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1. Conformal Metric Optimization on Surface (CMOS) for Deformation and Mapping in Laplace-Beltrami Embedding Space Yonggang Shi1, Rongjie Lai2, Raja Gill3, Daniel Pelletier4, David Mohr5, Nancy Sicotte6, Arthur W. Toga 1 1 Lab of Neuro Imaging (LONI), UCLA School of Medicine; 2Dept. of Mathematics, University of Southern California; 3Dept. Of Neurology, UCLA School of Medicine; 4Dept. of Neurology, Yale School of Medicine; 5Dept. Of Preventive Medicine, Northwestern University; 6Cedar Sinai Medical Center, Los Angeles, CA, USA Laplace-Beltrami Embedding withconformal metrics Surface mapping via deformation in Embedding space • Population study: hippocampal atrophy in multiple sclerosis (MS) patients with depression • 109 female patients split into two groups with the CES-D scale: low depression (CES-D ≤20) and high depression (CES-D>20) • Statistically significant group differences were localized on hippocampus (P=0.017) • The group differences correlate well with the clinical measure of depression • Comparison with SPHARM • Use SPHARM for mapping • No significant differences • Cortical mapping • Intrinsic mapping in high dimensional embedding space • Application: gyrallabelling • Let (M,g) be a genus–zero surface. The eigen-system of its Laplace-Beltrami operator is: • An embedding is defined as: • Conformal metrics • Numerical computation of the LB embedding with conformal metrics • Represent M as a triangular mesh • Represent the eigen-functions with Baricentric coordinate functions • Use the weak form to obtain the matrix formulation • We compute derivatives of each eigen-value and eigen-function with respect to the weight function • Derivative of the eigen-value • Derivatives of an eigen-function can be computed by solving • where • Using eigen-derivatives, we can optimize the metric in 3D shape analysis problems such as mapping and feature extraction. • Given two surfaces (M1,w1g1) and (M2,w2g2), we compute their maps by minimizing the following energy function • The distance functions are defined in the embedding space • The energy is minimized by iteratively updating the metrics: • Final maps: closest points in the embedding space Thickness p-value map Correlation with CES-D Correlation p-value map SPHARM Thickness p-value map Results • Example: mapping two hippocampal surfaces Optimized metric Atlas label Optimized embedding Left pial surface Right pial surface Default embedding www.loni.ucla.edu/LONIR NIH/NIBIB (P41-EB015922 / P41-RR013642)