Transcript talk

Multi-chart Geometry Images
Pedro Sander
Zoë Wood
Harvard
Caltech
Steven Gortler
John Snyder
Hugues Hoppe
Harvard
Microsoft Research
Microsoft Research
Geometry representation
irregular
semi-regular
completely regular
Basic idea
cut
parametrize
Basic idea
cut
sample
Basic idea
cut
store
simple traversal
to render
[r,g,b] = [x,y,z]
Benefits of regularity
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Simplicity in rendering
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No vertex indirection
No texture coordinate indirection
Hardware potential
Leverage image processing tools for geometric
manipulation
Limitations of single-chart
long extremities
high genus
 Unavoidable distortion and undersampling
Limitations of semi-regular
Base “charts” effectively constrained to be
equal size equilateral triangles
Multi-chart Geometry Images
irregular
400x160
piecewise regular
Multi-chart Geometry Images
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Simple reconstruction rules;
for each 2-by-2 quad of MCGIM samples:
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3 defined samples  render 1 triangle
4 defined samples  render 2 triangles
(using shortest diagonal)
undefined
defined
Multi-chart Geometry Images
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Simple reconstruction rules;
for each 2-by-2 quad of MCGIM samples:
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3 defined samples  render 1 triangle
4 defined samples  render 2 triangles
(using shortest diagonal)
Cracks in reconstruction
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Challenge: the discrete sampling will cause
cracks in the reconstruction between charts
“zippered”
MCGIM Basic pipeline
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Break mesh into charts
Parameterize charts
Pack the charts
Sample the charts
Zipper chart seams
Optimize the MCGIM
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.
Mesh chartification
Bootstrapping
 Start with single seed
 Run chartification using increasing number of
seeds each phase
 Until desired number reached
demo
Chartification Results
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Produces planar charts with compact boundaries
Sander et. al. 2001
80% stretch efficiency
Our method
99% stretch efficiency
Parameterization
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Goal: Penalizes undersampling
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L2 geometric stretch of Sander et. al. 2001
Hierarchical algorithm for solving minimization
Parameterization
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Goal: Penalizes undersampling
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L2 geometric stretch of Sander et. al. 2001
Hierarchical algorithm for solving minimization
Angle-preserving metric
(Floater)
Chart packing
Goal: minimize wasted space
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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
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pick best
Packing Results
Levy packing
efficiency 58.0%
Our packing
efficiency 75.6%
Sampling into a MCGIM
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Goal: discrete sampling of parameterized charts
into topological discs
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Rasterize triangles with scan conversion
Store geometry
Sampling into a MCGIM
Boundary
rasterization
Non-manifold dilation
Zippering the MCGIM
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Goal: to form a watertight reconstruction
Zippering the MCGIM
Algorithm:
Greedy (but robust) approach
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Identify cut-nodes and
cut-path samples.
Unify cut-nodes.
Snap cut-path samples
to geometric cut-path.
Unify cut-path samples.
Zippering: Snap
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Snap
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Snap discrete cut-path samples to
geometrically closest point on cut-path
Zippering: Unify
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Unify
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Greedily unify neighboring samples
How unification works
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Unify
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Test the distance of the next 3 moves
Pick smallest to unify then advance
How unification works
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Unify
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Test the distance of the next 3 moves
Pick smallest to unify then advance
How unification works
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Unify
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Test the distance of the next 3 moves
Pick smallest to unify then advance
Geometry image optimization
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Goal: align discrete samples with mesh features
 Hoppe et. al. 1993
 Reposition vertices to minimize distance to
the original surface
 Constrain connectivity
Multi-chart results
genus 2; 50 charts
478x133
Rendering
PSNR 79.5
Multi-chart results
genus 1; 40 charts
174x369
Rendering
PSNR 75.6
Multi-chart results
genus 0; 25 charts
281X228
Rendering
PSNR 84.6
Multi-chart results
genus 0; 15 charts
466x138
Rendering
PSNR 83.8
Multi-chart results
irregular
original
single
chart
PSNR 68.0
multichart
PSNR 79.5
demo
478x133
Comparison to semi-regular
Original irregular
Semi-regular
MCGIM
Comparison to semi-regular
Original irregular mesh
Semi-regular mesh
PSNR 87.8
MCGIM mesh
PSNR 90.2
Summary
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Contributions:
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Overall: MCGIM representation
– Rendering simplicity
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Major: zippering and optimization
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Minor: packing and chartification
Future work
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Provide:
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Compression
Level-of-detail rendering control
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Exploit rendering simplicity in hardware
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Improve zippering