Applications • Computer Vision: Non-isometric surface registration, tracking, and recognition - surface diffeomorphisms: Conformal mapping, Quasiconformal mapping (general) • Medical Imaging: Virtual colonoscopy, Brain mapping, Vestibular system analysis • Wireless Sensor Networks: Greedy routing, Load balancing, 3D Localization • Computer Graphics: Global surface conformal parameterization, Texture mapping • Geometric Modeling: Manifold spline, Shape indexing, Computational topology Selected Publications • W. Zeng, D. Samaras and X. Gu. Ricci Flow for 3D Shape Analysis. IEEE TPAMI, 2010. • X. Gu, W. Zeng, F. Luo and S.-T. Yau. Numerical Computation of Surface Conformal Mappings. Computational Methods and Functional Theory (CMFT), 11(2): 747-787 2011. • W. Zeng, L. M. Lui, F. Luo, T. Chan, S.-T. Yau and X. Gu. Computing Quasiconformal Maps Using an Auxiliary Metric and Discrete Curvature Flow. J. of Nume. Math., 2012. Acknowledgements • All the funding institutes: NSF, ONR, NIH • All the collaborators and coauthors Motivation • 3D geometric acquisition technology becomes mature • High resolution high speed medical imaging develops fast • Efficiently process massive geometric data (3D/4D) Highlights • Ricci curvature flow deforms Riemannian metric proportionally to curvature, such that curvature evolves like a heat diffusion, eventually becomes constant everywhere. • Ricci flow leads to conformal mapping of arbitrary surfaces. • Ricci flow can compute quasiconformal (general) mapping under auxiliary metric. Merits • Unification: All shapes 3 shapes (S 2 , E 2 or H 2 ) • Dimension reduction: 3D 2D (surface image) • Information preservation: Conformal factor + mean curvature + boundary condition • Capable for general mappings Theory • Hamilton’s Ricci flow • Uniformization theorem • Generalized uniformization • Conformal mapping • Quasiconformal mapping Algorithm • Discrete surface Ricci flow • Auxiliary metric associated with Beltrami coefficient Ricci Curvature Flow for General Shape Registration and Geometric Analysis Wei Zeng Xianfeng Gu Florida International University Stony Brook University [email protected] [email protected] ) ( 2 K u e K g u ) ( 2 K u e K g u u u n i i i i du K K u E 0 1 ) ( ) ( u u n i i i i du K K u E 0 1 ) ( ) ( i i K dt t du ) ( i i K dt t du ) ( FIU School of Computing and Information Sciences 25 th Anniversary Celebration, 1987-2012 November 9 th -10 th a 1 b 1 a 2 b 2 C 1 C 2 C 3 C 13 C 12 C 21 C 23 C 32 C 31 Ω Ω 1 Ω 2 Ω 3 2 3 1 3 1 2 1 2 3 1 2 3 1 1 3 2 2 3 1 1 2 2 3 3 2 2 || || || || z d dz dz ij ij g K K dt dg ) ( 2 || || 1 || || 1 K

Ricci Curvature Flow for General Shape Registration …Applications • Computer Vision: Non-isometric surface registration, tracking, and recognition - surface diffeomorphisms: Conformal

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Page 1: Ricci Curvature Flow for General Shape Registration …Applications • Computer Vision: Non-isometric surface registration, tracking, and recognition - surface diffeomorphisms: Conformal

Applications • Computer Vision: Non-isometric surface registration, tracking, and recognition

- surface diffeomorphisms: Conformal mapping, Quasiconformal mapping (general)

• Medical Imaging: Virtual colonoscopy, Brain mapping, Vestibular system analysis

• Wireless Sensor Networks: Greedy routing, Load balancing, 3D Localization

• Computer Graphics: Global surface conformal parameterization, Texture mapping

• Geometric Modeling: Manifold spline, Shape indexing, Computational topology

Selected Publications• W. Zeng, D. Samaras and X. Gu. Ricci Flow for 3D Shape Analysis. IEEE TPAMI, 2010. • X. Gu, W. Zeng, F. Luo and S.-T. Yau. Numerical Computation of Surface Conformal

Mappings. Computational Methods and Functional Theory (CMFT), 11(2): 747-787 2011. • W. Zeng, L. M. Lui, F. Luo, T. Chan, S.-T. Yau and X. Gu. Computing Quasiconformal

Maps Using an Auxiliary Metric and Discrete Curvature Flow. J. of Nume. Math., 2012.

Acknowledgements• All the funding institutes: NSF, ONR, NIH• All the collaborators and coauthors

Motivation• 3D geometric acquisition technology becomes mature• High resolution high speed medical imaging develops fast• Efficiently process massive geometric data (3D/4D)

Highlights• Ricci curvature flow deforms Riemannian metric proportionally to curvature, such

that curvature evolves like a heat diffusion, eventually becomes constant everywhere.• Ricci flow leads to conformal mapping of arbitrary surfaces. • Ricci flow can compute quasiconformal (general) mapping under auxiliary metric.

Merits• Unification: All shapes 3 shapes (S2, E2 or H2) • Dimension reduction: 3D 2D (surface image) • Information preservation: Conformal factor + mean curvature + boundary condition• Capable for general mappings

Theory • Hamilton’s Ricci flow• Uniformization theorem• Generalized uniformization• Conformal mapping• Quasiconformal mapping

Algorithm• Discrete surface Ricci flow• Auxiliary metric associated with Beltrami coefficient

Ricci Curvature Flow for General Shape Registration and Geometric AnalysisWei Zeng Xianfeng Gu

Florida International University Stony Brook [email protected] [email protected]

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