Shape transformation for polyhedral objects
SIGGRAPH '92 Proceedings of the 19th annual conference on Computer graphics and interactive techniques
Parametrization of closed surfaces for 3-D shape description
Computer Vision and Image Understanding
Signal-specialized parametrization
EGRW '02 Proceedings of the 13th Eurographics workshop on Rendering
Conformal Surface Parameterization for Texture Mapping
IEEE Transactions on Visualization and Computer Graphics
One-to-one piecewise linear mappings over triangulations
Mathematics of Computation
Spherical parametrization and remeshing
ACM SIGGRAPH 2003 Papers
Fundamentals of spherical parameterization for 3D meshes
ACM SIGGRAPH 2003 Papers
Practical Spherical Embedding of Manifold Triangle Meshes
SMI '05 Proceedings of the International Conference on Shape Modeling and Applications 2005
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We discuss how to map simply connected digital surfaces to the unit sphere using discrete harmonic functions. This technique is well known for constructing parametrizations of surfaces onto simpler domains. A common problem is the creation of dense clusters of vertices, which leads to numerical instability of methods operating on this mapping. By an explicit calculation, we quantify the cluster density for a simple model example. This example shows that distances between mapped vertices can decrease exponentially in regions on the sphere. By numerical examples, we show that severe clustering often occurs for natural objects. A computationally inexpensive algorithm, based on a bijective transformation of the unit sphere, is suggested for post-processing of clusters. Experiments indicate that this algorithm often improves the mapping such that convergence of a non-linear optimization program is achieved. The program aims at optimizing the parameterization, making it useful for global shape analysis. Examples of approximations in terms of spherical harmonics functions are presented.