| Surface parametrization is a fundamental problem in computer graphics. It is essential for operations such as texture mapping, texture synthesis, interactive 3D painting, remeshing, multi-resolution analysis and mesh compression. Conformal parameterization, which preserves angles, has many nice properties such as having no local distortion on textures, and being independent of triangulation or resolution. Existing conformal parameterization methods partition a mesh into several charts, each of which is then parametrized and packed to an atlas. These methods suffer from limitations such as difficulty in segmenting the mesh and artifacts caused by discontinuities between charts.; This work presents a method that was developed with collaboration with Professor Shing-Tung Yau to compute global conformal parameterizations 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 parameterization is stable in the sense that if the geometries are similar, then the parameterizations on the canonical domain are also close. The parameterization is conformal everywhere except on 2g − 2 number of points, where g is the number of genus. We prove the gradients of local conformal maps form a linear space and develop practical algorithms to compute its basis. By linearly combining the bases we can construct any global conformal parametrization of the surface. The algorithms only involve solving linear systems and are easy to implement.; In this work, we also propose to remesh an arbitrary surface onto a completely regular structure, we call a geometry image. This part of the thesis was completed under the supervision of Professor Steven Gortler. A geometry image 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 parameterization-texture coordinates are absent. To create a geometry image, we cut an arbitrary mesh along a network of edge paths, and parametrize the resulting single chart onto a square. Geometry images unify the geometry and image format and make it possible to apply many techniques in the image processing field to geometry directly. |
