Skip to main content
SpringerLink
Log in

Computational Conformal Geometric Methods for Vision

  • Reference work entry
  • First Online:
Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging

  • 2032 Accesses

Abstract

Conformal geometry studies the geometric properties of objects invariant under conformal transformation group. It is a powerful theoretic tool to study shape classification, surface deformation, and registration. Computational conformal geometry is an emerging field combining modern geometry and computer science and develops both theories in the discrete setting and computational algorithms. This work first briefly introduces the fundamental concepts, theorems in conformal geometry, such as the Riemann mapping theorem, the uniformization theorem, the Beltrami equation, Teichmüller space theory, and so on; then explains three categories of computational algorithms: discrete surface curvature flow, harmonic maps, and holomorphic differentials based on Hodge theory; and finally demonstrates practical applications in engineering and medical imaging fields. In computer vision, the work explains Teichmüller shape space for surface classification, landmark constrained surface registration based on Teichmüller map, and optimal transport map. In medical imaging, the work introduces brain mapping, brain morphology study, virtual colonoscopy, and so on.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 899.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 949.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Bobenko, A.I., Pinkall, U., Springborn, B.A.: Discrete conformal maps and ideal hyperbolic polyhedra. Geom. Topol. 19(4), 2155–2215 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, W., Zhang, M., Lei, N., Gu, D.X.: Dynamic unified surface ricci flow. Geom. Imag. Comput. 3(1), 31–56 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Chow, B., Luo, F.: Combinatorial Ricci flows on surfaces. J. Differ. Geom. 63(1), 97–129 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • de Verdière, Y.C.: Un principe variationnel pour les empilements de cercles. Invent. Math. 104(3), 655–669 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  • Glickenstein, D.: Discrete conformal variations and scalar curvature on piecewise flat two- and three-dimensional manifolds. J. Differ. Geom. 87(2), 201–237 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Gotsman, C., Gu, X., Sheffer, A.: Fundamentals of spherical parameterization for 3d meshes. ACM Trans. Graph. (TOG) 22(3), 358–363 (2003)

    Google Scholar 

  • Gu, X., Guo, R., Luo, F., Sun, J., Wu, T.: A discrete uniformization theorem for polyhedral surfaces (II). J. Differ. Geom. (JDG) 109(3), 431–466 (2018a)

    Google Scholar 

  • Gu, X., Luo, F., Sun, J., Wu, T.: A discrete uniformization theorem for polyhedral surfaces (I). J. Differ. Geom. (JDG) 109(2), 223–256 (2018b)

    Google Scholar 

  • Gu, X., Luo, F., Wu, T.: Convergence of discrete conformal geometry and computation of uniformization maps. Asian J. Math. (AJM) 23(1), 21–34 (2019)

    Google Scholar 

  • Gu, X., Wang, Y., Chan, T.F., Thompson, P.M., Yau, S.-T.: Genus zero surface conformal mapping and its application to brain surface mapping. IEEE Trans. Med. Imag. (TMI) 23(8), 949–958 (2004)

    Google Scholar 

  • Gu, X., Yau, S.-T.: Global conformal surface parameterization. In: Proceedings of the 2003 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing, pp. 127–137. Eurographics Association (2003)

    Google Scholar 

  • Gu, X., Yau, S.-T.: Computational Conformal Geometry. Advanced Lectures in Mathematics, vol. 3. International Press and Higher Education Press. Boston (2007)

    Google Scholar 

  • Gu, X., Yau, S.-T.: Computational Conformal Geometry – Theory. International Press and Higher Education Press. Boston (2020)

    MATH  Google Scholar 

  • Gu, X.D., Zeng, W., Luo, F., Yau, S.-T.: Numerical computation of surface conformal mappings. Comput. Methods Funct. Theory 11(2), 747–787 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Jin, M., Kim, J., Luo, F., Gu, X.: Discrete surface ricci flow. IEEE Trans. Vis. Comput. Graph. (TVCG) 14(5), 1030–1043 (2008)

    Google Scholar 

  • Jin, M., Wang, Y., Gu, X., Yau, S.-T., et al.: Optimal global conformal surface parameterization for visualization. Commun. Inf. Syst. 4(2), 117–134 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Jin, M., Zeng, W., Ding, N., Gu, X.: Computing fenchel-nielsen coordinates in teichmuller shape space. Commun. Inf. Syst. 9(2), 213–234 (2009a)

    Article  MathSciNet  MATH  Google Scholar 

  • Jin, M., Zeng, W., Luo, F., Gu, X.: Computing tëichmuller shape space. IEEE Trans. Vis. Comput. Graph. 15(3), 504–517 (2009b)

    Article  Google Scholar 

  • Lei, N., Zheng, X., Jiang, J., Lin, Y.-Y., Gu, D.X.: Quadrilateral and hexahedral mesh generation based on surface foliation theory. Comput. Methods Appl. Mech. Eng. 316, 758–781 (2017a)

    Article  MathSciNet  MATH  Google Scholar 

  • Lei, N., Zheng, X., Luo, Z., Gu, D.X.: Quadrilateral and hexahedral mesh generation based on surface foliation theory II. Comput. Methods Appl. Mech. Eng. 321, 406–426 (2017b)

    Article  MathSciNet  MATH  Google Scholar 

  • Lui, L.M., Gu, X., Yau, S.-T.: Convergence of an iterative algorithm for teichmüller maps via harmonic energy optimization. Math. Comput. 84(296), 2823–2842 (2015)

    Article  MATH  Google Scholar 

  • Lui, L.M., Wong, T.W., Zeng, W., Gu, X., Thompson, P.M., Chan, T.F., Yau, S.T.: Detection of shape deformities using yamabe flow and beltrami coefficients. Inverse Probl. Imag. 4(2), 311–333 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Lui, L.M., Wong, T.W., Zeng, W., Gu, X., Thompson, P.M., Chan, T.F., Yau, S.-T.: Optimization of surface registrations using beltrami holomorphic flow. J. Sci. Comput. 50(3), 557–585 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Luo, F.: Combinatorial Yamabe flow on surfaces. Commun. Contemp. Math. 6(5), 765–780 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Luo, F., Gu, X., Dai, J.: Variational Principles for Discrete Surfaces. Advanced Lectures in Mathematics, vol. 4. International Press and Higher Education Press. Boston (2007)

    Google Scholar 

  • Ma, M., Marino, J., Nadeem, S., Gu, X.: Supine to prone colon registration and visualization based on optimal mass transport. Graph. Models 104, 101031 (2019)

    Article  Google Scholar 

  • Ng, T.C., Gu, X., Lui, L.M.: Computing extremal teichmüller map of multiply-connected domains via beltrami holomorphic flow. J. Sci. Comput. 60(2), 249–275 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Peng, H., Wang, X., Duan, Y., Frey, S.H., Gu, X.: Brain morphometry on congenital hand deformities based on teichmüller space theory. Comput.-Aided Des. 58, 84–91 (2015)

    Article  Google Scholar 

  • Rodin, B., Sullivan, D.: The convergence of circle packings to the Riemann mapping. J. Differ. Geom. 26(2), 349–360 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  • Roček, M., Williams, R.M.: Quantum Regge calculus. Phys. Lett. B 104(1), 31–37 (1981)

    Article  MathSciNet  Google Scholar 

  • Saad Nadeem, J.M., Gu, X., Kaufman, A.: Corresponding supine and prone colon visualization using eigenfunction analysis and fold modeling. IEEE Trans. Vis. Comput. Graph. 23(1), 751–760 (2017)

    Article  Google Scholar 

  • Shi, R., Zeng, W., Su, Z., Jiang, J., Damasio, H., Lu, Z., Wang, Y., Yau, S.-T., Gu, X.: Hyperbolic harmonic mapping for surface registration. IEEE Trans. Pattern Anal. Mach. Intell. (2016)

    Google Scholar 

  • Su, Z., Wang, Y., Shi, R., Zeng, W., Sun, J., Luo, F., Gu, X.: Optimal mass transport for shape matching and comparison. IEEE Trans. Pattern Anal. Mach. Intell. 37(11), 2246–2259 (2015)

    Article  Google Scholar 

  • Thurston, W.P.: Three-Dimensional Geometry and Topology. Vol.1. Princeton Mathematical Series, vol. 35. Princeton University Press, Princeton (1997)

    Google Scholar 

  • Wang, Y., Gu, X., Hayashi, K.M., Chan, T.F., Thompson, P.M., Yau, S.-T.: Brain surface conformal parameterization. In: Proceedings of the Eighth IASTED International Conference, Computer Graphics and Imaging. Honolulu, Hawaii, pp. 76–81 (2005) Proceedings of the Eighth IASTED International Conference, Computer Graphics and Imaging, August, 2005, Honolulu, Hawaii, USA. Before references, a pa

    Google Scholar 

  • Wang, Y., Lui, L.M., Gu, X., Hayashi, K.M., Chan, T.F., Toga, A.W., Thompson, P.M., Yau, S.-T.: Brain surface conformal parameterization using riemann surface structure. IEEE Trans. Med. Imag. 26(6), 853–865 (2007)

    Article  Google Scholar 

  • Wong, T.W., Zhao, H.-K.: Computation of quasi-conformal surface maps using discrete beltrami flow. SIAM J. Imag. Sci. 7(4), 2675–2699 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Yin, X., Dai, J., Yau, S.-T., Gu, X.: Slit map: linear conformal parameterization for multiply connected domains. Comput.-Aided Geom. Des. (CAGD) 4975, 410–422 (2008)

    Google Scholar 

  • Yu, X., Lei, N., Wang, Y., Gu, X.: Intrinsic 3D dynamic surface tracking based on dynamic ricci flow and teichmuller map. In: Proceedings of the IEEE International Conference on Computer Vision, pp. 5390–5398 (2017)

    Google Scholar 

  • Zeng, W., Gu, X.: Ricci Flow for Shape Analysis and Surface Registration – Theories, Algorithms and Applications. Springer Briefs in Mathematics. Springer. Springer (2013)

    Book  MATH  Google Scholar 

  • Zeng, W., Marino, J., Gurijala, K.C., Gu, X., Kaufman, A.: Supine and prone colon registration using quasi-conformal mapping. IEEE Trans. Vis. Comput. Graph. 16(6), 1348–1357 (2010)

    Article  Google Scholar 

  • Zeng, W., Samaras, D., Gu, X. Ricci flow for 3D shape analysis. IEEE Trans. Pattern Anal. Mach. Intell. (TPAMI) 32(4), 662–677 (2010)

    Google Scholar 

  • Zeng, W., Yin, X., Zhang, M., Luo, F., Gu, X.: Generalized Koebe’s method for conformal mapping multiply connected domains. In: 2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling, pp. 89–100. ACM (2009)

    Google Scholar 

  • Zhang, M., Guo, R., Zeng, W., Luo, F., Yau, S.-T., Gu, X.: The unified discrete surface ricci flow. Graph. Models 76(5), 321–339 (2014)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dalian University of Technology, Dalian, China

    Na Lei

  2. Rutgers University, Piscataway, NJ, USA

    Feng Luo

  3. Harvard University, Cambridge, MA, USA

    Shing-Tung Yau

  4. Stony Brook University, Stony Brook, NY, USA

    Xianfeng Gu

Authors
  1. Na Lei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Feng Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shing-Tung Yau
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xianfeng Gu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xianfeng Gu .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, The University of Liverpool, Liverpool, UK

    Ke Chen

  2. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK

    Carola-Bibiane Schönlieb

  3. Hong Kong Center for Cerebrocardiovascular Health Engineering (COCHE), Shatin, Hong Kong

    Xue-Cheng Tai

  4. Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, MD, USA

    Laurent Younes

Rights and permissions

Reprints and permissions

Copyright information

© 2023 Springer Nature Switzerland AG

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

Lei, N., Luo, F., Yau, ST., Gu, X. (2023). Computational Conformal Geometric Methods for Vision. In: Chen, K., Schönlieb, CB., Tai, XC., Younes, L. (eds) Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging. Springer, Cham. https://doi.org/10.1007/978-3-030-98661-2_107

Download citation

Publish with us

Policies and ethics

Search

Navigation