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Recent Progress in Mesh Parameterization. Speaker : ZhangLei. Decade Retrospect. 75 papers. 2000. 2001. 2002. 2003. 2004. 2005. 2006. 2007. 1997. 1998. 1999. Research Blocks. Planar Parameterization MIPS, LSCM, Mean-value, ABF++,… Manifold Parameterization
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Recent Progress in Mesh Parameterization Speaker : ZhangLei
Decade Retrospect 75 papers 2000 2001 2002 2003 2004 2005 2006 2007 1997 1998 1999
Research Blocks • Planar Parameterization • MIPS, LSCM, Mean-value, ABF++,… • Manifold Parameterization • Spherical parameterization • Simplex parameterization • Inter-surface parameterization • Volumetric Parameterization • To be expected…
Alla Sheffer University of British Columbia M. S. Floater University of Oslo Kai Hormann Clausthal University of Technology Hugues Hoppe Microsoft Research Craig Gotsman Israel Institute of Technology Bruno Levy Pierre Alliez ALICE, GEOMETRICA@INRIA David Xianfeng Gu Stony Brook@State university of NewYOrk
Curvilinear Spherical Prameterization Zayer, R. MPI Rossl, C. INRIA Seidel, H.P. MPI Shape Modeling and Applications, 2006
Initial Parameterization pole date line pole
Secondary Parameterization • Angle or Area Distortion Control
Local Domain Distortion Reduction • Tangential Laplacian Smoothing
Conclusion • Pros • Easy-to-implement • Robust • Cons • Moderate distortion • Poles and date lines selection
Linear Angle Based Parameterization Levy, B. INRIA-Alice Seidel, H. P. MPI Zayer, R. MPI Eurographics Symposium on Geometry Processing, 2007
ABF&ABF++ • Sheffer, A., de Sturler, E. Parameterization of Faceted Surfaces for Meshing Using Angle Based Flattening. Engineering with Computers, 2001. • Sheffer, A., Levy, B., Mogilnitsky, M., Bogomyakov, A. ABF++: Fast and Robust Angle Based Flattening. ToG, 2005. Coordinate space Angle space Coordinate space
ABF • Planar Angle Constraints • Vertex consistency • Triangle consistency • Wheel consistency
ABF • Lagrange Multiplier Optimization Non-linear
Linearization Denote Logarithmic & Taylor expansion
Conclusion ABF++ Pros & Linear computation
Discrete Conformal Mappings via Circle Patterns Springborn, B. TU Berlin Kharevych, L. Caltech Schroder, P. Caltech ACM Transactions on Graphics, 2006
Circle Packing William Thurston combinatorics geometry
THEOREM (The Dirichlet Problem) Let K be a complex trangulating a closed topological disc, let A be an angle sum target function of K, and assume that is a function defined on the boundary vertices of K. Then there exists a unique Euclidean packing label R for K with the property that for each boundary vertex . CirclePack http://www.math.utk.edu/~kens/
Circle Pattern Problem To reconstruct a circle pattern from an abstract triangulation and the intersection angles.
Circle Pattern • Delaunay Triangulation for interior edges Edge weight for boundary edges
Circle Pattern • Delaunay Triangulation
Circle Pattern Problem • Local Geometry of an Edge For a flat triangle
Circle Pattern Problem vs Parameterization To reconstruct a circle pattern from an abstract triangulation and the intersection angles. Discrete conformal parameterization of triangular mesh ?
Parameterization Algorithm Setting the angles for each edge; Minimizing the energy; Generating the layout;
Parameterization Algorithm Setting the angles for each edge; Minimizing the energy; Generating the layout;
Parameterization Algorithm Setting the angles for each edge; Minimizing the energy; Generating the layout;
Conclusion • Pros • Circle version of ABF • … • Cons • Nonlinear
Periodic Global Parameterization Ray, N., Li, W. C., Levy, B. INRIA-Alice Sheffer, A. University of British Columbia Alliez, P. INRIA-Geometrica ACM Transactions on Graphics, 2006
Global Parameterization Given two charts C and C’, if their intersection is a topological disk, then the image of the intersection in parameterization space by and are linked by a geometric transition function : Translation: affine manifold General: complex manifold
Periodic Parameterization Translation Rotation
Problem Input: Output:
Formulation Objective
Application • Quad-Remeshing
Most Shape-preserving Mesh Parameteri-zation by Rigid Alignment
Complex Manifold Rigid transformation Translation Rotation
Local Shape-preserving Prameterization • 1-ring Patch: Geodesic Polar Map is an boundary vertex and otherwise
Global Shape-preserving Parameterization • Rigid Alignment
Least-squares Sense To minimize