Intrinsic Parameterization for Surface Meshes

Mathieu Desbrun, Mark Meyer, Pierre Alliez

CS598MJG Presented by Wei-Wen Feng

2004/10/5

What’s Parameterization?

Find a mapping between original surface and a target domain ( Planar in general )

What does it do?

Most significant : Texture Mapping Other applications include remeshing, morphing, etc.

Two Directions in Research

Define metric (energy) measuring distortion Minimize the energy to find mapping This paper’s main contribution

Two Directions in Research

Using the metric, and make it work on mesh Cut mesh into patches Considering arbitrary genus

Outline

Previous Work Intrinsic Properties DCP & DAP Boundary Control Future Work

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

Previous Work Shape Preserving Param. (Floater. 97):

Represent vertex as convex combination of neigobors Trivial choice : barycenter of neighbors Ensure valid embedding

Previous Work

Most Isometric Param. (MIPS) (K. Hormann . 99): Doesn’t need to fix boundary Conformal but need to minimize non-linear energ

y

MIPS Harmonic Map

Previous Work

Signal Specialized Param. (Sander. 02): Minimize signal stre

tch on the surface when reconstruct from parametrization

Intrinsic Parameterization

Motivation: Find good distortion measure only depending on

the intrinsic properties of mesh

Develop good tools for fast parameterization design

Intrinsic Properties

Defined at discrete suraface, restricted at 1-ring Notion:

Return the “score” of surface patch M E(M,U) : Distortion between mapping

Intrinsic Properties: Rotation & Translation Invariance Continuity : Converge to continuous surface Additivity : (A) + (B) = (AB) + (AB)

Intrinsic Properties

Minkowski Functional A = Area = Euler characteristic P = Perimeter

From Hadwiger, the only admissible intrinsic functional is : aA + b+ c P

Discrete Conformal Param.

0))(cot(cot)(

iNj

jiijiji

AE uuu

Measure of Area (Dirichlet Energy)

Conformality is attained when Dirichlet energy is minimum When fixed boundary, it is in fact discrete harmonic map

Discrete Authalic Param.

Measure of Euler characteristic (Angle) Integral of Gaussian curvature Derived as Chi Energy

0))(()(

2||

cotcot

iNj

jii

ji

ijijEuu

u xx

Comparing DCP & DAP

DCP (Dirichlet Energy) Measure area extension Minimized when angles preserved

DAP (Chi Energy) Measure angle excess Minimized when area preserved

Solving Parametrization

General distortion measure :

Fix the boundary, minimized the energy : Very sparse linear systems Conjugate gradient

EEE A

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.

A DΕ

0D Ε

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

AC DΕΕ

A DΕ

Extend to non-linear func.

All parametrization could be expressed as : U = UA + (1-) U

Substitute U in a non-linear function reduces the problem into solving

Ex :

Could be reduced into root finding

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

Boundary Optimization

Minimized arbitrary energy with respect to boundary parameterization

Using precomputed gradient to accelerate optimization

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

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

What’s Next ?

Spherical Parameterization (Praun. 03) Smooth parameterization for genus-0 model Using existing metric

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

Thank You

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.

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