Parametrization for surfaces with arbitrary topologies

Xianfeng Gu
Thesis, Computer Science, Harvard University, 2002
Surface parametrization is a fundamental problem in computer graphics. It is essential for operations such as texture mapping, texture synthesis on surface, interactive 3D painting, remeshing and multi-resolution analyis and mesh compression. Conformal parametrization, which preserves angles, has many nice properties such as having no local distortion on textures, and being independent of trangulation or resolution. Existing conformal parametrization methods partition the mesh into several charts, each of which is then parametrized and packed to an atlas. These methods surffer from limitations such as difficulty in segmenting the mesh and artifacts caused by discontinuites between charts. This paper presents a method to compute global conformal parametrization for triangulated surfaces with arbitrary topologies. Our method is boundary free, hence eliminating the need to chartify the mesh. We compute the natural conformal structure of the surface, which is determined solely by its geometry. The parametrization is stable in the sense that if the geometries are similar, then the parametrizations on canonical domain are also close. The parametrization is conformal everywhere except on $2g-2$ number of points, where $g$ is the number of genus. We first study special tangential vector fileds, the so called holomorphic differentials, which are the gradients of local conformal maps. It is proven that all discrete holomorphic differentials form a $2g$ dimensional linear space. We derive practical algorithms of computing the bases of this linear space from the constructive proofs. Then by integrating the linear combination of a set of bases, we can construct any global conformal parametrization of the surface. The algorithms only involve solving linear systems and is easy to implement. In this work, we also propose to remesh an arbitrary surface onto a completely regular structure, we call \emph{geometry image}. It captures geometry as a simple 2D array of quantized points. Surface signals like normals and colors are stored in similar 2D arrays using the same implicit surface parametrization-texture coordinates are absent. To create a geometry image, we cut an arbitary mesh along a network of edge paths, and parametrize the resulting single chart onto a square. Geometry images unifies the geometry and image format. Many techinques in image processing field can be applied to geometry directly. The methods introduced in this paper are very general and can be applied to many other topologic and geomeric problems in computer graphics.