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Spherical parametrization and remeshing

Spherical parametrization and remeshing
Emil Praun, Hugues Hoppe.
ACM Trans. Graphics (SIGGRAPH), 22(3), 2003.
Robust mapping of a surface onto a sphere, allowing 2D-grid resampling.
Abstract: 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 scale-distortion 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 semi-regular 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-of-detail, morphing, compression, and smooth surface subdivision.
Hindsights: The execution times listed in the paper were reduced by a factor of 5 after further code optimization (see Talk slides).

Our approach avoids a fundamental limitation of the traditional conformal parametrization, which is that the solution may simply run out of numerical precision due to the severe scale distortion.

Inter-surface mapping allows direct optimization of the map over the octahedron, rather than through the sphere intermediary, resulting in improved parametrization efficiency and reconstruction accuracy. Polycube maps are an interesting generalization of our cube parametrization.

Our flat-octahedron parametrization can be seen as using a Euclidean domain with 4 cone singularities, each of 180 degrees.

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