Isocharts: Stretchdriven Mesh Parameterization using Spectral Analysis

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Iso-charts Stretch-driven Mesh Parameterization
using Spectral Analysis
Kun Zhou, John Snyder, Baining Guo,
Heung-Yeung Shum Microsoft Research
Asia Microsoft Research
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Parameterizing Arbitrary 3D Meshes
Chartification
Texture Atlas
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Goals of Mesh Parameterization
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Iso-chart Algorithm Overview
Input 3D mesh, user-specified stretch threshold

• Surface spectral analysis

• Stretch optimization

• Surface spectral clustering

• Optimize chart boundaries

• Recursively split charts
• until stretch criterion is met

Output atlas having large charts with bounded
stretch
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IsoMap
Tenenbaum et al, 2000
Data points in high dimensional space
Data points in low dimensional space
Neighborhood graph
Analyze geodesic distance to uncover nonlinear
manifold structure
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Surface Spectral Analysis
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Surface Spectral Analysis
Construct matrix of squared geodesic distances DN
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Surface Spectral Analysis
Perform centering and normalization to DN
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Surface Spectral Analysis
Perform eigenanalysis on BN to get embedding
coords yi
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GDD-minimizing Parameterization
Zigelman et al, 2002
Parametric coordinates
Texture mapping

• Produces triangle flips

• Only handles single-chart (disk-topology) models

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Stretch-minimizing Parameterization
Sander et al, 2001
2D texture domain
surface in 3D
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Stretch Optimization
Sander01, L2 1.04, 222s
Sander02, L2 1.03, 39s
IsoMap, L2 1.04, 2s
IsoMapOptimization, L2 1.03, 6s
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Surface Spectral Clustering
Analysis
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Surface Spectral Clustering

• Get top n ( 3) eigenvalues/eigenvectors
• where n maximizes
• For each vertex
• compute n-dimensional embedding coordinates
• For each of the n dimensions
• find two extreme vertices
• set them as representatives
• Remove representatives that are too close
• Grow charts from representatives

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Surface Spectral Clustering
n4
n3
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Surface Spectral Clustering
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Optimizing Partition Boundaries

• create nonjaggy cut, through crease edges
  Katz2003

• minimize embedding distortion

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Optimizing Partition Boundaries
Angular capacity alone Katz et al, 2003
Distortion capacity alone
Combined capacity
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Special Spectral Clustering

• Avoid excessive partition for simple shapes

• Special clustering for tabular shapes

n 2 1st dimension
n 2 2nd dimension
n 2 3rd dimension
n gt 2
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Signal-Specialized Atlas Creation

• Signal-specialized parameterization Sander02

geometry stretch
signal stretch

• Combine geodesic and signal distances

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Implementation Details

• Acceleration
• Landmark IsoMap Silva et al, 2003
• Only compute the top 10 eigenvalues

• Merge small charts as a post-process

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Partition Process
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Results
19 charts, L21.03, running time 98s, 97k faces
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Results
38 charts, L21.07, running time 287s, 150k faces
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Results
23 charts, L21.06, running time 162s, 112k faces
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Results
11 charts, L21.01, running time 4s, 10k faces
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Results
11 charts, L21.02, running time 90s, 90k faces
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Results
6 charts, L21.03, running time 17s, 40k faces
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Geometry Remeshing
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Remeshing Comparison
Sander03, 79.5dB
Iso-chart, 82.9dB
Original model
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LOD Generation for Texture Synthesis
32x32
64x64
128x128
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Texture Synthesis Results
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Texture Synthesis Results
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Signal-Specialized Atlas Creation
Original
Geometry stretch SAE 20.8
Signal param SAE 17.9
Signal chartparam SAE 16.5
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Signal-Specialized Atlas Creation
Original
Geometry stretch SAE 18.7
Signal param SAE 11.5
Signal chartparam SAE 9.7
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Conclusion

• Iso-chart a fast and effective atlas generator

• Surface spectral analysis
• for parameterization
• provides good starting point for stretch
  minimization
• for chartification
• separates global features well
• optimizes chart boundaries
• yields special partition for tubular shapes

• Signal-specialized atlas creation