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Design and analysis of optimization methods for subdivision surface fitting.

Kin-Shing Cheng1, Wenping Wang, Hong Qin

  • 1Department of Computer Science, The University of Hong Kong, Hong Kong. ksdcheng@cs.hku.hk

IEEE Transactions on Visualization and Computer Graphics
|July 12, 2007
PubMed
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This study introduces a framework for creating subdivision surfaces from point data. Squared and tangent distance minimization methods offer more efficient convergence than point distance minimization.

Area of Science:

  • Computer Graphics
  • Numerical Analysis
  • Optimization

Background:

  • Approximating unorganized point sample data with subdivision surfaces is a key problem in computer graphics.
  • Existing optimization schemes for this problem exhibit varying convergence properties and efficiencies.

Purpose of the Study:

  • To present a comprehensive framework for computing subdivision surfaces from point data.
  • To analyze the convergence and stability of different optimization schemes.
  • To compare the efficiency of point distance minimization, tangent distance minimization, and squared distance minimization.

Main Methods:

  • Formulating surface approximation as a separable nonlinear least squares problem.
  • Investigating three geometrically-motivated optimization schemes: point distance minimization, tangent distance minimization, and squared distance minimization.

Related Experiment Videos

  • Analyzing the convergence properties by relating them to standard nonlinear optimization methods (gradient descent, Gauss-Newton, Newton).
  • Examining the impact of step size control methods like Levenberg-Marquardt regularization and the Armijo rule.

    • Main Results:

    • Point distance minimization exhibits linear convergence.
    • Tangent distance minimization shows near quadratic convergence under specific conditions.
    • Squared distance minimization, derived from the Newton method, offers improved efficiency.
    • Both tangent and squared distance minimization are more efficient than point distance minimization with proper regularization.
    • Levenberg-Marquardt and Armijo rules enhance convergence stability and efficiency.

    Conclusions:

    • Squared distance minimization and tangent distance minimization are superior to point distance minimization for subdivision surface approximation.
    • The choice of optimization scheme and step size control significantly impacts computational efficiency and stability.
    • This framework provides a robust approach for accurate and efficient surface reconstruction from point data.