Intrinsic Parameterization for Surface Meshes

1. Intrinsic Parameterization for Surface Meshes Mathieu Desbrun, Mark Meyer, Pierre Alliez CS598MJG Presented by Wei-Wen Feng 2004/10/5

2. What’s Parameterization? • Find a mapping between original surface and a target domain ( Planar in general )

3. What does it do? • Most significant : Texture Mapping • Other applications include remeshing, morphing, etc.

4. Two Directions in Research • Define metric (energy) measuring distortion • Minimize the energy to find mapping • This paper’s main contribution

5. Two Directions in Research • Using the metric, and make it work on mesh • Cut mesh into patches • Considering arbitrary genus

6. Outline • Previous Work • Intrinsic Properties • DCP & DAP • Boundary Control • Future Work

7. Previous Work • Discrete Harmonic Map (Eck. 95): • Minimize Eharm[h] = ½ ΣKi,j |h(i) – h(j)|2 • K : Spring constant • The same as minimize Dirichlet energy

8. Previous Work • Shape Preserving Param. (Floater. 97): • Represent vertex as convex combination of neigobors • Trivial choice : barycenter of neighbors • Ensure valid embedding

9. Previous Work • Most Isometric Param. (MIPS) (K. Hormann . 99): • Doesn’t need to fix boundary • Conformal but need to minimize non-linear energy MIPS Harmonic Map

10. Previous Work • Signal Specialized Param. (Sander. 02): • Minimize signal stretch on the surface when reconstruct from parametrization

11. Intrinsic Parameterization • Motivation: • Find good distortion measure only depending on the intrinsic properties of mesh • Develop good tools for fast parameterization design

12. Intrinsic Properties • Defined at discrete suraface, restricted at 1-ring • Notion: • F : Return the “score” of surface patch M • E(M,U) : Distortion between mapping • Intrinsic Properties: • Rotation & Translation Invariance • Continuity : Converge to continuous surface • Additivity : f (A) + f (B) = f (AB) + f (AB)

13. Intrinsic Properties • Minkowski Functional • fA = Area • fc = Euler characteristic • fP = Perimeter • From Hadwiger, the only admissible intrinsic functional is : • f = a fA + b fc+ c fP

14. Discrete Conformal Param. • Measure of Area (Dirichlet Energy) • Conformality is attained when Dirichlet energy is minimum • When fixed boundary, it is in fact discrete harmonic map

15. Discrete Authalic Param. • Measure of Euler characteristic (Angle) • Integral of Gaussian curvature • Derived as Chi Energy

16. Comparing DCP & DAP • DCP (Dirichlet Energy) • Measure area extension • Minimized when angles preserved • DAP (Chi Energy) • Measure angle excess • Minimized when area preserved

17. Solving Parametrization • General distortion measure : • Fix the boundary, minimized the energy : • Very sparse linear systems  Conjugate gradient

18. Natural Boundary • Instead fixed the boundary, solve for optimal conformal mapping which yields “best” boundary. • For interior points • For boundary points : • Constrain two points to avoid degeneracy.

19. Compare with LSCM • Least Square Conformal Map (Levy. ’02) • Start from Cauchy-Riemann Equation • Theoretically equivalent to Natural Boundary Map • Minimize conformal energy • Natural Conformal Map • Imposing boundary constraint for boundary points

20. Extend to non-linear func. • All parametrization could be expressed as : • U = l UA + (1-l) Uc • Substitute U in a non-linear function reduces the problem into solving l • Ex : • Could be reduced into root finding

21. Boundary Control • Precompute the “impulse response” parameterization for each boundary points • New parameterization could be obtained by projecting boundary parameter onto its “impulse response” parameterization

22. Boundary Optimization • Minimized arbitrary energy with respect to boundary parameterization • Using precomputed gradient to accelerate optimization

23. Summary of Contributions • A linear system solution for Natural Conformal Map • A new geometric metric for parameterization (DAP) • Real-time boundary control for better parameterization design

24. What’s Next ? • Mean Value Coordinate (Floater. 03) • The same property of convex combination • Approximating Harmonic Map but ensure a valid embedding Tutte Harmonic Shape Preserving Mean Value

25. What’s Next ? • Spherical Parameterization (Praun. 03) • Smooth parameterization for genus-0 model • Using existing metric

26. Conclusion • There seems to be less paper directly about finding metrics (or find a better way to model them) for parameterization. • Now more efforts in finding globally smooth parameterization on arbitrary meshes

27. Thank You

28. References • (Eck. 95) Multiresolution Analysis of Arbitrary Meshes. Proceedings of SIGGRAPH 95\ • (Floater. 97) Parametrization and Smooth Approximationof Surface Triangulations. Computer Aided Geometric Design 14, 3 (1997) • (K. Hormann . 99) MIPS: An Efficient Global Parametrization Method. In Curve and Surface Design: Saint-Malo 1999 (2000) • (Sander. 02) Signal-Specialized Parameterization. In Eurographics Workshop on Rendering, 2002.

29. References • (Floater, Hormann 03) Surface Parameterization : A Tutorial and Survey • (Levy. ’02) Least Squares Conformal Maps for Automatic Texture Atlas Generation. ACM SIGGRAPH Proceedings • (Floater. 03) Mean Value Coordinates. Computer Aided Geometric Design 20, 2003