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
Simplicity in rendering
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
Simple reconstruction rules;
for each 2-by-2 quad of MCGIM samples:
3 defined samples render 1 triangle
4 defined samples render 2 triangles
(using shortest diagonal)
undefined
defined
Multi-chart Geometry Images
Simple reconstruction rules;
for each 2-by-2 quad of MCGIM samples:
3 defined samples render 1 triangle
4 defined samples render 2 triangles
(using shortest diagonal)
Cracks in reconstruction
Challenge: the discrete sampling will cause
cracks in the reconstruction between charts
“zippered”
MCGIM Basic pipeline
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
Produces planar charts with compact boundaries
Sander et. al. 2001
80% stretch efficiency
Our method
99% stretch efficiency
Parameterization
Goal: Penalizes undersampling
L2 geometric stretch of Sander et. al. 2001
Hierarchical algorithm for solving minimization
Parameterization
Goal: Penalizes undersampling
L2 geometric stretch of Sander et. al. 2001
Hierarchical algorithm for solving minimization
Angle-preserving metric
(Floater)
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
Packing Results
Levy packing
efficiency 58.0%
Our packing
efficiency 75.6%
Sampling into a MCGIM
Goal: discrete sampling of parameterized charts
into topological discs
Rasterize triangles with scan conversion
Store geometry
Sampling into a MCGIM
Boundary
rasterization
Non-manifold dilation
Zippering the MCGIM
Goal: to form a watertight reconstruction
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.
Zippering: Snap
Snap
Snap discrete cut-path samples to
geometrically closest point on cut-path
Zippering: Unify
Unify
Greedily unify neighboring samples
How unification works
Unify
Test the distance of the next 3 moves
Pick smallest to unify then advance
How unification works
Unify
Test the distance of the next 3 moves
Pick smallest to unify then advance
How unification works
Unify
Test the distance of the next 3 moves
Pick smallest to unify then advance
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
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
Contributions:
Overall: MCGIM representation
– Rendering simplicity
Major: zippering and optimization
Minor: packing and chartification
Future work
Provide:
Compression
Level-of-detail rendering control
Exploit rendering simplicity in hardware
Improve zippering