Conformal Invariants for Multiply Connected Surfaces: Application to Landmark Curve-Based Brain Morphometry Analysis,

Jie Shi, Wen Zhang, Miao Tang, Richard J. Caselli, Yalin Wang


Abstract

Landmark curves were widely adopted in neuroimaging research for surface correspondence computation and quanti ed morphometry analysis. However, most of the landmark based morphometry studies only focused on landmark curve shape di erence. Here we propose to compute a set of conformal invariant-based shape indices, which are associated with the landmark curve induced boundary lengths in the hyperbolic parameter domain. Such shape indices may be used to identify which surfaces are conformally equivalent and further quantitatively measure surface deformation. With the surface Ricci ow method, we can conformally map a multiply connected surface to the Poincare disk. Our algorithm provides a stable method to compute the shape index values in the 2D (Poincare Disk) parameter domain. The proposed shape indices are succinct, intrinsic and informative. Experimental results with synthetic data and 3D MRI data demonstrate that our method is invariant under isometric transformations and able to detect brain surface abnormalities. We also applied the new shape indices to analyze brain morphometry abnormalities associated with Alzheimer's disease (AD). We studied the baseline MRI scans of a set of healthy control and AD patients from the Alzheimer's Disease Neuroimaging Initiative (ADNI: 30 healthy control subjects vs. 30 AD patients). Although the lengths of the landmarks in Euclidean space, cortical surface area, and volume features did not di er between the two groups, our conformal invariant based shape indices revealed signi cant di erences by Hotelling's T2 test. The novel conformal invariant shape indices may o er a new sensitive biomarker and enrich our brain imaging analysis toolset for studying diagnosis and prognosis of AD.

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