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Multi-chart Geometry Images

This paper introduces a multi-chart representation called Multi-chart Geometry Images (MCGIM) that simplifies rendering and zippering for irregular meshes. It presents a pipeline for breaking meshes into charts, parameterizing and packing the charts, and sampling and zippering the charts to create a watertight reconstruction. The proposed method improves stretch efficiency, packing efficiency, and sampling accuracy compared to semi-regular approaches. The results demonstrate the effectiveness of MCGIM for various genus and number of charts.

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Multi-chart Geometry Images

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  1. Multi-chart Geometry Images Pedro Sander Harvard Zoë Wood Caltech Steven Gortler Harvard John Snyder Microsoft Research Hugues Hoppe Microsoft Research

  2. Geometry representation irregular semi-regular completely regular

  3. Basic idea cut parametrize

  4. Basic idea cut sample

  5. Basic idea cut store simple traversal to render [r,g,b] = [x,y,z]

  6. Benefits of regularity • Simplicity in rendering • No vertex indirection • No texture coordinate indirection • Hardware potential • Leverage image processing tools for geometric manipulation

  7. Limitations of single-chart long extremities high genus  Unavoidable distortion and undersampling

  8. Limitations of semi-regular Base “charts” effectively constrained to be equal size equilateral triangles

  9. Multi-chart Geometry Images irregular 400x160 piecewise regular

  10. 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)

  11. 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)

  12. Cracks in reconstruction • Challenge: the discrete sampling will cause cracks in the reconstruction between charts “zippered”

  13. MCGIM Basic pipeline • Break mesh into charts • Parameterize charts • Pack the charts • Sample the charts • Zipper chart seams • Optimize the MCGIM

  14. 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.

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

  16. Chartification Results • Produces planar charts with compact boundaries Sander et. al. 2001 80% stretch efficiency Our method 99% stretch efficiency

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

  18. Parameterization • Goal: Penalizes undersampling • L2 geometric stretch of Sander et. al. 2001 • Hierarchical algorithm for solving minimization Angle-preserving metric (Floater)

  19. 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

  20. Packing Results Levy packing efficiency 58.0% Our packing efficiency 75.6%

  21. Sampling into a MCGIM • Goal: discrete sampling of parameterized charts into topological discs • Rasterize triangles with scan conversion • Store geometry

  22. Sampling into a MCGIM Boundary rasterization Non-manifold dilation

  23. Zippering the MCGIM • Goal: to form a watertight reconstruction

  24. 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.

  25. Zippering: Snap • Snap • Snap discrete cut-path samples to geometrically closest point on cut-path

  26. Zippering: Unify • Unify • Greedily unify neighboring samples

  27. How unification works • Unify • Test the distance of the next 3 moves • Pick smallest to unify then advance

  28. How unification works • Unify • Test the distance of the next 3 moves • Pick smallest to unify then advance

  29. How unification works • Unify • Test the distance of the next 3 moves • Pick smallest to unify then advance

  30. 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

  31. Multi-chart results genus 2; 50 charts Rendering PSNR 79.5 478x133

  32. Multi-chart results RenderingPSNR 75.6 genus 1; 40 charts 174x369

  33. Multi-chart results RenderingPSNR 84.6 genus 0; 25 charts 281X228

  34. Multi-chart results RenderingPSNR 83.8 genus 0; 15 charts 466x138

  35. Multi-chart results irregularoriginal singlechart PSNR 68.0 multi-chart PSNR 79.5 478x133 demo

  36. Comparison to semi-regular Original irregular Semi-regular MCGIM

  37. Comparison to semi-regular Original irregular mesh Semi-regular mesh PSNR 87.8 MCGIM mesh PSNR 90.2

  38. Summary • Contributions: • Overall: MCGIM representation • Rendering simplicity • Major: zippering and optimization • Minor: packing and chartification

  39. Future work • Provide: • Compression • Level-of-detail rendering control • Exploit rendering simplicity in hardware • Improve zippering

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