Journal of Systems Science and Complexity

, Volume 31, Issue 6, pp 1618–1632 | Cite as

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

  • Hongwei LinEmail author
  • Qi Cao
  • Xiaoting Zhang


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.


Data fitting geometric modeling LSPIA singular least-squares fitting system 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Pereyra V and Scherer G, Least squares scattered data fitting by truncated svds, Applied Numerical Mathematics, 2002, 40(1): 73–86.zbMATHGoogle Scholar
  2. [2]
    Pereyra V and Scherer G, Large scale least squares scattered data fitting, Applied Numerical Mathematics, 2003, 44(1): 225–239.MathSciNetzbMATHGoogle Scholar
  3. [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. [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.MathSciNetGoogle Scholar
  5. [5]
    Brandt C, Seidel H P, and Hildebrandt K, Optimal spline approximation via l0-minimization, Computer Graphics Forum, 2015, 34: 617–626.Google Scholar
  6. [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.MathSciNetGoogle Scholar
  7. [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.MathSciNetzbMATHGoogle Scholar
  8. [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.MathSciNetzbMATHGoogle Scholar
  9. [9]
    Lin H and Zhang Z, An extended iterative format for the progressive-iteration approximation, Computers & Graphics, 2011, 35(5): 967–975.Google Scholar
  10. [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.MathSciNetzbMATHGoogle Scholar
  11. [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.MathSciNetGoogle Scholar
  12. [12]
    Fan F, Cheng F, and Lai S, Subdivision based interpolation with shape control, Computer Aided Design & Applications, 2008, 5(1–4): 539–547.Google Scholar
  13. [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.Google Scholar
  14. [14]
    Maekawa T, Matsumoto Y, and Namiki K, Interpolation by geometric algorithm, Computer-Aided Design, 2007, 39: 313–323.Google Scholar
  15. [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.Google Scholar
  16. [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.MathSciNetzbMATHGoogle Scholar
  17. [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.Google Scholar
  18. [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.Google Scholar
  19. [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.Google Scholar
  20. [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.MathSciNetzbMATHGoogle Scholar
  21. [21]
    Horn R A and Johnson C R, Matrix Analysis, Volume 1, Cambridge University Press, Cambridge, 1985.zbMATHGoogle Scholar
  22. [22]
    James M, The generalised inverse, The Mathematical Gazette, 1978, 62(420): 109–114.MathSciNetzbMATHGoogle Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical Science, State Key Laboratory of CAD&CGZhejiang UniversityHangzhouChina

Personalised recommendations