Sampling-efficient mesh parametrization

Texture-mapping is a traditional graphics technique that has several applications in computer graphics rendering. It uses surface signals to achieve a variety of rendering effects, including color mapping, bump mapping (where surface normals are the signal), displacement mapping (geometry), and self-shadowing. While these rendering effects can also be computed in vertex shaders, texture-mapping is advantageous because storing and processing texture images is generally more efficient than refining the geometry to represent the detailed signal at the vertices of a dense mesh.; To allow texture-mapping, a surface must be parametrized onto a texture domain by assigning texture coordinates to its vertices. This thesis presents new metrics and algorithms to parametrize a 3D surface onto the 2D domain for the purpose of texture-mapping. The main contributions of this thesis are two “stretch-based” parametrization algorithms. As opposed to previous methods, our new geometric-stretch metric directly minimizes geometry undersampling in the texture domain. Our novel signal-stretch metric constructs a parametrization that is specialized to store a given surface signal while attempting to minimize a measure of signal approximation error. This latter metric allocates more texture samples to regions of the surface where there is higher signal variation. These metrics allow for more efficient use of the texture-map, yielding significantly better results than previous approaches for storing signals over surfaces.; Two mesh parametrization algorithms that optimize the abovementioned metrics are presented. The first algorithm is based on iterative local optimization of the texture coordinates, while the second algorithm employs hierarchical techniques to accelerate the process and improve results. These algorithms can also be used to optimize other parametrization metrics.; In order to handle complex meshes that cannot be parametrized onto the plane without incurring severe distortion, we present methods to partition a complex mesh into multiple charts that are suitable for parametrization. We also present methods to efficiently pack the parametrized 2D charts onto a texture atlas. All of the above methods allow us to create efficient parametrizations of complex meshes.; We also describe additional techniques to parametrize progressive meshes so that the same texture-map can be used for all levels of detail. Finally, we show some recent results of applying our parametrization methods to resample the model's geometry itself.