Skip to main content
SpringerLink
Log in

The Convergence of Least-Squares Progressive Iterative Approximation for Singular Least-Squares Fitting System

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

Data fitting is an extensively employed modeling tool in geometric design. With the advent of the big data era, the data sets to be fitted are made larger and larger, leading to more and more leastsquares fitting systems with singular coefficient matrices. LSPIA (least-squares progressive iterative approximation) is an efficient iterative method for the least-squares fitting. However, the convergence of LSPIA for the singular least-squares fitting systems remains as an open problem. In this paper, the authors showed that LSPIA for the singular least-squares fitting systems is convergent. Moreover, in a special case, LSPIA converges to the Moore-Penrose (M-P) pseudo-inverse solution to the leastsquares fitting result of the data set. This property makes LSPIA, an iterative method with clear geometric meanings, robust in geometric modeling applications. In addition, the authors discussed some implementation detail of LSPIA, and presented an example to validate the convergence of LSPIA for the singular least-squares fitting systems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Pereyra V and Scherer G, Least squares scattered data fitting by truncated svds, Applied Numerical Mathematics, 2002, 40(1): 73–86.

    Article  MATH  Google Scholar 

  2. Pereyra V and Scherer G, Large scale least squares scattered data fitting, Applied Numerical Mathematics, 2003, 44(1): 225–239.

    Article  MathSciNet  MATH  Google Scholar 

  3. Lin H and Zhang Z, An efficient method for fitting large data sets using T-splines, SIAM Journal on Scientific Computing, 2013, 35(6): A3052–A3068.

    Google Scholar 

  4. Deng C and Lin H, Progressive and iterative approximation for least squares B-spline curve and surface fitting, Computer-Aided Design, 2014, 47: 32–44.

    Article  MathSciNet  Google Scholar 

  5. Brandt C, Seidel H P, and Hildebrandt K, Optimal spline approximation via l0-minimization, Computer Graphics Forum, 2015, 34: 617–626.

    Article  Google Scholar 

  6. Lin H, Jin S, Hu Q, et al., Constructing B-spline solids from tetrahedral meshes for isogeometric analysis, Computer Aided Geometric Design, 2015, 35: 109–120.

    Article  MathSciNet  MATH  Google Scholar 

  7. Lin H, Wang G, and Dong C, Constructing iterative non-uniform B-spline curve and surface to fit data points, Science in China, Series F, 2004, 47(3): 315–331.

    MathSciNet  MATH  Google Scholar 

  8. Lin H, Bao H, and Wang G, Totally positive bases and progressive iteration approximation, Computers and Mathematics with Applications, 2005, 50(3): 575–586.

    Article  MathSciNet  MATH  Google Scholar 

  9. Lin H and Zhang Z, An extended iterative format for the progressive-iteration approximation, Computers & Graphics, 2011, 35(5): 967–975.

    Article  Google Scholar 

  10. Shi L and Wang R, An iterative algorithm of nurbs interpolation and approximation, Journal of Mathematical Research and Exposition, 2006, 26(4): 735–743.

    MathSciNet  MATH  Google Scholar 

  11. Cheng F, Fan F, Lai S, et al., Loop subdivision surface based progressive interpolation, Journal of Computer Science and Technology, 2009, 24(1): 39–46.

    Article  MathSciNet  Google Scholar 

  12. Fan F, Cheng F, and Lai S, Subdivision based interpolation with shape control, Computer Aided Design & Applications, 2008, 5(1–4): 539–547.

    Article  Google Scholar 

  13. Chen Z, Luo X, Tan L, et al., Progressive interpolation based on catmull-clark subdivision surfaces, Computer Grahics Forum, 2008, 27(7): 1823–1827.

    Article  Google Scholar 

  14. Maekawa T, Matsumoto Y, and Namiki K, Interpolation by geometric algorithm, Computer-Aided Design, 2007, 39: 313–323.

    Article  Google Scholar 

  15. Kineri Y, Wang M, Lin H, et al., B-spline surface fitting by iterative geometric interpolation/ approximation algorithms, Computer-Aided Design, 2012, 44(7): 697–708.

    Article  Google Scholar 

  16. Yoshihara H, Yoshii T, Shibutani T, et al., Topologically robust B-spline surface reconstruction from point clouds using level set methods and iterative geometric fitting algorithms, Computer Aided Geometric Design, 2012, 29(7): 422–434.

    Article  MathSciNet  MATH  Google Scholar 

  17. Okaniwa S, Nasri A, Lin H, et al., Uniform B-spline curve interpolation with prescribed tangent and curvature vectors, IEEE Transactions on Visualization and Computer Graphics, 2012, 18(9): 1474–1487.

    Article  Google Scholar 

  18. Lin H, Qin Y, Liao H, et al., Affine arithmetic-based B-spline surface intersection with gpu acceleration, IEEE Transactions on Visualization and Computer Graphics, 2014, 20(2): 172–181.

    Article  Google Scholar 

  19. Sederberg T W, Cardon D L, Finnigan G T, et al., T-spline simplification and local refinement, ACM Transactions on Graphics, 2004, 23: 276–283.

    Article  Google Scholar 

  20. Zhang Y, Wang W, and Hughes T J, Solid T-spline construction from boundary representations for genus-zero geometry, Computer Methods in Applied Mechanics and Engineering, 2012, 249: 185–197.

    Article  MathSciNet  MATH  Google Scholar 

  21. Horn R A and Johnson C R, Matrix Analysis, Volume 1, Cambridge University Press, Cambridge, 1985.

    Book  MATH  Google Scholar 

  22. James M, The generalised inverse, The Mathematical Gazette, 1978, 62(420): 109–114.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Science, State Key Laboratory of CAD&CG, Zhejiang University, Hangzhou, 310027, China

    Hongwei Lin, Qi Cao & Xiaoting Zhang

Authors
  1. Hongwei Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qi Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaoting Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hongwei Lin.

Additional information

This research was supported by the Natural Science Foundation of China under Grant No. 61379072.

This paper was recommended for publication by Editor-in-Chief GAO Xiao-Shan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lin, H., Cao, Q. & Zhang, X. The Convergence of Least-Squares Progressive Iterative Approximation for Singular Least-Squares Fitting System. J Syst Sci Complex 31, 1618–1632 (2018). https://doi.org/10.1007/s11424-018-7443-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-018-7443-y

Keywords

Search

Navigation