Unconstrained Spherical Parameterization

Ilja Friedel, Peter Schröder, and Mathieu Desbrun
California Institute of Technology

This paper appears in issue Volume 12, Number 1.
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Abstract

We introduce a novel approach for the construction of spherical parameterizations based on energy minimization. The energies are derived in a general manner from classic formulations well known in the planar parameterization setting (e.g., conformal, Tutte, area, stretch energies, etc.), based on the following principles: the energy should (1) be a measure of spherical triangles; (2) treat energies independently of the triangle location on the sphere; and (3) converge to the continuous energy from above under refinement. Based on these considerations, we give a very simple nonlinear modification of standard formulas that fulfills all these requirements. The method avoids the usual collapse of flat energies when they are transferred to the spherical setting without additional constraints (e.g., fixing three or more vertices). Our unconstrained energy minimization problem is amenable to the use of standard solvers. Consequently, the implementation effort is minimal while still achieving excellent robustness and performance through the use of widely available numerical minimization software.

Author Information

Ilja Friedel, NVIDIA, 2701 San Tomas Expressway, Santa Clara, CA 95050 ifriedel@nvidia.com

Peter Schröder, California Institute of Technology, MS 256-80, Pasadena, CA 91125 ps@cs.caltech.edu

Mathieu Desbrun, California Institute of Technology, MS 256-80, Pasadena, CA 91125 mathieu@cs.caltech.edu

BibTeX Entry

@article{FriedelEtAl07,
  author = "Ilja Friedel and Peter Schröder and Mathieu Desbrun",
  title = "Unconstrained Spherical Parameterization",
  journal = "journal of graphics tools",
  volume = "12",
  number = "1",
  pages = "17-26",
  year = "2007",
}