Notes On Mesh Parametrization

Parameterization has lots of applications in computer graphics such as texture mapping, remeshing and morphing. Texture mapping a 2D image onto a 3D mesh would previously cost lots of efforts and time for an experienced modeler to tweak the u,v texture coordinates of the mesh. By parameterizing the mesh onto a rectangular 2D domain, this becomes an easy task of mapping a 2D image onto an rectangle. Parameterizing a 3D mesh would also benefits the task of remeshing since it will be easier to redistribute the samples and to perform re-triangulation on 2D domain. Moreover, by transforming the 3D irregular mesh data into 2D regular resampled data [9], a good number of well-established algorithms which works generally on 2D regular grids could then also be effectively performed on the mesh, like JPEG compression. There are roughly two research directions in parameterization, planar and non planar. The planar parameterization focuses on minimizing the specific stretching when mapping the 3D patch onto the 2D planar domain. The usual process of these methods is first defining an error metric which measure a specific error caused by the mapping. It then formulates the mapping as the energy minimization using the error metric and solves the system to obtain results in parameter domain. The non-planar parameterization tries to find parameterization on the domain which is topologically equivalent to the mesh. These methods in general focus on constructing a target domain suitable for building the parameterization for the whole mesh globally.