Brain surface conformal parameterization with the ricci flow

Yalin Wang*, Jie Shi, Xiaotian Yin, Xianfeng Gu, Tony F. Chan, Shing Tung Yau, Arthur W. Toga, Paul M. Thompson

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

67 Scopus citations


In brain mapping research, parameterized 3-D surface models are of great interest for statistical comparisons of anatomy, surface-based registration, and signal processing. Here, we introduce the theories of continuous and discrete surface Ricci flow, which can create Riemannian metrics on surfaces with arbitrary topologies with user-defined Gaussian curvatures. The resulting conformal parameterizations have no singularities and they are intrinsic and stable. First, we convert a cortical surface model into a multiple boundary surface by cutting along selected anatomical landmark curves. Secondly, we conformally parameterize each cortical surface to a parameter domain with a user-designed Gaussian curvature arrangement. In the parameter domain, a shape index based on conformal invariants is computed, and inter-subject cortical surface matching is performed by solving a constrained harmonic map. We illustrate various target curvature arrangements and demonstrate the stability of the method using longitudinal data. To map statistical differences in cortical morphometry, we studied brain asymmetry in 14 healthy control subjects. We used a manifold version of Hotelling's T 2 test, applied to the Jacobian matrices of the surface parameterizations. A permutation test, along with the cumulative distribution of p-values, were used to estimate the overall statistical significance of differences. The results show our algorithm's power to detect subtle group differences in cortical surfaces.

Original languageEnglish (US)
Article number6020804
Pages (from-to)251-264
Number of pages14
JournalIEEE Transactions on Medical Imaging
Issue number2
StatePublished - Feb 2012
Externally publishedYes

Bibliographical note

Funding Information:
Manuscript received April 11, 2011; accepted August 29, 2011. Date of publication September 15, 2011; date of current version February 03, 2012. This work was supported by the National Institutes of Health (NIH) through the NIH Roadmap for Medical Research, Grant U54 RR021813 entitled Center for Computational Biology (CCB). Additional support was provided by the National Institute on Aging (AG016570 to PMT), the National Library of Medicine, the National Institute for Biomedical Imaging and Bioengineering, and the National Center for Research Resources (LM05639, EB01651, RR019771 to PMT), National Science Foundation (NSF) (Nets-1016829, IIS-0916286, CCF-1081424 to XG), and Office of Naval Research (ONR N000140910228 to XG). Asterisk indicates corresponding author. *Y. Wang was with the Laboratory of Neuro Imaging, School of Medicine, University of California, Los Angeles, CA 90095 USA. He is now with the School of Computing, Informatics, and Decision Systems Engineering, Arizona State University, Tempe, AZ 85281 USA (e-mail: [email protected]).


  • Brain surface parameterization
  • Ricci flow
  • conformal mapping
  • multivariate tensor-based morphometry

ASJC Scopus subject areas

  • Software
  • Radiological and Ultrasound Technology
  • Computer Science Applications
  • Electrical and Electronic Engineering


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