The Community for Technology Leaders
RSS Icon
Subscribe
Issue No.03 - May/June (2009 vol.15)
pp: 504-517
Wei Zeng , Stony Brook University, Stony Brook
Feng Luo , Rutgers University, Piscataway
Xianfeng Gu , Stony Brook University, Stony Brook
ABSTRACT
Shape indexing, classification, and retrieval are fundamental problems in computer graphics. This work introduces a novel method for surface indexing and classification based on Teichmuller theory. The Teichmuller space for surfaces with the same topology is a finite dimensional manifold, where each point represents a conformal equivalence class, a curve represents a deformation process from one class to the other. We apply Teichmuller space coordinates as shape descriptors, which are succinct, discriminating and intrinsic; invariant under the rigid motions and scalings, insensitive to resolutions. Furthermore, the method has solid theoretic foundation, and the computation of Teichmuller coordinates is practical, stable and efficient. This work focuses on the surfaces with negative Euler numbers, which have a unique conformal Riemannian metric with -1 Gaussian curvature. The coordinates which we will compute are the lengths of a special set of geodesics under this special metric. The metric can be obtained by the curvature flow algorithm, the geodesics can be calculated using algebraic topological method. We tested our method extensively for indexing and comparison of about one hundred of surfaces with various topologies, geometries and resolutions. The experimental results show the efficacy and efficiency of the length coordinate of the Teichmuller space.
INDEX TERMS
Curve, surface, solid, and object representations, Geometric algorithms, languages, and systems
CITATION
Wei Zeng, Feng Luo, Xianfeng Gu, "Computing Teichmüller Shape Space", IEEE Transactions on Visualization & Computer Graphics, vol.15, no. 3, pp. 504-517, May/June 2009, doi:10.1109/TVCG.2008.103
REFERENCES
[1] W.P. Thurston, Geometry and Topology of Three-Manifolds, Princeton lecture notes, 1976.
[2] M. Seppala and T. Sorvali, Geometry of Riemann Surfaces and Teichmüller Spaces, North-Holland Math. Studies, 1992.
[3] X. Gu and S.-T. Yau, “Global Conformal Parameterization,” Proc. Symp. Geometry Processing (SGP '03), pp. 127-137, 2003.
[4] X. Gu and B.C. Vemuri, “Matching 3D Shapes Using 2D Conformal Representations,” Proc. Seventh Int'l Conf. Medical Image Computing and Computer Assisted Intervention (MICCAI '04), pp.771-780, 2004.
[5] Y. Wang, M. Chiang, and P.M. Thompson, “Mutual Information-Based 3D Surface Matching with Applications to Face Recognition and Brain Mapping,” Proc. 10th IEEE Int'l Conf. Computer Vision (ICCV '05), vol. 1, pp. 527-534, 2005.
[6] S. Wang, Y. Wang, M. Jin, X. Gu, and D. Samaras, “3D Surface Matching and Recognition Using Conformal Geometry,” Proc. IEEE Conf. Computer Vision and Pattern Recognition (CVPR '06), pp.2453-2460, 2006.
[7] M. Jin, F. Luo, S.-T. Yau, and X. Gu, “Computing Geodesic Spectra of Surfaces,” Proc. ACM Symp. Solid and Physical Modeling (SPM '07), pp. 387-393, 2007.
[8] X. Gu and S.-T. Yau, “Surface Classification Using Conformal Structures,” Proc. Ninth IEEE Int'l Conf. Computer Vision (ICCV '03), pp. 701-708, 2003.
[9] F. Luo, “Geodesic Length Functions and Teichmüller Spaces,” J.Differential Geometry, vol. 48, p. 275, 1998.
[10] R.S. Hamilton, “The Ricci Flow on Surfaces,” Math. General Relativity, vol. 71, pp. 237-262, 1988.
[11] B. Chow and F. Luo, “Combinatorial Ricci Flows on Surfaces,” J.Differential Geometry, vol. 63, no. 1, pp. 97-129, 2003.
[12] P. Buser, Geometry and Spectra of Compact Riemann Surfaces. Birkhauser, 1992.
[13] W.P. Thurston, Three-Dimensional Geometry and Topology. Princeton Univ. Press, 1997.
[14] J.R. Munkres, Elements of Algebraic Topology. Addison-Wesley, 1984.
[15] J.W. Tangelder and R.C. Veltkamp, “A Survey of Content Based 3D Shape Retrieval Methods,” Multimedia Tools and Applications, inpress, 2008.
[16] C. Carner, M. Jin, X. Gu, and H. Qin, “Topology-Driven Surface Mappings with Robust Feature Alignment,” Proc. IEEE Visualization Conf., pp. 543-550, 2005.
[17] N. Iyer, S. Jayanti, K. Lou, Y. Kalyanaraman, and K. Ramani, “Three-Dimensional Shape Searching: State-of-the-Art Review and Future Trends,” Computer-Aided Design, vol. 37, no. 5, pp.509-530, 2005.
[18] X. Gu, S. Wang, J. Kim, Y. Zeng, Y. Wang, H. Qin, and D. Samaras, “Ricci Flow for 3D Shape Analysis,” Proc. 11th IEEE Int'l Conf. Computer Vision (ICCV '07), pp. 1-8, 2007.
[19] R.-M. Rustamov, “Laplace-Beltrami Eigenfunctions for Deformation Invariant Shape Representation,” Proc. Symp. Geometry Processing (SGP), 2007.
[20] M. Reuter, F.-E. Wolter, and N. Petnecke, “Laplace-Spectra as Fingerprints for Shape Matching,” Proc. ACM Symp. Solid and Physical Modeling (SPM '05), pp. 101-106, 2005.
[21] P. Xiang, C.-O. Hua, F.-X. Gang, and Z.-B. Chuan, “Pose Insensitive 3D Retrieval by Poisson Shape Histogram,” Lecture Notes in Computer Science, vol. 4488, pp. 25-32, 2007.
[22] V. Jain and H. Zhang, “A Spectral Approach to Shape-Based Retrieval of Articulated 3D Models,” Computer Aided Design, vol. 39, pp. 398-407, 2007.
[23] M. Ben-Chen and C. Gotsman, “Characterizing Shape Using Conformal Factors,” Proc. Eurographics Workshop Shape Retrieval, 2008.
[24] K. Stephenson, Introduction to Circle Packing. Cambridge Univ. Press, 2005.
[25] A. Elad and R. Kimmel, “On Bending Invariant Signatures for Surfaces,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 25, no. 10, pp. 1285-1295, Oct. 2003.
[26] T. Tung and F. Schmitt, “The Augmented Multiresolution Reeb Graph Approach for Content-Based Retrieval of 3D Shapes,” Int'l J. Shape Modeling, vol. 11, no. 1, pp. 91-120, 2005.
[27] R. Gal, A. Shamir, and D. Cohen-Or, “Pose Oblivious Shape Signature,” IEEE Trans. Visualization and Computer Graphics, vol. 13, no. 2, pp. 261-271, Mar./Apr. 2007.
[28] P. Shilane, P. Min, M. Kazhdan, and T. Funkhouser, “The Princeton Shape Benchmark,” Proc. IEEE Int'l Conf. Shape Modeling Int'l (SMI '04), pp. 167-178, 2004.
[29] R.S. Hamilton, “Three Manifolds with Positive Ricci Curvature,” J.Differential Geometry, vol. 17, pp. 255-306, 1982.
[30] B. Chow, P. Lu, and L. Ni, “Hamilton's Ricci Flow,” Am. Math. Soc., vol. 77, 2006.
[31] B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, F. Luo, T. Ivey, D. Knopf, P. Lu, and L. Ni, “The Ricci Flow: Techniques and Applications. Part I: Mathematical Surveys and Monographs,” Am. Math. Soc., vol. 135, 2007.
[32] C. de Verdiere Yves, “Un Principe Variationnel pour les Empilements de Cercles,” Inventiones Math., vol. 104, no. 3, pp.655-669, 1991.
[33] Braegger and Walter, “Kreispackungen und Triangulierungen,” Enseign. Math., vol. (2)38, no. 3-4, pp. 201-217, 1992.
[34] I. Rivin, “Euclidean Structures on Simplicial Surfaces and Hyperbolic Volume,” Ann. Math., vol. 2, no. 3, pp. 553-580, 1994.
[35] F. Luo, “Combinatorial Yamabe Flow on Surfaces,” Comm. Contemporary Math., vol. 6, no. 5, pp. 765-780, 2004.
[36] F. Luo, On Teichmüller Space of Surface with Boundary, preprint, 2005.
[37] G. Perelman, “The Entropy Formula for the Ricci Flow and Its Geometric Applications,” technical report arXiv.org, Nov. 2002.
[38] G. Perelman, “Ricci Flow with Surgery on Three-Manifolds,” technical report arXiv.org, Mar. 2003.
[39] G. Perelman, “Finite Extinction Time for the Solutions to the Ricci Flow on Certain Three-Manifolds,” technical report arXiv.org, July 2003.
[40] G. Leibon, “Characterizing the Delaunay Decompositions of Compact Hyperbolic Surfaces,” Geometry and Topology, vol. 6, pp. 361-391, 2002.
[41] A.I. Bobenko and B.A. Springborn, “Variational Principles for Circle Patterns and Koebe's Theorem,” Trans. Am. Math. Soc., vol. 256, no. 2, pp. 659-689, 2004.
[42] M. Jin, J. Kim, F. Luo, and X. Gu, “Discrete Surface Ricci Flow,” IEEE Trans. Visualization and Computer Graphics, vol. 14, no. 5, Sept./Oct. 2008.
[43] L. Kharevych, B. Springborn, and P. Schröder, “Discrete Conformal Mappings via Circle Patterns,” ACM Trans. Graphics, vol. 25, no. 2, pp. 412-438, 2006.
[44] M. Ben-Chen, C. Gotsman, and G. Bunin, “Conformal Flattening by Curvature Prescription and Metric Scaling,” Computer Graphics Forum (Proc. Eurographics '08), vol. 27, no. 2, 2008.
[45] Y.-L. Yang, J. Kim, F. Luo, and X. Gu, “Optimal Surface Parameterization Using Inverse Curvature Map,” IEEE Trans. Visualization and Computer Graphics, vol. 14, no. 5, pp. 1054-1066, Sept./Oct. 2008.
[46] A.I. Bobenko and B.A. Springborn, “Variational Principles for Circle Patterns and Koebe's Theorem,” Trans. Am. Math. Soc., vol. 356, pp. 659-689, 2004.
[47] A. Bobenko and P. Schröder, “Discrete Willmore Flow,” Proc. Symp. Geometry Processing (SGP '05), pp. 101-110, 2005.
[48] Y.C. de Verdière, “Un Principe Variationnel pour les Empilements de Cercles. [a Variational Principle for Circle Packings],” Invent. Math., vol. 104, no. 3, pp. 655-669, 1991.
[49] P.L. Bowers and M.K. Hurdal, “Planar Conformal Mappings of Piecewise Flat Surfaces,” Visualization and Math. III, pp.3-34, Springer, 2003.
[50] B. Springborn, P. Schröder, and U. Pinkall, “Conformal Equivalence of Triangle Meshes,” Proc. ACM SIGGRAPH, 2008.
[51] M.S. Floater and K. Hormann, “Surface Parameterization: A Tutorial and Survey,” Advances in Multiresolution for Geometric Modelling, pp. 157-186, Springer, 2005.
[52] A. Sheffer, E. Praun, and K. Rose, “Mesh Parameterization Methods and Their Applications,” Foundations and Trends in Computer Graphics and Vision, vol. 2, no. 2, 2006.
50 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool