Brain surface conformal parameterization using riemann surface structure

Yalin Wang*, Lok Ming Lui, Xianfeng Gu, Kiralee M. Hayashi, Tony F. Chan, Arthur W. Toga, Paul M. Thompson, Shing Tung Yau

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

75 Scopus citations


In medical imaging, parameterized 3-D surface models are useful for anatomical modeling and visualization, statistical comparisons of anatomy, and surface-based registration and signal processing. Here we introduce a parameterization method based on Riemann surface structure, which uses a special curvilinear net structure (conformal net) to partition the surface into a set of patches that can each be conformally mapped to a parallelogram. The resulting surface subdivision and the parameterizations of the components are intrinsic and stable (their solutions tend to be smooth functions and the boundary conditions of the Dirichlet problem can be enforced). Conformal parameterization also helps transform partial differential equations (PDEs) that may be defined on 3-D brain surface manifolds to modified PDEs on a two-dimensional parameter domain. Since the Jacobian matrix of a conformal parameterization is diagonal, the modified PDE on the parameter domain is readily solved. To illustrate our techniques, we computed parameterizations for several types of anatomical surfaces in 3-D magnetic resonance imaging scans of the brain, including the cerebral cortex, hippocampi, and lateral ventricles. For surfaces that are topologically homeomorphic to each other and have similar geometrical structures, we show that the parameterization results are consistent and the subdivided surfaces can be matched to each other. Finally, we present an automatic sulcal landmark location algorithm by solving PDEs on cortical surfaces. The landmark detection results are used as constraints for building conformal maps between surfaces that also match explicitly defined landmarks.

Original languageEnglish (US)
Pages (from-to)853-865
Number of pages13
JournalIEEE Transactions on Medical Imaging
Issue number6
StatePublished - Jun 2007
Externally publishedYes

Bibliographical note

Funding Information:
Manuscript received June 12, 2006; revised January 30, 2007. This work was supported in part by the National Institutes of Health through the NIH Roadmap for Medical Research under Grant U54 RR021813, in part by NIH/NCRR under Resource Grant P41 RR013642, in part by the National Science Foundation (NSF) under Contract DMS-0610079 and ONR Contract N0014-06-1-0345. The work of X. Gu was supported in part by the NSF under CAREER Award CCF-0448339, Grant DMS-0528363, and Grant NSF DMS-0626223. The work of P. M. Thompson was supported by the National Institute for Biomedical Imaging and Bioengineering, National Center for Research Resources, National Institute for Neurological Disorders and Stroke, National Institute on Aging, and National Institute for Child Health, and Development under Grant EB01651, Grant RR019771, AG016570, Grant NS049194, and Grant HD050735.Asterisk indicates corresponding author. *Y. Wang is with the Laboratory of Neuro Imaging, Department of Neurology, University of California—Los Angeles School of Medicine, Los Angeles, CA 90095 USA and with the Department of Mathematics, University of California, Los Angeles, CA 90095 USA (e-mail:


  • Brain mapping
  • Conformal parameterization
  • Partial differential equation
  • Riemann surface structure

ASJC Scopus subject areas

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


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