Surface parameterization establishes bijective maps from a surface onto a topologically equivalent standard domain. It is well known that the spherical parameterization is limited to genus-zero surfaces. In this work, we design a new parameter domain, two-layered sphere, and present a framework for mapping high genus surfaces onto sphere. This setup allows us to trans- fer the existing applications based on general spherical parameterization to the field of high genus surfaces, such as remeshing, consistent parameterization, shape analysis, and so on. Our method is based on Riemann surface theory. We construct meromorphic functions on surfaces: for genus one surfaces, we apply Weierstrass $P$-functions; for high genus surfaces, we compute the quotient between two holomorphic one-forms. Our method of spherical parameterization is theoretically sound and practically efficient. It makes the subsequent applications on high genus surfaces very promising.
Wei Zeng. Xin Li. Shing-Tung Yau. Xianfeng Gu. "Conformal Spherical Parametrization for High Genus Surfaces." Commun. Inf. Syst. 7 (3) 273 - 286, 2007.