Conformal Metric Optimization on Surface (CMOS) for Deformation and Mapping in Laplace-Beltrami Embedding Space

Lab of Neuro Imaging, UCLA School of Medicine, Los Angeles, CA, USA.
Medical image computing and computer-assisted intervention : MICCAI ... International Conference on Medical Image Computing and Computer-Assisted Intervention 09/2011; 14(Pt 2):327-34. DOI: 10.1007/978-3-642-23629-7_40
Source: PubMed


In this paper we develop a novel technique for surface deformation and mapping in the high-dimensional Laplace-Beltrami embedding space. The key idea of our work is to realize surface deformation in the embedding space via optimization of a conformal metric on the surface. Numerical techniques are developed for computing derivatives of the eigenvalues and eigenfunctions with respect to the conformal metric, which is then applied to compute surface maps in the embedding space by minimizing an energy function. In our experiments, we demonstrate the robustness of our method by applying it to map hippocampal atrophy of multiple sclerosis patients with depression on a data set of 109 subjects. Statistically significant results have been obtained that show excellent correlation with clinical variables. A comparison with the popular SPHARM tool has also been performed to demonstrate that our method achieves more significant results.

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Available from: Rongjie Lai, Dec 19, 2013
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    • "For numerical implementation, we first built the signed distance function [30], [31] of the white matter surface in 3D space and then computed the CT as the value on the signed distance function at those locations. The cortical surfaces and the corresponding CT maps were registered to the International Consortium for Brain Mapping (ICBM) brain surface [32] and then vertex-wise correspondences were established between all cortical surface models using a Conformal Metric Optimization method [33]. An experienced human brain researcher rated each brain surface reconstruction by visually inspecting the surfaces using LONI ShapeViewer ( "
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    • "For anatomical shapes without perfect symmetry, we find this assumption always holds in our experience. For synthetic shapes with perfect symmetry, we can perturb the metric [37], [43] and make sure this assumption is valid. To accurately represent the partition of the surface by neighboring saddle points on the Reeb graph, which could have very subtle differences in the function values, we will augment the original mesh by splitting its triangles along the level contours during the Reeb graph construction process. "
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    ABSTRACT: In this paper we present a novel system for the automated reconstruction of cortical surfaces from T1-weighted magnetic resonance images. At the core of our system is a unified Reeb analysis framework for the detection and removal of geometric and topological outliers on tissue boundaries. Using intrinsic Reeb analysis, our system can pinpoint the location of spurious branches and topological outliers, and correct them with localized filtering using information from both image intensity distributions and geometric regularity. In this system, we have also developed enhanced tissue classification with Hessian features for improved robustness to image inhomogeneity, and adaptive interpolation to achieve sub-voxel accuracy in reconstructed surfaces. By integrating these novel developments, we have a system that can automatically reconstruct cortical surfaces with improved quality and dramatically reduced computational cost as compared with the popular FreeSurfer software. In our experiments, we demonstrate on 40 simulated MR images and the MR images of 200 subjects from two databases: the Alzheimers Disease Neuroimaging Initiative (ADNI) and International Consortium of Brain Mapping (ICBM), the robustness of our method in large scale studies. In comparisons with FreeSurfer, we show that our system is able to generate surfaces that better represent cortical anatomy and produce thickness features with higher statistical power in population studies.
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    • "In addition, a new surface quadrangulation can be also obtained using the Morse-Smale complex of certain LB eigenfunction [13]. Moreover, LB eigenfunctions can be viewed as either global or local embedding to analyze surface geometric structures [30] [7] [28] [17] [18]. Furthermore, the LB operator is also closely related to harmonic maps between two surfaces. "
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    ABSTRACT: In this paper, we propose a general framework for approximating differential operator directly on point clouds and use it for geometric understanding on them. The discrete approximation of differential operator on the underlying manifold represented by point clouds is based only on local approximation using nearest neighbors, which is simple, efficient and accurate. This allows us to extract the complete local geometry, solve partial differential equations and perform intrinsic calculations on surfaces. Since no mesh or parametrization is needed, our method can work with point clouds in any dimensions or co-dimensions or even with variable dimensions. The computation complexity scaled well with the number of points and the intrinsic dimensions (rather than the embedded dimensions). We use this method to define the Laplace-Beltrami (LB) operator on point clouds, which links local and global information together. With this operator, we propose a few key applications essential to geometric understanding for point clouds, including the computation of LB eigenvalues and eigenfunctions, the extraction of skeletons from point clouds, and the extraction of conformal structures from point clouds.
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