Brain Surface Conformal Parameterization
Yalin Wang, Xianfeng Gu, Kiralee M. Hayashi, Tony F. Chan, Paul M. Thompson and Shing-Tung Yau
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. We can then automatically
partition the surface using a critical graph that connects
zero points in the 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 de.ned on the surface. 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 dif.cult to parameterize. Compared with other
variational approaches based on surface in.ation, our technique
works on surfaces with arbitrary complexity while
guaranteeing minimal distortion in the parameterization. It
also generates grids on surfaces for PDE-based signal processing.