Computational Conformal Geometry and Its Applications

Conformal geometry has deep roots in pure mathematics. It is the intersection of complex analysis, Riemann surface theory, algebraic geometry, differential geometry and algebraic topology. Computational conformal geometry plays an important role in digital geometry processing. Recently, theory of discrete conformal geometry and algorithms of computational conformal geometry have been developed. A series of practical algorithms are presented to compute conformal mapping, which has been broadly applied in a lot of practical fields, including computer graphics, computer vision, medical imaging, visualization, and so on. The thesis focuses on computational conformal geometry and its applications on computer graphics and visualization, including surface conformal spherical parameterization, 3D shape space descriptor, quasiconformal mapping, surface remeshing, and consistent matching. Practical conformal parameterization methods are generated for specified popular applications, like human face expressions matching, and colon flattening. The initial experimental results are very promising.