Intrinsic Parameterization for Surface Meshes Mathieu Desbrun, Mark Meyer, Pierre Alliez CS 598 MJG Presented by Wei-Wen Feng 2004/10/5
What’s Parameterization? l Find a mapping between original surface and a target domain ( Planar in general )
What does it do? l l Most significant : Texture Mapping Other applications include remeshing, morphing, etc.
Two Directions in Research l Define metric (energy) measuring distortion l Minimize the energy to find mapping l This paper’s main contribution
Two Directions in Research l Using the metric, and make it work on mesh l Cut mesh into patches l Considering arbitrary genus
Outline l l l Previous Work Intrinsic Properties DCP & DAP Boundary Control Future Work
![Previous Work l Discrete Harmonic Map (Eck. 95): l Minimize Eharm[h] = ½ ΣKi,](https://present5.com/presentation/8c1e79fd6d6cd12eb2ec033cab25c9b4/image-7.jpg "Previous Work l Discrete Harmonic Map (Eck. 95): l Minimize Eharm[h] = ½ ΣKi,")
Previous Work l Discrete Harmonic Map (Eck. 95): l Minimize Eharm[h] = ½ ΣKi, j |h(i) – h(j)|2 l K : Spring constant l The same as minimize Dirichlet energy
Previous Work l Shape Preserving Param. (Floater. 97): l Represent vertex as convex combination of neigobors l Trivial choice : barycenter of neighbors l Ensure valid embedding
Previous Work l Most Isometric Param. (MIPS) (K. Hormann. 99): l Doesn’t need to fix boundary l Conformal but need to minimize non-linear energy MIPS Harmonic Map
Previous Work l Signal Specialized Param. (Sander. 02): l Minimize signal stretch on the surface when reconstruct from parametrization
Intrinsic Parameterization l Motivation: l Find good distortion measure only depending on the intrinsic properties of mesh l Develop good tools for fast parameterization design
Intrinsic Properties l Defined at discrete suraface, restricted at 1 -ring l Notion: l l l F : Return the “score” of surface patch M E(M, U) : Distortion between mapping Intrinsic Properties: l l l Rotation & Translation Invariance Continuity : Converge to continuous surface Additivity : f (A) + f (B) = f (A B) + f (A B)
Intrinsic Properties l Minkowski Functional l l f. A = Area fc = Euler characteristic f. P = Perimeter From Hadwiger, the only admissible intrinsic functional is : l f = a f. A + b fc + c f. P
Discrete Conformal Param. l Measure of Area (Dirichlet Energy) l l Conformality is attained when Dirichlet energy is minimum When fixed boundary, it is in fact discrete harmonic map
Discrete Authalic Param. l Measure of Euler characteristic (Angle) l l Integral of Gaussian curvature Derived as Chi Energy
Comparing DCP & DAP l DCP (Dirichlet Energy) l Measure area extension l Minimized when angles preserved l DAP (Chi Energy) l Measure angle excess l Minimized when area preserved
Solving Parametrization l General distortion measure : l Fix the boundary, minimized the energy : l Very sparse linear systems Conjugate gradient
Natural Boundary l Instead fixed the boundary, solve for optimal conformal mapping which yields “best” boundary. For interior points l For boundary points : l Constrain two points to avoid degeneracy. l
Compare with LSCM l Least Square Conformal Map (Levy. ’ 02) l l Start from Cauchy-Riemann Equation Theoretically equivalent to Natural Boundary Map Minimize conformal energy Natural Conformal Map l Imposing boundary constraint for boundary points
Extend to non-linear func. l All parametrization could be expressed as : l U = l UA + (1 -l) Uc l Substitute U in a non-linear function reduces the problem into solving l Ex : l Could be reduced into root finding l
Boundary Control l Precompute the “impulse response” parameterization for each boundary points l New parameterization could be obtained by projecting boundary parameter onto its “impulse response” parameterization
Boundary Optimization l l Minimized arbitrary energy with respect to boundary parameterization Using precomputed gradient to accelerate optimization
Summary of Contributions l l l A linear system solution for Natural Conformal Map A new geometric for parameterization (DAP) Real-time boundary control for better parameterization design
What’s Next ? l Mean Value Coordinate (Floater. 03) l l The same property of convex combination Approximating Harmonic Map but ensure a valid embedding Tutte Harmonic Shape Preserving Mean Value
What’s Next ? l Spherical Parameterization (Praun. 03) l l Smooth parameterization for genus-0 model Using existing metric
Conclusion l There seems to be less paper directly about finding metrics (or find a better way to model them) for parameterization. l Now more efforts in finding globally smooth parameterization on arbitrary meshes
Thank You
References l (Eck. 95) Multiresolution Analysis of Arbitrary Meshes. Proceedings of SIGGRAPH 95 l (Floater. 97) Parametrization and Smooth Approximationof Surface Triangulations. Computer Aided Geometric Design 14, 3 (1997) l (K. Hormann. 99) MIPS: An Efficient Global Parametrization Method. In Curve and Surface Design: Saint-Malo 1999 (2000) l (Sander. 02) Signal-Specialized Parameterization. In Eurographics Workshop on Rendering, 2002.
References l (Floater, Hormann 03) Surface Parameterization : A Tutorial and Survey l (Levy. ’ 02) Least Squares Conformal Maps for Automatic Texture Atlas Generation. ACM SIGGRAPH Proceedings l (Floater. 03) Mean Value Coordinates. Computer Aided Geometric Design 20, 2003
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