Surface parameterization using riemann surface structure

Yalin Wang, Tony F. Chan, Xianfeng Gu, Paul M. Thompson, Kiralee M. Hayashi, Shing Tung Yau

Research output: Chapter in Book/Report/Conference proceedingConference contribution

20 Scopus citations


We propose a general method that parameterizes general surfaces with complex (possible branching) topology using Riemann surface structure. Rather than evolve the surface geometry to a plane or sphere, we instead use the fact that all orientable surfaces are Riemann surfaces and admit conformal structures, which induce special curvilinear coordinate systems on the surfaces. We can then automatically partition the surface using a critical graph that connects zero points in the global conformal structure on the surface. The trajectories of iso-parametric curves canonically partition a surface into patches. Each of these patches is either a topological disk or a cylinder and can be conformally mapped to a parallelogram by integrating a holomorphic 1-form defined on the surface. The resulting surface subdivision and the parameterizations of the components are intrinsic and stable. For surfaces with similar topology and geometry, we show that the parameterization results are consistent and the subdivided surfaces can be matched to each other using constrained harmonic maps. The surface similarity can be measured by direct computation of distance between each pair of corresponding points on two surfaces. To illustrate the technique, we computed conformal structures for anatomical surfaces in MRI scans of the brain and human face surfaces. We found that the resulting parameterizations were consistent across subjects, even for branching structures such as the ventricles, which are otherwise difficult to parameterize. Our method provides a surface-based framework for statistical comparison of surfaces and for generating grids on surfaces for PDE-based signal processing.

Original languageEnglish (US)
Title of host publicationProceedings - 10th IEEE International Conference on Computer Vision, ICCV 2005
Number of pages6
StatePublished - 2005
Externally publishedYes
EventProceedings - 10th IEEE International Conference on Computer Vision, ICCV 2005 - Beijing, China
Duration: Oct 17 2005Oct 20 2005

Publication series

NameProceedings of the IEEE International Conference on Computer Vision


OtherProceedings - 10th IEEE International Conference on Computer Vision, ICCV 2005

ASJC Scopus subject areas

  • Software
  • Computer Vision and Pattern Recognition


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