We present a geometric method for parameterization, matching, and analysis of surface shapes. Surfaces are parameterized and represented by intrinsic coordinate maps derived from the conformal structure of the shape. This parameterization is invariant to rigid transformations of the shape, as well as angle-preserving parameterizations of the surface. Shape matching between coordinate maps of two surfaces is achieved by i) deforming the isothermal curves of the intrinsic parameterization under a nonlinear transformation, and ii) locally reparameterizing the isothermal curves to yield invariant diffeomorphic matchings. We show experimental results for open surfaces such as facial geometries, as well as closed surfaces representing neuroanatomical shapes such as the hippocampus and the cortex. Lastly, we show significant statistical effects of age on the morphology of the hippocampus for a population of healthy individuals.
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