# Teichmüller Shape Descriptor and Its Application to Alzheimer’s Disease Study

- 780 Downloads
- 1 Citations

## 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 surface-based cortical morphometry measurement in AD research.

## Keywords

Teichmüller space Conformal welding Shape analysis## Notes

### Acknowledgments

This 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.; Bristol-Myers Squibb Company; Eisai Inc.; Elan Pharmaceuticals Inc.; Eli Lilly and Company; F. Hoffmann-La Roche Ltd and its affiliated company Genentech, Inc.; GE Healthcare; Innogenetics, N.V.; 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://www.fnih.org). 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 CCF-0448399, NSF DMS-0528363, NSF DMS-0626223, NSF CCF-0830550, NSF IIS-0916286, NSF CCF-1081424, and ONR N000140910228. Data used in preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (adni.loni.ucla.edu). 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://adni.loni.ucla.edu/wp-content/uploads/how_to_apply/ADNI_Acknowledgement_List.pdf.

## References

- Angenent, S., Haker, S., Kikinis, R.,& Tannenbaum, A. (2000). Nondistorting flattening maps and the 3D visualization of colon CT images.
*IEEE Transactions on Medical Imaging, 19*, 665–671.Google Scholar - Ashburner, J., Hutton, C., Frackowiak, R., Johnsrude, I., Price, C.,& Friston, K. (1998). Identifying global anatomical differences: Deformation-based morphometry.
*Human Brain Mapping*,*6*, 348–357.CrossRefGoogle Scholar - Chincarini, A., Bosco, P., Calvini, P., Gemme, G., Esposito, M., Olivieri, C., et al. (2011). Local MRI analysis approach in the diagnosis of early and prodromal Alzheimer’s disease.
*Neuroimage,**58*(2), 469–480.Google Scholar - Chow, B., Lu, P.,& Ni, L. (2006).
*Hamilton’s Ricci flow*. Providence: American Mathematical Society.MATHGoogle Scholar - Chung, M. K., Dalton, K. M.,& Davidson, R. J. (2008). Tensor-based cortical surface morphometry via weighted spherical harmonic representation.
*IEEE Transactions on Medical Imaging, 27*, 1143–1151.Google Scholar - Chung, M. K., Robbins, S. M., Dalton, K. M., Davidson, R. J., Alexander, A. L.,& Evans, A. C. (May 2005). Cortical thickness analysis in autism with heat kernel smoothing.
*Neuroimage,**25*, 1256–1265.Google Scholar - Cuingnet, R., Gerardin, E., Tessieras, J., Auzias, G., Lehericy, S., Habert, M., et al. (2011). Automatic classification of patients with Alzheimer’s disease from structural MRI: A comparison of ten methods using the ADNI database.
*Neuroimage,**56*(2), 766–781.Google Scholar - Dale, A. M., Fischl, B.,& Sereno, M. I. (1999). Cortical surface-based analysis I: Segmentation and surface reconstruction.
*Neuroimage,**27*, 179–194.Google Scholar - Davies, R. H., Twining, C. J., Allen, P. D., Cootes, T. F.,& Taylor, C. J. (2003). Shape discrimination in the hippocampus using an MDL model. In
*International conference on information processing in medical imaging (IPMI)*. Ambleside.Google Scholar - Desikan, R. S., Segonne, F., Fischl, B., Quinn, B. T., Dickerson, B. C., Blacker, D., et al. (2006). An automated labeling system for subdividing the human cerebral cortex on MRI scans into gyral based regions of interest.
*Neuroimage*,*31*, 968–980.CrossRefGoogle Scholar - Farkas, H. M.,& Kra, I. (1991).
*Riemann surfaces (Graduate texts in mathematics)*. New York: Springer.Google Scholar - Fischl, B., Sereno, M. I.,& Dale, A. M. (1999). Cortical surface-based analysis II: Inflation, flattening, and a surface-based coordinate system.
*NeuroImage,**9*, 195–207.Google Scholar - Fox, N., Scahill, R., Crum, W.,& Rossor, M. (1999). Correlation between rates of brain atrophy and cognitive decline in AD.
*Neurology*,*52*(8), 1687–1689.CrossRefGoogle Scholar - Frisoni, G., Fox, N., Jack, C., Scheltens, P.,& Thompson, P. (2010). The clinical use of structural MRI in Alzheimer disease.
*Nature Reviews Neurology,**6*(2), 67–77.Google Scholar - Gardiner, F. P.,& Lakic, N. (2000).
*Quasiconformal Teichmüller theory*. Providence: American Mathematical Society.MATHGoogle Scholar - Gerig, G., Styner, M., Jones, D., Weinberger, D.,& Lieberman, J. (2001). Shape analysis of brain ventricles using SPHARM. In
*Proceedings of MMBIA 2001*(pp. 171–178).Google Scholar - Gorczowski, K., Styner, M., Jeong, J.-Y., Marron, J. S., Piven, J., Hazlett, H. C., Pizer, S. M.,& Gerig, G. (2007). Statistical shape analysis of multi-object complexes.
*IEEE computer society conference on computer vision and pattern recognition, CVPR ’07*(pp. 1–8). Minneapolis.Google Scholar - Gu, X., Wang, Y., Chan, T. F., Thompson, P. M.,& Yau, S.-T. (2004). Genus zero surface conformal mapping and its application to brain surface mapping.
*IEEE Transactions on Medical Imaging, 23*, 949–958.Google Scholar - Guo, X., Wang, Z., Li, K., Li, Z., Qi, Z., Jin, Z., et al. (2010). Voxel-based assessment of gray and white matter volumes in Alzheimer’s disease.
*Neuroscience Letters*,*468*, 146–150.CrossRefGoogle Scholar - Hamilton, R. S. (1988). The Ricci flow on surfaces.
*Mathematics and General Relativity*,*71*, 237–262.MathSciNetCrossRefGoogle Scholar - Henrici, P. (1988). Applied and computational complex analysis (Vol. 3). New York: Wiley-Intersecience.Google Scholar
- Hua, X., Lee, S., Hibar, D. P., Yanovsky, I., Leow, A. D., Toga, A. W., et al. (2010). Mapping Alzheimer’s disease progression in 1309 MRI scans: Power estimates for different inter-scan intervals.
*Neuroimage,**51*, 63–75.Google Scholar - Hurdal, M. K.,& Stephenson, K. (2004). Cortical cartography using the discrete conformal approach of circle packings.
*NeuroImage,**23*, S119–S128.Google Scholar - Jack, C. R. J., Bernstein, M. A., Fox, N. C., Thompson, P. M., Alexander, P. M., Harvey, D., et al. (2007). The Alzheimer’s disease neuroimaging initiative (ADNI): MRI methods.
*Journal of Magnetic Resonance Imaging, 27*, 685–691.Google Scholar - Jack, C. R, Jr, Shiung, M. M., Gunter, J. L., O’Brien, P. C., Weigand, S. D., Knopman, D. S., et al. (2004). Comparison of different MRI brain atrophy rate measures with clinical disease progression in AD.
*Neurology*,*62*, 591–600.Google Scholar - Jin, M., Kim, J., Luo, F.,& Gu, X. (September 2008). Discrete surface Ricci flow.
*IEEE Transactions on Visualization and Computer Graphics*,*14*, 1030–1043.Google Scholar - Lai, R., Shi, Y., Scheibel, K., Fears, S., Woods, R., Toga, A.,& Chan, T. (2010). Metric-induced optimal embedding for intrinsic 3D shape analysis. In
*2010 IEEE conference on computer vision and pattern recognition (CVPR)*(pp. 2871–2878). San Francisco.Google Scholar - Liu, X., Shi, Y., Dinov, I.,& Mio, W. (2010). A computational model of multidimensional shape.
*International Journal of Computer Vision*,*89*, 69–83.MathSciNetCrossRefGoogle Scholar - Lui, L. M., Zeng, W., Yau, S.-T.& Gu, X. (2010). Shape analysis of planar objects with arbitrary topologies using conformal geometry. In
*11th European conference on computer vision (ECCV 2010)*. Heraklion.Google Scholar - Mueller, S. G., Weiner, M. W., Thal, L. J., Petersen, R. C., Jack, C., Jagust, W., et al. (2005). The Alzheimer’s disease neuroimaging initiative.
*Neuroimaging Clinics of North America*,*15*, 869– 877.CrossRefGoogle Scholar - Pizer, S., Fritsch, D., Yushkevich, P., Johnson, V.,& Chaney, E. (1999). Segmentation, registration, and measurement of shape variation via image object shape.
*IEEE Transactions on Medical Imaging, 18*, 851–865.Google Scholar - Qiu, A.,& Miller, M. I. (2008). Multi-structure network shape analysis via normal surface momentum maps.
*NeuroImage,**42*, 1430–1438.Google Scholar - Schoen, R.,& Yau, S.-T. (1994).
*Lectures on differential geometry*. Boston: International Press of Boston.MATHGoogle Scholar - Schwartz, E. L., Shaw, A.,& Wolfson, E. (1989). A numerical solution to the generalized Mapmaker’s problem: Flattening nonconvex polyhedral surfaces.
*IEEE Transactions on Pattern Analysis and Machine Intelligence, 11*, 1005–1008.Google Scholar - Seppala, M.,& T.S. (1992).
*Geometry of Riemann surfaces and Teichmüller spaces. North-Holland mathematics studies*. Amsterdam: North-Holland.Google Scholar - Sharon, E.,& Mumford, D. (October 2006). 2D-shape analysis using conformal mapping.
*International Journal of Computer Vision,**70*, 55–75.Google Scholar - Shen, L., Saykin, A. J., Chung, M. K.,& Huang, H. (2007). Morphometric analysis of hippocampal shape in mild cognitive impairment: An imaging genetics study. In
*IEEE 7th international conference bioinformatics and bioengineering*. Boston.Google Scholar - Shi, Y., Lai, R.,& Toga, A. (2011). Corporate: cortical reconstruction by pruning outliers with Reeb analysis and topology-preserving evolution.
*Information Process Medical Imaging,**22*, 233–244.Google Scholar - Thompson, P. M. (1996). A surface-based technique for warping 3-dimensional images of the brain.
*IEEE Transactions on Medical Imaging*,*15*, 1–16.CrossRefGoogle Scholar - Thompson, P. M., Hayashi, K. M., Zubicaray, G. D., Janke, A. L., Rose, S. E., Semple, J., et al. (2003). Dynamics of gray matter loss in Alzheimer’s disease.
*Journal of Neuroscience*,*23*, 994–1005 .Google Scholar - Thurston, W. P. (1980).
*Geometry and topology of three-manifolds*. Princeton: Princeton university.Google Scholar - Timsari, B.,& Leahy, R. M. (2000). Optimization method for creating semi-isometric flat maps of the cerebral cortex. In
*SPIE symposium on medical imaging 2000: image processing*(Vol. 3979, pp. 698–708). San Diego.Google Scholar - Tosun, D., Reiss, A., Lee, A. D., Dutton, R. A., Hayashi, K. M., Bellugi, U., et al. (2006). Use of 3-D cortical morphometry for mapping increased cortical gyrification and complexity in Williams syndrome. In
*3rd IEEE international symposium on biomedical imaging: From nano to macro 2006*(pp. 1172–1175). Arlington.Google Scholar - Trouve, A.,& Younes, L. (2005). Metamorphoses through Lie group action.
*Foundations of Computational Mathematics*,*5*, 173–198.MathSciNetMATHCrossRefGoogle Scholar - Wang, Y., Gu, X., Chan, T. F.,& Thompson, P. M. (2009). Shape analysis with conformal invariants for multiply connected domains and its application to analyzing brain morphology.
*IEEE computer society conference on computer vision and pattern recognition, CVPR ’09*(pp. 202–209). Miami.Google Scholar - Wang, Y., Gu, X., Chan, T. F., Thompson, P. M.,& Yau, S.-T. (2006). Brain surface conformal parameterization with algebraic functions.
*Proceedings of medical image computing and computer-assisted intervention Part II*(pp. 946–954). Copenhagen.Google Scholar - Wang, Y., Gu, X., Chan, T. F., Thompson, P. M.,& Yau, S.-T. (2008). Conformal slit mapping and its applications to brain surface parameterization. In
*Proceedings of international conference on medical image computing and computer-assisted intervention: Part I*(pp. 585–593). New York.Google Scholar - Wang, Y., Lui, L., Gu, X., Hayashi, K. M., Chan, T. F., Toga, A. W., et al. (2007). Brain surface conformal parameterization using Riemann surface structure.
*IEEE Transactions on Medical Imaging, 26*, 853–865.Google Scholar - Wang, Y., Shi, J. Yin, X., Gu, X., Chan, T. F., Yau, S.-T., Toga, A. W.& Thompson, P. M. (2012). Brain surface conformal parameterization with the Ricci flow.
*IEEE Transcations on Medical Imaging,**31*, 251–264.Google Scholar - Wang, Y., Song, Y., Rajagopalan, P., An, K. L. T., Chou, Y., Gutman, B., et al. (2011). Surface-based TBM boosts power to detect disease effects on the brain: An N=804 ADNI study.
*Neuroimage,**56*(4), 1993–2010.Google Scholar - Winkler, A. M., Kochunov, P., Blangero, J., Almasy, L., Zilles, K., Fox, P. T., et al. (2010). Cortical thickness or grey matter volume? The importance of selecting the phenotype for imaging genetics studies.
*NeuroImage, 53*(3), 1135–1146.Google Scholar - Zeng, W., Lui, L. M., Gu, X.,& Yau, S.-T. (2008). Shape analysis by conformal modules.
*International Journal of Methods and Applications of Analysis (MAA),15*(4), 539–556.Google Scholar - Zeng, W., Samaras, D.,& Gu, X. D. (2010). Ricci flow for 3D shape analysis.
*The IEEE Transactions on Pattern Analysis and Machine Intelligence,**32*(4), 662–677.Google Scholar