Computational Conformal Geometric Methods for Vision

Conformal geometry studies the geometric properties of objects invariant under conformal transformation group. It is a powerful theoretic tool to study shape classification, surface deformation, and registration. Computational conformal geometry is an emerging field combining modern geometry and computer science and develops both theories in the discrete setting and computational algorithms. This work first briefly introduces the fundamental concepts, theorems in conformal geometry, such as the Riemann mapping theorem, the uniformization theorem, the Beltrami equation, Teichmüller space theory, and so on; then explains three categories of computational algorithms: discrete surface curvature flow, harmonic maps, and holomorphic differentials based on Hodge theory; and finally demonstrates practical applications in engineering and medical imaging fields. In computer vision, the work explains Teichmüller shape space for surface classification, landmark constrained surface registration based on Teichmüller map, and optimal transport map. In medical imaging, the work introduces brain mapping, brain morphology study, virtual colonoscopy, and so on.