Brain Surface Conformal Parameterization using Riemann Surface Structure

Yalin Wang, Lok Ming Lui, Xianfeng Gu, Kiralee M. Hayashi, Tony F. Chan, Arthur W. Toga,Paul M. Thompson, Shing-Tung Yau
IEEE Transaction on Medical Imaging 2007
In medical imaging, parameterized 3D surface models are useful for anatomical modeling and visualization, statistical comparisons of anatomy, as well as surface-based registration and signal processing. Here we introduce a parameterization method based on Riemann surface structure, which uses a special curvilinear net structure (conformal net) to partition the surface into a set of patches that can each be conformally mapped to a parallelogram. The resulting surface subdivision and the parameterizations of the components are intrinsic and stable (their solutions tend to be smooth functions and the boundary conditions of the Dirichlet problem can be enforced). Conformal parameterization also helps transform partial differential equations (PDEs) that may be defined on 3D brain surface manifolds to modified PDEs on a 2D parameter domain. Since the Jacobian matrix of a conformal parameterization is diagonal, the modified PDE on the parameter domain is readily solved. To illustrate our techniques, we computed parameterizations for several types of anatomical surfaces in 3D MRI scans of the brain, including the cerebral cortex, hippocampi, and lateral ventricles. For surfaces that are topologically homeomorphic to each other and have similar geometrical structures, we show that the parameterization results are consistent and the subdivided surfaces can be matched to each other. Finally, we present an automatic sulcal landmark location algorithm by solving PDEs on cortical surfaces. The landmark detection results are used as constraints for building conformal maps between surfaces that also match explicitly defined landmarks.