Brain Surface Parameterization using Riemann Surface Structure
Yalin Wang, Xianfeng Gu, Kiralee M. Hayashi, Tony F. Chan,
Paul M. Thompson and ShingTung Yau
Abstract
We develop a general approach that uses holomorphic 1forms 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
surfacebased framework to compare anatomy statistically
and to generate grids on surfaces for PDEbased signal processing.
Figures (click on each for a larger version):
Related Publications

Y. Wang, L.M. Lui, X. Gu, K.M. Hayashi, T.F. Chan, A.W. Toga, P.M. Thompson and S.T. Yau,
Brain Surface Conformal Parameterization Using Riemann Surface Structure",
IEEE Transactions on Medical Imaging, Vol. 26, Nov. 6, June 2007 pp. 853865

Y. Wang, X. Gu, K.M. Hayashi, T.F. Chan, P.M. Thompson and S.T. Yau,
"Surface Parameterization
using Riemann Surface Structure",
10th IEEE International Conference on Computer
Vision (ICCV), Beijing, China, Oct. 2005, pp. 10611066