Errorbounded Simplification of Triangle Meshes with Multivariate Functions

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Error-bounded Simplification of Triangle Meshes
with Multivariate Functions

• Daniel R. Schikore
• Chandrajit L. Bajaj

Shastra Lab and Center for Data
Visualization Purdue University West Lafayette,
IN 47907
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Outline

• Background
• Problem Definition
• Related Work
• Outline of Algorithm
• Results
• Additional Applications
• Future Work

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Interrogative Visualization
Data Synthesis Computation
Workstation
Workstation
Servers
Display, Querying Analysis
Domain Data Acquisition
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Mesh Simplification

• Input Mesh Functions defined on mesh Bounds
  on error in geometry and functions
• Output Simplified mesh within specified error
  constraints

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Problem Definition

• Original mesh Simplified
  approximation

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Problem Definition

• Original mesh Simplified
  approximation

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Related Work

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

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Point Deletion/Insertion
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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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Vertex Classification
Interior Vertex

• Edge Vertex

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Mapping Between Triangulations

• Projection of New
• Triangulation

Interior Vertex
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Mapping Between Triangulations

• Projection of New
• Triangulation

Edge Vertex
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Error Propagation

• Projection of New
• Triangulation

Segmentation
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Error Representation

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

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Updating Errors
Vj
p
Vi
Vj
p
Vi

• Error bound in Original Triangulation
• Cumulative Error Bound to Edge from Vi to Vj

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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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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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Pion Collision
12 functions, 51 timesteps
3 Error (60-85 reduced)
Data Courtesy Lawrence Livermore National
Laboratory
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Heart MRI
Original Data (256x256)
6 Error (85 reduced, 19460 tri)
Data Courtesy Tsuyoshi Yamamoto and Hiroyuki
Fukuda, Hokkaido University
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Austin DEM
Original Data (1201x1201)
6 Error (96 reduced, 125K tri)
Data Courtesy US Geological Survey
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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 Courtesy Space Science and Engineering Center
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Earth Science Data
Original Data (4846 tri)
3 Error (2384 tri)
Data Courtesy Space Science and Engineering Center
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Isosurface of Visible Woman
Original Data (275,048 tri)
3 Error (24,211 tri)
Data Courtesy National Library of Medicine
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Application Scattered Data Fitting
Triangle mesh
Points
Reduced mesh
Smooth model
3D Delaunay tri. and ??solid
Mesh reduction
A-patch fitting
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Phase 1 ?-solid
Build an interpolating triangle mesh from the
points. Technique inspired by ??shapes??Edelsbrunn
er?????and 3D Delaunay triangulation sculpturing
Boissonnat 84

• Three steps
• Compute the 3D Delaunay triangulation
• Extract a subset of tetrahedra whose boundary is
  a closed two-manifold that best approximates the
  points
• Remove additional tetrahedra until all points on
  boundary

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Phase 2 Mesh reduction

• Based on incremental deletion of vertices and
  retriangulation
• Guaranteed, global error-bound
• Sharp feature recognition and preservation

• Mesh reduction technique for triangle meshes with
  multivariate data Bajaj, Schikore 96

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Phase 3 Surface Fitting

• Uses cubic A-patches (algebraic patches)
• C1 continuity

• Sharp features (corners, sharp curved edges)
• Singularities

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Example
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Example
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Conclusions

• Fast Error-bounded Simplification of Planar
  and Surface Meshes
• Multiple Scalar Functions

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Scalar Field Topology
Topology of wind speed in a climate model
simulation
Topology of a scalar field
Topology of a mathematical function reveals
information hidden in contour display
Close-up of the above
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Future Work

• Improved measures of error (Linear/ Quadratic
  fitting)
• Better initial triangulation
• Simulated annealing in edge flipping and
  vertex removal
• Different norms for error vector
• Time domain simplification
• Surface normal consideration