Teichmüller Shape Descriptor and Its Application t

Authors: Wei Zeng Rui Shi Yalin Wang ShingTung Yau Xianfeng Gu Alzheimer’s Disease Neuroimaging Initiative

Publish Date: 2012/11/09

Volume: 105, Issue: 2, Pages: 155-170

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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 cortical morphometry Experimental results with 3D MRI 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 surfacebased cortical morphometry measurement in AD researchThis work was supported by NIH R01EB007530 0A1 NSF IIS0916286 NSF CCF0916235 NSF CCF0830550 NSF III0713145 and ONR N000140910228 NSFC 61202146 and SDC BS2012DX014 Data collection and sharing for this project was funded by the Alzheimer’s Disease Neuroimaging Initiative ADNI National Institutes of Health Grant U01 AG024904 ADNI is funded by the National Institute on Aging the National Institute of Biomedical Imaging and Bioengineering and through generous contributions from the following Abbott Alzheimer’s Association Alzheimer’s Drug Discovery Foundation Amorfix Life Sciences Ltd AstraZeneca Bayer Healthcare BioClinica Inc Biogen Idec Inc BristolMyers Squibb Company Eisai Inc Elan Pharmaceuticals Inc Eli Lilly and Company F HoffmannLa Roche Ltd and its affiliated company Genentech Inc GE Healthcare Innogenetics NV Janssen Alzheimer Immunotherapy Research Development LLC Johnson Johnson Pharmaceutical Research Development LLC Medpace Inc Merck Co Inc Meso Scale Diagnostics LLC Novartis Pharmaceuticals Corporation Pfizer Inc Servier Synarc Inc and Takeda Pharmaceutical Company The Canadian Institutes of Health Research is providing funds to support ADNI clinical sites in Canada Private sector contributions are facilitated by the Foundation for the National Institutes of Health http//wwwfnihorg The grantee organization is the Northern California Institute for Research and Education and the study is coordinated by the AD Cooperative Study at the University of California San Diego ADNI data are disseminated by the Laboratory for Neuro Imaging at the University of California Los Angeles This research was also supported by NIH grants P30 AG010129 K01 AG030514 and the Dana Foundation This work has been supported by NSF CCF0448399 NSF DMS0528363 NSF DMS0626223 NSF CCF0830550 NSF IIS0916286 NSF CCF1081424 and ONR N000140910228 Data used in preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative ADNI database adniloniuclaedu As such the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this report A complete listing of ADNI investigators can be found at http//adniloniuclaedu/wpcontent/uploads/how to apply/ADNI Acknowledgement ListpdfWe want to find a map from the Riemann sphere S to the original Riemann sphere Omega Phi Srightarrow Omega The Beltramicoefficient mu S rightarrow mathbb C is the union of mu k’s each segments mu z = mu kz forall z in S k The solution exists and is unique up to a Möbius transformation according to Quasiconformal Mapping theorem Gardiner and Lakic 2000 square Note that the discrete computational method is more direct without explicitly solving the Beltrami equation From the Beltrami coefficient mu one can deform the conformal structure of S k to that of Omega k under the conformal structures of Omega k Phi Srightarrow Omega becomes a conformal mapping The conformal structure of Omega k is equivalent to that of D k therefore one can use the conformal structure of D k directly In discrete case the conformal structure is represented as the angle structure Therefore in our algorithm we copy the angle structures of D k’s to S and compute the conformal map Phi directly

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