Yalin Wang, X. Gu, K.M. Hayashi, T.F. Chan, P.M. Thompson, and S.-T. Yau (USA)
Brain Mapping, Riemann Surface Structure, Conforaml Net, Critical Graph
We develop a general approach that uses holomorphic 1 forms to parameterize anatomical surfaces with complex (possibly branching) topology. Rather than evolve the sur face geometry to a plane or sphere, we instead use the fact that all orientable surfaces are Riemann surfaces and ad mit conformal structures, which induce special curvilinear coordinate systems on the surfaces. We can then automat ically partition the surface using a critical graph that con nects zero points in the conformal structure on the surface. The trajectories of iso-parametric curves canonically parti tion 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 subdi vision and the parameterizations of the components are in trinsic and stable. To illustrate the technique, we computed conformal structures for several types of anatomical sur faces in MRI scans of the brain, including the cortex, hip pocampus, and lateral ventricles. We found that the result ing 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 tech nique works on surfaces with arbitrary complexity while guaranteeing minimal distortion in the parameterization. It also generates grids on surfaces for PDE-based signal pro cessing.
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