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Brain Surface Parameterization Using Riemann Surface Structure

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Abstract

We develop a general approach that uses holomorphic 1-forms to parameterize anatomical surfaces with complex (possibly branching) topology. 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. Based on Riemann surface structure, we can then canonically partition the surface into patches. Each of these patches can be conformally mapped to a parallelogram. The resulting surface subdivision and the parameterizations of the components are intrinsic and stable. To illustrate the technique, we computed conformal structures for several types of anatomical surfaces in MRI scans of the brain, including the cortex, hippocampus, and lateral ventricles. We found that the resulting parameterizations were consistent across subjects, even for branching structures such as the ventricles, which are otherwise difficult to parameterize. Compared with other variational approaches based on surface inflation, our technique works on surfaces with arbitrary complexity while guaranteeing minimal distortion in the parameterization. It also offers a way to explicitly match landmark curves in anatomical surfaces such as the cortex, providing a surface-based framework to compare anatomy statistically and to generate grids on surfaces for PDE-based signal processing.

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Author information

Authors and Affiliations

  1. Mathematics Department, UCLA, Los Angeles, CA, 90095, USA

    Yalin Wang & Tony F. Chan

  2. Comp. Sci. Department, SUNY at Stony Brook, Stony Brook, NY, 11794, USA

    Xianfeng Gu

  3. Lab. of Neuro Imaging, UCLA School of Medicine, Los Angeles, CA, 90095, USA

    Kiralee M. Hayashi & Paul M. Thompson

  4. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Shing-Tung Yau

Authors
  1. Yalin Wang

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  2. Xianfeng Gu

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  3. Kiralee M. Hayashi

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  4. Tony F. Chan

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  5. Paul M. Thompson

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  6. Shing-Tung Yau

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Editor information

Editors and Affiliations

  1. Department of Diagnostic Radiology, Yale University, USA

    James S. Duncan

  2. Department of Psychiatry, University of North Carolina, USA

    Guido Gerig

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© 2005 Springer-Verlag Berlin Heidelberg

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Wang, Y., Gu, X., Hayashi, K.M., Chan, T.F., Thompson, P.M., Yau, ST. (2005). Brain Surface Parameterization Using Riemann Surface Structure. In: Duncan, J.S., Gerig, G. (eds) Medical Image Computing and Computer-Assisted Intervention – MICCAI 2005. MICCAI 2005. Lecture Notes in Computer Science, vol 3750. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11566489_81

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