Brain Mapping with the Ricci Flow Conformal Parameterization and Multivariate Statistics on Deformation Tensors

By solving the Yamabe equation with the discrete surface Ricci flow method, we can conformally parameterize a multiple boundary surface by a multi-hole disk. The resulting parameterizations do not have any singularities and they are intrinsic and stable. For applications in brain mapping research, first, we convert a cortical surface model into a multiple boundary surface by cutting along selected anatomical landmark curves. Secondly, we conformally parameterize each cortical surface using a multi-hole disk. Inter-subject cortical surface matching is performed by solving a constrained harmonic map in the canonical parameter domain. To map group differences in cortical morphometry, we then compute a manifold version of Hotelling’s T 2 test on the Jacobian matrices. Permutation testing was used to estimate statistical significance. We studied brain morphology in 21 patients with Williams Syndrome (WE) and 21 matched healthy control subjects with the proposed method. The results demonstrate our algorithm’s potential power to effectively detect group differences on cortical surfaces.