Conformal Structure is a natural geometric structure on surfaces, which governs many physics phenomena, such as heat diffusion, electric-magnetic fields, etc. Conformal field theory plays fundamental role in string theory. All metric orientable surfaces are Riemann surfaces and admit conformal structure. We develop general approaches that conformally parameterize brain anatomical surfaces with Riemann surface structure. With harmonic energy minimization, holomorphic 1-form and the Ricci flow methods, we can parameterize brain surfaces with various canonical surfaces such as sphere, Euclidean plane and punched hole disks. The resulting surface subdivision and the parameterizations of the components are intrinsic and stable. Our parameterization scheme offers ways to apply harmonic analysis to compare anatomy, explicitly match landmark curves in anatomical surfaces such as the cortex, and generate grids on surfaces for PDE-based signal processing. Various applications of our research will also be discussed.
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