Shape Analysis with Conformal Invariants for Multiply Connected Domains and its Application to Analyzing Brain Morphology
Yalin Wang, Xianfeng Gu, Tony F. Chan, and Paul M. Thompson
Abstract
Here we propose to compute a conformal invariant, a shape index, that is associated with the perimeter of the inner circle in the hyperbolic parameter plane and may be used to identify which surfaces are conformally equivalent and measure surface deformation. With the surface Ricci flow method, we can conformally map a multiply connected domain to a multi-hole disk. Our algorithm provides a stable method to compute the values of this shape index in the 2D (Poincaré Disk) parameter domain. We also applied this new shape index for analyzing abnormalities in brain morphology in Alzheimer\textquoteright s disease (AD) and Williams syndrome (WS). After cutting along various landmark curves on surface models of the cerebral cortex or hippocampus, we obtained multiply connected domains. We conformally projected the surfaces to hyperbolic plane, accurately computed the proposed conformal invariant for each selected landmark curve, and assembled these into a feature vector. We also detected group differences in brain structure based on multivariate analysis of the surface deformation tensors induced by these Ricci flow mappings. Experimental results with 3D MRI data from 80 subjects demonstrate that our method powerfully detects brain surface abnormalities when combined with a constrained harmonic map based surface registration method.
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