Hierarchical Optimization for Spherical Parameterization

Abstract

Spherical parameterization is a key enabling technology in geometric modeling and processing. We develop an efficient algorithm to optimize the spherical parameterization. The objective function aims to reduce both angle and area distortion. Compared with other existing spherical mapping computation algorithms, our hierarchical spherical optimization is efficient and has the guarantee of local convergence; our mapping results in a less distorted parameterization with guaranteed bijectivity. We also demonstrate the effectiveness of our spherical parameterization in spherical harmonics computation and shape analysis.

Illustrations of Spherical Parameterization Results

Bunny Model

Head Model

Spherical Mapping Results with Angle and Area Distortions Color-encoded (Red: Large, Blue: Small).
Mapping Results Illustrated on Spheres.

Statistical Results Compared to Other Spherical Parameterization Methods

References: [1: Gu and Yau 03], [2: Zayer at al. 06], [3: Praun and Hoppe 03]. E_D is Dirichlet energy indicating the angle distortion and E_A is area distortion. The execution time is measured on a desktop with AMD Athlon X2 2.9GHz CPU with 2GB RAM. Our method is very efficient which makes it suitable for large-scale models.

Related Publication

  • Efficient Spherical Parametrization Using Progressive Optimization, Shenghua Wan, Tengfei Ye, Maoqing Li, Hongchao Zhang, Xin Li, Proc. of Computational Visual Media Conference, pp. 170-177, 2012. [Paper] [Slides] [Bibtex]
  • An Efficient Spherical Mapping Algorithm and its Application on Spherical Harmonics, Shenghua Wan, Tengfei Ye, Maoqing Li, Hongchao Zhang, Xin Li, invited paper from CVM 2012 to Science China Information Science.
  • Efficient Hierarchical Optimization for Spherical Mapping, Shenghua Wan, Tengfei Ye, Maoqing Li, Hongchao Zhang, Xin Li, Techincal Report, School of EECS, LSU 2012. [Paper] [Bibtex]
  • Downloadable Program/Data

    Please contact us "xinli@cct.lsu.edu" for the program.

    .OBJ Version: Mapping Mesh in OBJ Format

    Usage: SphMapOBJ.exe inputfile.obj outputfile.obj

    The "inputfile.obj" is the OBJ format triangle mesh. It should be genus-0 and has no boundary.

    .m Version: Mapping Mesh in Hugues Hoppe's M Format

    .m File Format

  • The input .m file:
  • The three floating numbers (x, y, z) after the vertex id are the coordinates of this vertex.
  • The resultant mesh encodes both the original mesh and its coordinates on the sphere:
  • The three floating numbers (x, y, z) after the vertex id are the coordinates of this vertex. It is mapped to the Opos coordinates (ox, oy, oz) on the unit sphere.

    Usage: HOSphMap.exe in.m out.m

    where in.m is the input mesh for a closed genus-0 surface and out.m is the output spherical mapping.
    Some available mapping results here : Bimba Venus Bunny Armadillo Head

    References

    1. [Gu and Yau 03]        X. Gu and S.-T. Yau. Global conformal surface parameterization. In Proceedings of 2003 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing, pp 127-137.
    2. [Zayer et al 06]         R. Zayer, C. Rossl, and H.-P. Seidel. Curvilinear spherical parameterization. In Proceedings of IEEE International Conference on Shape Modeling and Applications 2006.
    3. [Praun and Hoppe 03] E. Praun and H. Hoppe. Spherical parametrization and remeshing. ACM Transactions on Graphics, 22:340-349, July 2003.

    Acknowledgements

    Data Sources

  • The head model is from Robert W. Sumner, Jovan Popovic. Deformation Transfer for Triangle Meshes. ACM Transactions on Graphics. 23, 3. August 2004.
  • Other models are from Stanford Shape Repository.
  • Funding Agencies

    LA Board of Regents RCS, LA Board of Regents PFunds.

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