Teichmüller Shape Descriptor and Its Application to Alzheimerr's Diseasee Study

Wei Zeng, Rui Shi, Yalin Wang, Shing-Tung Yau, Xianfeng Gu


Abstract

We propose a novel method to apply Teichmüller space theory to study the signature of a family of nonintersecting closed 3D curves on a general genus zero closed surface. Our algorithm provides an efficient method to encode both global surface and local contour shape information. The signature - Teichmüller shape descriptor - is computed by surface Ricci flow method, which is equivalent to solving an elliptic partial differential equation on surfaces and is numerically stable. We propose to apply the new signature to analyze abnormalities in brain corticalmorphometry.Experimental results with 3DMRI data from Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset (152 healthy control subjects versus 169 Alzheimer's disease (AD) patients) demonstrate the effectiveness of our method and illustrate its potential as a novel surface-based cortical morphometry measurement in AD research.

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'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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