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

Abstract : By solving the Yamabe equation with the discrete surface Ricci flow method, we can conformally parameterize a multiple bound- ary 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 T2 test on the Ja- cobian 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 pro- posed method. The results demonstrate our algorithm's potential power to effectively detect group differences on cortical surfaces.