Multi-chart Geometry Images

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# Multi-chart Geometry Images - PowerPoint PPT Presentation

Multi-chart Geometry Images. Pedro Sander Harvard. Zo ë Wood Caltech. Steven Gortler Harvard. John Snyder Microsoft Research. Hugues Hoppe Microsoft Research. Geometry representation. irregular. semi-regular. completely regular. Basic idea. cut. parametrize. Basic idea. cut.

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### Multi-chart Geometry Images

Pedro Sander

Harvard

Zoë Wood

Caltech

Steven Gortler

Harvard

John Snyder

Microsoft Research

Hugues Hoppe

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

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)
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

• 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

Rendering

PSNR 79.5

478x133

Multi-chart results

RenderingPSNR 75.6

genus 1; 40 charts

174x369

Multi-chart results

RenderingPSNR 84.6

genus 0; 25 charts

281X228

Multi-chart results

RenderingPSNR 83.8

genus 0; 15 charts

466x138

Multi-chart results

irregularoriginal

singlechart

PSNR 68.0

multi-chart

PSNR 79.5

478x133

demo

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