Multichart Geometry Images

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Multi-chart Geometry Images

• Pedro Sander
• Harvard

Zoë Wood Caltech
Hugues Hoppe Microsoft Research
Steven Gortler Harvard
John Snyder Microsoft Research
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Geometry representation
semi-regular
irregular
completely regular
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Basic idea
cut
parametrize
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Basic idea
cut
sample
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Basic idea
cut
store
simple traversal to render
r,g,b x,y,z
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Benefits of regularity

• Simplicity in rendering
• No vertex indirection
• No texture coordinate indirection
• Hardware potential
• Leverage image processing tools for geometric
  manipulation

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Limitations of single-chart
long extremities
high genus
? Unavoidable distortion and undersampling
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Limitations of semi-regular
Base charts effectively constrained to be
equal size equilateral triangles
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Multi-chart Geometry Images
piecewise regular
400x160
irregular
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Multi-chart Geometry Images

• Simple reconstruction rulesfor each 2-by-2 quad
  of MCGIM samples
• 3 defined samples ? render 1 triangle
• 4 defined samples ? render 2 triangles
  (using shortest diagonal)

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Multi-chart Geometry Images

• Simple reconstruction rulesfor each 2-by-2 quad
  of MCGIM samples
• 3 defined samples ? render 1 triangle
• 4 defined samples ? render 2 triangles
  (using shortest diagonal)

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Cracks in reconstruction

• Challenge the discrete sampling will cause
  cracks in the reconstruction between charts

zippered
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MCGIM Basic pipeline

• Break mesh into charts
• Parameterize charts
• Pack the charts
• Sample the charts
• Zipper chart seams
• Optimize the MCGIM

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Mesh chartification

• Goal planar charts with compact boundaries
• Clustering optimization - Lloyd-Max (Shlafman
  2002)
• Iteratively grow chart from given seed
  face.(metric is a product of distance and
  normal)
• Compute new seed face for each chart.(face that
  is farthest from chart boundary)
• Repeat above steps until convergence.

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Mesh chartification

• Bootstrapping
• Start with single seed
• Run chartification using increasing number of
  seeds each phase
• Until desired number reached

demo
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Chartification Results

• Produces planar charts with compact boundaries

Sander et. al. 2001 80 stretch efficiency
Our method 99 stretch efficiency
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Parameterization

• Goal Penalizes undersampling
• L2 geometric stretch of Sander et. al. 2001
• Hierarchical algorithm for solving minimization

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Parameterization

• Goal Penalizes undersampling
• L2 geometric stretch of Sander et. al. 2001
• Hierarchical algorithm for solving minimization

Angle-preserving metric (Floater)
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Chart packing

• Goal minimize wasted space
• Based on Levy et al. 2002
• Place a chart at a time (from largest to
  smallest)
• Pick best position and rotation (minimize
  wasted space)
• Repeat above for multiple MCGIM rectangle shapes
• pick best

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Packing Results
Levy packing efficiency 58.0
Our packing efficiency 75.6
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Sampling into a MCGIM

• Goal discrete sampling of parameterized charts
  into topological discs
• Rasterize triangles with scan conversion
• Store geometry

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Sampling into a MCGIM
Boundary rasterization
Non-manifold dilation
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Zippering the MCGIM

• Goal to form a watertight reconstruction

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Zippering the MCGIM

• Algorithm Greedy (but robust) approach
• Identify cut-nodes and cut-path samples.
• Unify cut-nodes.
• Snap cut-path samples to geometric cut-path.
• Unify cut-path samples.

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Zippering Snap

• Snap
• Snap discrete cut-path samples to geometrically
  closest point on cut-path

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Zippering Unify

• Unify
• Greedily unify neighboring samples

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How unification works

• Unify
• Test the distance of the next 3 moves
• Pick smallest to unify then advance

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How unification works

• Unify
• Test the distance of the next 3 moves
• Pick smallest to unify then advance

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How unification works

• Unify
• Test the distance of the next 3 moves
• Pick smallest to unify then advance

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Geometry image optimization

• Goal align discrete samples with mesh features
• Hoppe et. al. 1993
• Reposition vertices to minimize distance to
  the original surface
• Constrain connectivity

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Multi-chart results
genus 2 50 charts
478x133
Rendering PSNR 79.5
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Multi-chart results
RenderingPSNR 75.6
genus 1 40 charts
174x369
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Multi-chart results
RenderingPSNR 84.6
genus 0 25 charts
281X228
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Multi-chart results
RenderingPSNR 83.8
genus 0 15 charts
466x138
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Multi-chart results
irregularoriginal
singlechart PSNR 68.0
multi-chart PSNR 79.5
478x133
demo
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Comparison to semi-regular
Original irregular
Semi-regular
MCGIM
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Comparison to semi-regular
Original irregular mesh
Semi-regular mesh PSNR 87.8
MCGIM mesh PSNR 90.2
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Summary

• Contributions
• Overall MCGIM representation
• Rendering simplicity
• Major zippering and optimization
• Minor packing and chartification

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Future work

• Provide
• Compression
• Level-of-detail rendering control
• Exploit rendering simplicity in hardware
• Improve zippering