Brain Surface Conformal Parameterization with Algebraic Functions

Yalin Wang, Xianfeng Gu, Paul Thompson, Tony F. Chan, Shing-Tung Yau
In medical imaging, parameterized 3D surface models are of great interest for anatomical modeling and visualization, statistical comparisons of anatomy, and surface-based registration and signal pro- cessing. Here we introduce a parameterization method based on algebraic functions. By solving the Yamabe equation with the Ricci °ow method, we can conformally map a brain surface to a multi-hole disk. The re- sulting parameterizations do not have any singularities and are intrinsic and stable. To illustrate the technique, we computed parameterizations of several types of anatomical surfaces in MRI scans of the brain, in- cluding the hippocampi and the cerebral cortices with various landmark curves labeled. For the cerebral cortical surfaces, we show the parameter- ization results are consistent with selected landmark curves and can be matched to each other using constrained harmonic maps. Unlike previous planar conformal parameterization methods, our algorithm does not in- troduce any singularity points. It also oŽers a method to explicitly match landmark curves between anatomical surfaces such as the cortex, and to compute conformal invariants for statistical comparisons of anatomy.