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Progressive-iterative approximation method

In mathematics, the progressive-iterative approximation method is an iterative method of data fitting with geometric meanings.[1] Given a set of data points to be fitted, the method obtains a series of fitting curves (or surfaces) by iteratively updating the control points, and the limit curve (surface) can interpolate or approximate the given data points.[2] It avoids solving a linear system of equations directly and allows flexibility in adding constraints during the iterative process.[3] Therefore, it has been widely used in geometric design and related fields.[2]

The study of the iterative method with geometric meaning can be traced back to the work of scholars such as Dongxu Qi and Carl de Boor in the 1970s.[4][5] In 1975, Qi et al. developed and proved the "profit and loss" algorithm for uniform cubic B-spline curves,[4] and in 1979, de Boor independently proposed this algorithm.[5] In 2004, Hongwei Lin and coauthors proved that non-uniform cubic B-spline curves and surfaces have the "profit and loss" property.[3] Later, in 2005, Lin et al. proved that the curves and surfaces with normalized and totally positive basis all have this property and named it progressive iterative approximation (PIA).[1] In 2007, Maekawa et al. changed the algebraic distance in PIA to geometric distance and named it geometric interpolation (GI).[6] In 2008, Cheng et al. extended it to subdivision surfaces and named the method progressive interpolation (PI).[7] Since the iteration steps of the PIA, GI, and PI algorithms are similar and all have geometric meanings, they are collectively referred to as geometric iterative methods (GIM).[2]

PIA is now extended to several common curves and surfaces in the geometric design field,[8] including NURBS curves and surfaces,[9] T-spline surfaces,[10] and implicit curves and surfaces.[11]

  1. ^ a b Lin, Hong-Wei; Bao, Hu-Jun; Wang, Guo-Jin (2005). "Totally positive bases and progressive iteration approximation". Computers & Mathematics with Applications. 50 (3–4): 575–586. doi:10.1016/j.camwa.2005.01.023. ISSN 0898-1221.
  2. ^ a b c Lin, Hongwei; Maekawa, Takashi; Deng, Chongyang (2018). "Survey on geometric iterative methods and their applications". Computer-Aided Design. 95: 40–51. doi:10.1016/j.cad.2017.10.002. ISSN 0010-4485.
  3. ^ a b Lin, Hongwei; Wang, Guojin; Dong, Chenshi (2004). "Constructing iterative non-uniform B-spline curve and surface to fit data points". Science in China Series F. 47 (3): 315. doi:10.1360/02yf0529. ISSN 1009-2757. S2CID 966980.
  4. ^ a b Qi, Dongxu; Tian, Zixian; Zhang, Auxin; Feng, Jiabin (1975). "The method of numeric polish in curve fitting". Acta Math Sinica. 18 (3): 173–184.
  5. ^ a b Carl, de Boor (1979). "How does Agee's smoothing method work?". Proceedings of the 1979 Army Numerical Analysis and Computers Conference, ARO Report.
  6. ^ Maekawa, Takashi; Yasunori, Matsumoto; Ken, Namiki (2007). "Interpolation by geometric algorithm". Computer-Aided Design. 39 (4): 313–323. doi:10.1016/j.cad.2006.12.008.
  7. ^ Cheng, Fuhua; Fan, Fengtao; Lai, Shuhua; Huang, Conglin; Wang, Jiaxi; Yong, Junhai (2008). "Progressive Interpolation Using Loop Subdivision Surfaces". Advances in Geometric Modeling and Processing. Lecture Notes in Computer Science. Vol. 4975. pp. 526–533. doi:10.1007/978-3-540-79246-8_43. ISBN 978-3-540-79245-1.
  8. ^ Hoschek, Josef (February 1993). Fundamentals of computer aided geometric design. USA: A. K. Peters, Ltd. ISBN 978-1-56881-007-2.
  9. ^ Shi, Limin; Wang, Renhong (2006). "An iterative algorithm of NURBS interpolation and approximation". Journal of Mathematical Research with Applications. 26 (4): 735–743.
  10. ^ Lin, Hongwei; Zhang, Zhiyu (2013). "An Efficient Method for Fitting Large Data Sets Using T-Splines". SIAM Journal on Scientific Computing. 35 (6): A3052 – A3068. Bibcode:2013SJSC...35A3052L. doi:10.1137/120888569. ISSN 1064-8275.
  11. ^ Hamza, Yusuf Fatihu; Lin, Hongwei; Li, Zhao (2020). "Implicit progressive-iterative approximation for curve and surface reconstruction". Computer Aided Geometric Design. 77: 101817. arXiv:1909.00551. doi:10.1016/j.cagd.2020.101817. S2CID 202540812.

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