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# Brain Surface Conformal Parameterization With the Ricci Flow

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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

ABSTRACT 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. "
##### Article: Impact of Early and Late Visual Deprivation on the Structure of the Corpus Callosum: A Study Combining Thickness Profile with Surface Tensor-Based Morphometry
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ABSTRACT: Blindness represents a unique model to study how visual experience may shape the development of brain organization. Exploring how the structure of the corpus callosum (CC) reorganizes ensuing visual deprivation is of particular interest due to its important functional implication in vision (e.g., via the splenium of the CC). Moreover, comparing early versus late visually deprived individuals has the potential to unravel the existence of a sensitive period for reshaping the CC structure. Here, we develop a novel framework to capture a complete set of shape differences in the CC between congenitally blind (CB), late blind (LB) and sighted control (SC) groups. The CCs were manually segmented from T1-weighted brain MRI and modeled by 3D tetrahedral meshes. We statistically compared the combination of local area and thickness at each point between subject groups. Differences in area are found using surface tensor-based morphometry; thickness is estimated by tracing the streamlines in the volumetric harmonic field. Group differences were assessed on this combined measure using Hotelling’s T 2 test. Interestingly, we observed that the total callosal volume did not differ between the groups. However, our fine-grained analysis reveals significant differences mostly localized around the splenium areas between both blind groups and the sighted group (general effects of blindness) and, importantly, specific dissimilarities between the LB and CB groups, illustrating the existence of a sensitive period for reorganization. The new multivariate statistics also gave better effect sizes for detecting morphometric differences, relative to other statistics. They may boost statistical power for CC morphometric analyses.
Neuroinformatics 02/2015; 13(3). DOI:10.1007/s12021-014-9259-9 · 2.83 Impact Factor
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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. "
##### Article: Metric Optimization for Surface Analysis in the Laplace-Beltrami Embedding Space
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ABSTRACT: In this paper we present a novel approach for the intrinsic mapping of anatomical surfaces and its application in brain mapping research. Using the Laplace-Beltrami eigensystem, we represent each surface with an isometry invariant embedding in a high dimensional space. The key idea in our system is that we realize surface deformation in the embedding space via the iterative optimization of a conformal metric without explicitly perturbing the surface or its embedding. By minimizing a distance measure in the embedding space with metric optimization, our method generates a conformal map directly between surfaces with highly uniform metric distortion and the ability of aligning salient geometric features. Besides pairwise surface maps, we also extend the metric optimization approach for group-wise atlas construction and multi-atlas cortical label fusion. In experimental results, we demonstrate the robustness and generality of our method by applying it to map both cortical and hippocampal surfaces in population studies. For cortical labeling, our method achieves excellent performance in a crossvalidation experiment with 40 manually labeled surfaces, and successfully models localized brain development in a pediatric study of 80 subjects. For hippocampal mapping, our method produces much more significant results than two popular tools on a multiple sclerosis study of 109 subjects.
03/2014; 33(7). DOI:10.1109/TMI.2014.2313812
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• "Computing Teichmüller maps is also an important problem in the fields of medical imaging, computer graphics and vision. In medical imaging, conformal and quasiconformal mapping has been applied for brain cortical surface registration ( [24], [16]). In computer vision, conformal geometry has been applied for shape analysis and dynamic surface registration and tracking ( [25]), [27]), and in computer graphics, conformal geometry has been applied for surface parameterization ( [15]). "
##### Article: Computing Teichm\"{u}ller Maps between Polygons
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ABSTRACT: By the Riemann-mapping theorem, one can bijectively map the interior of an $n$-gon $P$ to that of another $n$-gon $Q$ conformally. However, (the boundary extension of) this mapping need not necessarily map the vertices of $P$ to those $Q$. In this case, one wants to find the best" mapping between these polygons, i.e., one that minimizes the maximum angle distortion (the dilatation) over \textit{all} points in $P$. From complex analysis such maps are known to exist and are unique. They are called extremal quasiconformal maps, or Teichm\"{u}ller maps. Although there are many efficient ways to compute or approximate conformal maps, there is currently no such algorithm for extremal quasiconformal maps. This paper studies the problem of computing extremal quasiconformal maps both in the continuous and discrete settings. We provide the first constructive method to obtain the extremal quasiconformal map in the continuous setting. Our construction is via an iterative procedure that is proven to converge quickly to the unique extremal map. To get to within $\epsilon$ of the dilatation of the extremal map, our method uses $O(1/\epsilon^{4})$ iterations. Every step of the iteration involves convex optimization and solving differential equations, and guarantees a decrease in the dilatation. Our method uses a reduction of the polygon mapping problem to that of the punctured sphere problem, thus solving a more general problem. We also discretize our procedure. We provide evidence for the fact that the discrete procedure closely follows the continuous construction and is therefore expected to converge quickly to a good approximation of the extremal quasiconformal map.