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Surface Quasi-Conformal Mapping by Solving Beltrami Equations

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Mathematics of Surfaces XIII (Mathematics of Surfaces 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5654))

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Abstract

We consider the problem of constructing quasi-conformal mappings between surfaces by solving Beltrami equations. This is of great importance for shape registration.

In the physical world, most surface deformations can be rigorously modeled as quasi-conformal maps. The local deformation is characterized by a complex-value function, Beltrami coefficient, which describes the deviation from conformality of the deformation at each point.

We propose an effective algorithm to solve the quasi-conformal map from the Beltrami coefficient. The major strategy is to deform the conformal structure of the original surface to a new conformal structure by the Beltrami coefficient, such that the quasi-conformal map becomes a conformal map. By using holomorphic differential forms, conformal maps under the new conformal structure are calculated, which are the desired quasi-conformal maps.

The efficiency and efficacy of the algorithms are demonstrated by experimental results. Furthermore, the algorithms are robust for surfaces scanned from real life, and general for surfaces with different topologies.

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Author information

Authors and Affiliations

  1. Computer Science Department, State University of New York at Stony Brook, USA

    W. Zeng & X. D. Gu

  2. Mathematics Department, Rutgers University, USA

    F. Luo

  3. Mathematics Department, Harvard University, USA

    S. -T. Yau

Authors
  1. W. Zeng
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  2. F. Luo
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  3. S. -T. Yau
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  4. X. D. Gu
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Editor information

Editors and Affiliations

  1. Computer Vision and Pattern Recognition Group,Computer Science, University of York Heslington, YO10-5DD, York, United Kingdom

    Edwin R. Hancock

  2. School of Computer Science, Cardiff University, Cardiff, UK

    Ralph R. Martin

  3. Numerical Geometry Ltd, 26 Abbey Lane, Lode, CB5 9EP, Cambridge, UK

    Malcolm A. Sabin

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© 2009 Springer-Verlag Berlin Heidelberg

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Zeng, W., Luo, F., Yau, S.T., Gu, X.D. (2009). Surface Quasi-Conformal Mapping by Solving Beltrami Equations. In: Hancock, E.R., Martin, R.R., Sabin, M.A. (eds) Mathematics of Surfaces XIII. Mathematics of Surfaces 2009. Lecture Notes in Computer Science, vol 5654. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03596-8_23

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  • DOI: https://doi.org/10.1007/978-3-642-03596-8_23

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03595-1

  • Online ISBN: 978-3-642-03596-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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