Brain Surface Parameterization Using Riemann Surface Structure
Yalin Wang, Xianfeng Gu, Kiralee M. Hayashi, Tony F. Chan,Paul M. Thompson, and Shing-Tung Yau
MICCAI, 2005.
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.