Errorbounded Reduction of Triangle Meshes with Multivariate Data

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Error-bounded Reduction of Triangle Meshes with
Multivariate Data

• Chandrajit L. Bajaj
• Daniel R. Schikore

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Outline

• Problem Definition
• Related Work
• Motivation
• Outline of Algorithm
• Results
• Future Work

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Problem Definition

• Geometric surface defined by triangles
• Multivariate data defined at vertices
• Compute a reduced size mesh for display which
  adapts to both the geometry and the data defined

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Related Work

• Geometric Reduction - Point/Edge/Triangle
  Deletion (Hamann, Rossignac et al., Schroeder
  et al.) Energy Optimization (Hoppe et al.,
  Turk) Resampling (He et al.) Remeshing (Eck et
  al.)
• Data Reduction - Point Insertion/Deletion
  (DeFloriani et al., Fowler et al., Lee, Puppo
  et. al, Silva et al., Tsai, Ware et al.)

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Motivation

• Data is often defined on surfaces
• Existing mesh reduction methods satisfy error
  criteria in geometry or data, but not both.

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Algorithm Outline

• Consider each vertex v for deletion
• Compute a triangulation of the neighbors of
  vertex v
• Determine mapping between triangulations
• Iteratively attempt to improve triangulation
  (reduce error) by edge-flipping
• Delete v if error criteria are satisfied

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Mapping Between Triangulations

• Interior Projection of New
  Triangulation

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Error Representation

• Two error values per face - Easy to update
  - Combines information about all deleted
  vertices

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Computing Geometric Error

• Geometric error is quantified by the signed
  distance from the old triangulation to the new
  triangulation
• Displacement in the direction of normal is
  positive error by convention

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Computing Data Errors

• Errors in data are quantified by the signed
  difference between interpolated data values in
  the old and new triangulations

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Error Propagation

• Projection of New Triangulation
  Segmentation

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Retriangulation

• Initial triangulation is arbitrary
• Cost of an edge may be computed as a) Maximum
  error in geometry or data b) Maximum error
  introduced in geometry or data
• Edges are flipped if cost of the flipped edge is
  lower than the cost of the current edge

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Results
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Projectile Impact
Original Data (2869 tri)
3 Error (1356 tri)
Data Courtesy Lawrence Livermore National
Laboratory
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Projectile Impact
Original Data (2869 tri)
3 Error (1356 tri)
Data Courtesy Lawrence Livermore National
Laboratory
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Earth Science Data
Original Data (4846 tri)
3 Error (2384 tri)
Data Courtesty Space Science and Engineering
Center
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Earth Science Data
Original Data (4846 tri)
3 Error (2384 tri)
Data Courtesty Space Science and Engineering
Center
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Future Work

• Improved measures of error (Linear/Quadratic
  fitting)
• Better initial triangulation
• Improved edge-flipping for optimization

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For Further Information

• Daniel R. Schikore
• drs_at_cs.purdue.edu
• Chandrajit L. Bajaj
• bajaj_at_cs.purdue.edu
• http//www.cs.purdue.edu/research/shastra/shastra.
  html