Brain Surface Conformal Parameterization With the Ricci Flow

Laboratory of Neuro Imaging, School of Medicine, University of California, Los Angeles, CA 90095 USA.
IEEE transactions on medical imaging 09/2011; 31(2):251-64. DOI: 10.1109/TMI.2011.2168233
Source: PubMed


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.

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Available from: Paul Thompson, Dec 16, 2013
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    • "We use multivariate surface tensor-based morphometry to analyze differences in local area of the CC between subject groups. In the field of computational anatomy, tensor-based morphometry (TBM) (Davatzikos et al. 1996; Chung et al. 2008; Thompson et al. 2000) and more recently its multivariate extension, multivariate TBM (mTBM) (Leporé et al. 2008; Wang et al. 2010), have been used extensively to detect regional differences in surface and volume brain morphology between two groups of subjects (Wang et al. 2011, 2012c, Neuroinform 2013c; Shi et al. 2013a, c, 2014). Prior work (Wang et al. 2011; Shi et al. 2014) combining mTBM with other statistics such as the radial distance significantly improved statistical power. "
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    • "After that, a customized warping process can be applied to obtain the final map [2], [3], [12], [13]. To map a surface to the canonical domain, conformal maps are among the most popular tools because they have the mathematical guarantee of being diffeomorphic and the angle-preserving property [7]–[9], [14], but large metric distortions in these maps could affect the computational efficiency and mapping quality of the downstream warping process. During the customized warping on the canonical domain, different choices were made in previous works according to the specific brain structure under study. "
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