The Convergence of Least-Squares Progressive Iterative Approximation for Singular Least-Squares Fitting System
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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.
KeywordsData fitting geometric modeling LSPIA singular least-squares fitting system
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- 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
- Brandt C, Seidel H P, and Hildebrandt K, Optimal spline approximation via l0-minimization, Computer Graphics Forum, 2015, 34: 617–626.Google Scholar
- Lin H and Zhang Z, An extended iterative format for the progressive-iteration approximation, Computers & Graphics, 2011, 35(5): 967–975.Google Scholar
- 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
- 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
- Maekawa T, Matsumoto Y, and Namiki K, Interpolation by geometric algorithm, Computer-Aided Design, 2007, 39: 313–323.Google Scholar
- 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
- 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
- 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
- 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