triangle count drops below
  limit
• by loading grid cells that neighbor current
  patch, edges can be collapsed across patch
  boundaries
• no need to keep patch boundaries at full
  resolution
• outputs progressive mesh, e.g. for view-dependent
  rendering

37
OEMM Results

• speed
• one-time OEMM construction 10K triangles per
  second
• simplification 10K triangles per second
• quality

38
Error-Driven Surface SegmentationEl-Sana
Chiang 00

• disk-based implementation of edge collapse
• uses B-tree disk data structure to index mesh,
  priority queue
• simplification done in batches by loading pieces
  of mesh
• surface patches given by faces spanned by edges
  to be collapsed

figure courtesy of El-Sana Chiang
39
Error-Driven Surface Segmentation

• algorithm overview
• compute mesh connectivity, initialize priority
  queue on disk
• for each batch of lowest-error edges
• dequeue edges and load incident faces into RAM
• merge faces into larger patches whenever possible
• collapse edges, recompute errors, write back to
  disk
• repeat until desired accuracy/size reached
• features
• entirely error-driven surface segmentation
• drawbacks
• limited scalability patches small when
  input-to-RAM ratio large
• non-trivial error metrics may force out-of-order
  edge collapses

40
Other Surface Segmentation Methods

• Prince 00
• extension of Hoppe 98 terrain simp. to arbitrary
  surfaces
• uses octree partition and edge collapse
• contrary to Cignoni et al. 03, patch boundaries
  frozen
• 1K triangles per second simplification speed
  reported
• Choudhury Watson 02
• simplification in reverse (i.e. refinement from
  coarse to fine)
• segments surface into face clusters via binary
  splitting
• face clusters grouped together on disk
• increased refinement yields increased locality of
  reference
• hence virtual memory disk access possible with
  little thrashing

41
Talk Overview

• introduction
• spatial clustering
• surface segmentation
• streaming
• summary

42
Streaming

• mesh as a sequential stream of triangles
• read/write one triangle at a time, process mesh
  in single pass
• build up indexed submesh in finite in-core stream
  buffer
• fixed-size sliding window over mesh triangle
  list
• perform conventional in-core processing on
  buffer
• e.g. error-driven edge collapse
• supports most simplification operators
• requires coherent linear layout of mesh triangles
  to ensure buffer contains sufficiently large and
  connected mesh patches
• most large meshes already fairly coherent,
  otherwise
• geometric (external) sort
• topological sort/region growing (e.g. Isenburg
  Gumhold 03)

43
Stream Buffer Properties

• stream boundary
• one or more loops of edges that divide mesh into
  processed and unprocessed regions
• indexed triangles read frominput boundary,
  written to output boundary (may be the same)
• stream boundary sweepscontinuouslyover mesh
• therefore no stitching needed,unlike
  conventional surface segmentation

processed region
bufferedregion
output boundary
input boundary
unprocessed region
44
Advancing Stream Boundary
45
Stream-Based Spatial ClusteringIsenburg et al.
03

• algorithm overview
• spatial clustering on uniform grid (as in OoCS)
• compressed mesh stream provides geometry,
  connectivity
• maintain only cells intersected by stream
  boundary
• compute and output cluster vertex when boundary
  exits cell
• create new cluster if boundary re-enters cell
• maintain small temporary in-corebuffer of
  surviving triangles adjacentto input boundary
• output triangle as soon as its verticeshave been
  output
• collapse only edge-connected layers
• clustering reduced to edge collapse,i.e.
  collapse all edges whose verticesfall in the
  same cell

46
Stream-Based Spatial Clustering

• algorithm characteristics
• features
• streams input and output through small in-core
  buffer
• connectivity information available, therefore
  quality improved
• surface boundaries preserved
• edge-connected clustering reduces pinching and
  other non-manifold cases
• drawbacks
• cluster grid not adaptive
• but easily could be, e.g. as in Shaffer Garland
  01
• non-manifolds still possible
• as in unconstrained edge collapse methods

47
Stream-Based Spatial Clustering
ResultsTopological Quality
OoCS 3,366 of 33,053 vertices non-manifold
Stream-Based 897 of 35,134 vertices non-manifold
48
Stream-Based Spatial Clustering ResultsResource
Usage

• memory usage
• simplifying St. Matthew from 373M to 23M
  triangles
• OoCS 3,282 MB of RAM
• stream-based clustering 121 MB (27 times less)
• simplification speed (compressed mesh input)
• OoCS 67 minutes
• stream-based clustering 83 minutes
• overhead due to maintaining per-cell cluster trees

49
Stream-Based Edge CollapseWu Kobbelt 03

• algorithm overview
• read coherent triangle soup
• reconstruct connectivity in-core usinggeometric
  hashing on vertex coordinates
• use fixed-size in-core triangle buffer
• interleave batches of read, decimate, and write
  operations
• collapse edges not adjacent to input/output
  stream boundaries
• avoid priority queues using multiple choice
  selection
• choose best of N randomly selected candidates

figure courtesy of Leif Kobbelt
50
Stream-Based Edge Collapse

• algorithm characteristics
• features
• out-of-core error-driven edge collapse with no
  stitching artifacts
• fast 30K triangles per second
• easy to implement
• can specify exact target size or error tolerance
• fixed target size implies nearly uniform
  coarsening
• drawbacks
• geometric hashing adds overhead
• cannot distinguish stream boundary from surface
  boundary
• surface boundaries not simplified until entire
  mesh read
• may clog stream buffer

51
Stream-Based Edge CollapseIsenburg et al. 03

• algorithm overview
• extension of Wu Kobbelt 03
• reads compressed mesh instead of triangle soup
• no geometric hashing needed ? faster execution
• 50-70K triangles per second on this 2 GHz laptop
• stream and surface boundariesdistinguished ?
  less memory and/orhigher quality
• 100 MB to simplify St. Matthew
• output triangle order chosen topreserve mesh
  coherence forsubsequent stream-based processing

52
Talk Overview

• introduction
• spatial clustering
• surface segmentation
• streaming
• summary

53
Summary of Simplification Methods

• spatial clustering
• clusters vertices based on spatial proximity
• adaptive space partitioning possible
• features
• very fast (50400K triangles per second)
• simple to implement
• drawbacks
• low quality results
• topology not preserved, non-manifold simplices
  possible
• use when
• time or space is at a premium
• topology simplification is desired

54
Summary (continued)

• surface segmentation
• clusters vertices primarily based on error
• iterative simplification lends itself to
  progressive mesh output
• features
• high quality results
• can preserve topology
• drawbacks
• rather slow (1-10K triangles per second)
• must implement iterative method plus external
  memory support
• use when
• quality is more important than speed
• progressive streaming or view-dependent rendering
  is needed

55
Summary (continued)

• streaming
• clusters vertices either based on spatial or
  error-based criteria
• features
• mesh quality governed by size of user-specified
  stream buffer
• low memory requirements
• weakly input sensitive O(?n)
• fast and highly parallelizable via pipelining
• delay between input and output small
• single-pass, linear time complexity
• works well in tandem with other out-of-core
  processing,e.g. compression, denoising,
  analysis,
• drawbacks
• not obvious how to process multiresolution meshes
  (progressive streaming)

56
Future Work

• improved spatial clustering methods
• metric spaces other than Euclidean
• cluster based on surface geometry (normals,
  curvature)
• hybrid methods
• Garland Shaffer 02
• first-phase spatial clustering using quadric
  error metric
• pass quadric information to second-phase edge
  collapse
• efficient construction of stream-based
  representations
• alternatives to conventional simplification
• out-of-core remeshing and similar techniques

57
Acknowledgements

• thanks to
• the LLNL ASCI VIEWS team for datasets and support
• the Digital Michelangelo Project for sharing
  their datasets
• Yi-Jen Chiang, Paolo Cignoni, Jihad El-Sana,
  Michael Garland, Hugues Hoppe, and Leif Kobbelt
  for contributing figures and images
• Eric Shaffer for providing simplified meshes

58
References

• F. Bernardini et al. Building a Digital Model
  of Michelangelos Florentine Pietà. Computer
  Graphics and Applications 22(1) 2002, pp. 59-67.
• P. Choudhury B. Watson. Completely Adaptive
  Simplification of Massive Meshes. Northwestern
  University tech. report CS-02-09, 2002.
• P. Cignoni et al. External Memory Management
  and Simplification of Huge Meshes. Transactions
  on Vis. and Computer Graphics 9(4) 2003, pp.
  525-537.
• J. El-Sana Y.-J. Chiang. External Memory
  View-Dependent Simplification. Computer
  Graphics Forum 19(3) 2000, pp. 139-150.
• Fei et al. An Adaptive Sampling Scheme for
  Out-of-Core Simplification. Computer Graphics
  Forum 21(2) 2002, pp. 111-119.
• Fei et al. Detail Calibration for Out-of-Core
  Model Simplification through Interlaced
  Sampling. SIGGRAPH 2002 tech. sketch, p. 166.
• M. Garland E. Shaffer. A Multiphase Approach
  to Efficient Surface Simplification.
  Visualization 02, pp. 117-124.
• H. Hoppe. Smooth View-Dependent Level-of-Detail
  Control and its Application to Terrain
  Rendering. Visualization 98, pp. 35-42.

59
References (continued)

• M. Isenburg et al. Large Mesh Simplification
  using Processing Sequences. Visualization 03,
  to appear.
• P. Lindstrom. Out-of-Core Simplification of
  Large Polygonal Models. SIGGRAPH 2000, pp.
  259-262.
• P. Lindstrom C. T. Silva. A Memory
  Insensitive Technique for Large Model
  Simplification. Visualization 01, pp. 121-126.
• C. Prince. Progressive Meshes for Large Models
  of Arbitrary Topology. Masters Thesis,
  University of Washington, 2000.
• J. Wu L. Kobbelt. A Stream Algorithm for the
  Decimation of Massive Meshes. Graphics
  Interface 03, pp. 185-192.