Spherical parametrization and remeshing

Recently, Gu et al. [2002] introduced geometry images, in which geometry is resampled into a completely regular 2D grid. The process involves cutting the surface into a disk using a network of cut paths, and then mapping the boundary of this disk to a square. Both geometry and other signals are stored as 2D grids, with grid samples in implicit correspondence, obviating the need to store a parametrization. Also, the boundary parametrization makes both geometry and textures seamless. The traditional approach for parametrizing a surface involves cutting it into charts and mapping these piecewise onto a planar domain. We introduce a robust technique for directly parametrizing a genus-zero surface onto a spherical domain. A key ingredient for making such a parametrization practical is the minimization of a stretch-based measure, to reduce scaledistortion and thereby prevent undersampling. Our second contribution is a scheme for sampling the spherical domain using uniformly subdivided polyhedral domains, namely the tetrahedron, octahedron, and cube. We show that these particular semiregular samplings can be conveniently represented as completely regular 2D grids, i.e. geometry images. Moreover, these images have simple boundary extension rules that aid many processing operations. Applications include geometry remeshing, level-ofdetail, morphing, compression, and smooth surface subdivision. In all three approaches, the surface is first cut into one or more disk-like charts using a network of cut paths, and a parametrization is formed piecewise on each chart. The a priori construction of the chart boundaries or cut paths is heuristic, and constrains the quality of the attainable parametrization. In texture atlases, both the number of charts and their surface extents are selected heuristically to minimize parametric distortion onto planar polygons, while also maintaining good packing efficiency. In semi-regular remeshing, surface charts are selected to have low-distortion maps onto regular domain faces, and to have approximately the same size. Finally, in geometry images, the surface is heuristically cut into a disk that hopefully maps well onto a square.