Progressive iterative approximation for regularized least square bivariate B-spline surface fitting

Liu, Ming-Zeng; Li, Baojun; Guo, Qingjie; Zhu, Chuanjie; Hu, Ping; Shao, Yuan‐Hai

doi:10.1016/j.cam.2017.06.013

Cited by 31 publications

(12 citation statements)

References 21 publications

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“…The iteration (5.1) is the conventional iterative method for tensor product surfacefitting and always converges for all normalized totally positive bases. Therefore, the iteration (5.1) is referred to as the progressive iterative approximation, briefly denoted by PIA in CAGD, see [19,20] and references therein. Furthermore, let A and B be the collocation matrices of the bases (φ 0 , .…”

Section: Example 41 In This Example We Havementioning

confidence: 99%

“…where A ∈ R n×n , B ∈ R m×m and X, C ∈ R n×m . This matrix equation plays an important role in many practical applications such as surfaces fitting in computer aided geometric design (CAGD), signal and image processing, photogrammetry, etc., see for example [14,19,20,30,31] and a large literature therein. Given the existence of many and important applications, the theory and algorithms for matrix equations have intrigued the researchers for decades, see [7,9,15,21,28,29,30,31,34,37].…”

Section: Introduction Consider the Matrix Equationmentioning

confidence: 99%

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Stationary splitting iterative methods for the matrix equation AXB=C

Liu

Ferreira

et al. 2020

Applied Mathematics and Computation

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Stationary splitting iterative methods for solving AXB = C are considered in this paper. The main tool to derive our new method is the induced splitting of a given nonsingular matrix A = M − N by a matrix H such that (I − H) −1 exists. Convergence properties of the proposed method are discussed and numerical experiments are presented to illustrate its computational efficiency and the effectiveness of some preconditioned variants. In particular, for certain surface-fitting applications our method is much more efficient than the progressive iterative approximation (PIA), a conventional iterative method often used in computer aided geometric design (CAGD).

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Section: Example 41 In This Example We Havementioning

confidence: 99%

Section: Introduction Consider the Matrix Equationmentioning

confidence: 99%

Stationary splitting iterative methods for the matrix equation AXB=C

Liu

Ferreira

et al. 2020

Applied Mathematics and Computation

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“…]. By replacing the B-spline basis with the generalized B-spline basis, it is generalized in [9] to the weighted least square fitting curve, and, by replacing the tensor product B-spline basis with the non-tensor product bivariate B-spline basis, extended in [13] to the regularized least square fitting surface.…”

Section: Related Workmentioning

confidence: 99%

“…Since it is linear, the construction of the method may be simpler. One can look for more details about these two kinds of fittings, for examples, in [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and references therein.…”

Section: Introductionmentioning

confidence: 99%

On a progressive and iterative approximation method with memory for least square fitting

Huang

Wang

2019

Preprint

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In this paper, we present a progressive and iterative approximation method with memory for least square fitting (MLSPIA). It adjusts the control points and the weighted sums iteratively to construct a series of fitting curves (surfaces) with three weights. For any normalized totally positive basis even when the collocation matrix is of deficient column rank, we obtain a condition to guarantee that these curves (surfaces) converge to the least square fitting curve (surface) to the given data points. It is proved that the theoretical convergence rate of the method is faster than the one of the progressive and iterative approximation method for least square fitting (LSPIA) in [Deng C-Y, Lin H-W. Progressive and iterative approximation for least squares B-spline curve and surface fitting. Computer-Aided Design 2014;47:32-44] under the same assumption. Examples verify this phenomenon.

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“…基于广义 B 样条, 另一种具有不同权重的 LSPIA 算法被提出以拟合更复杂的数据点 [12][13] . 此外, Liu 等 [14] 提出了用于正则化最小二乘双变量 B 样条曲面拟合算法 (progressive iterative approximation for regularized least square bivariate B-spline surface fitting, RLSPIA), 将单变量 NTP 的 PIA 性质推广至线性相关的非张量积型二元 B 样条基. Ebrahimi 等 [15] 提出了一种复合型 PIA 算法, 采用 Schulz 迭代方法计算调整向量.…”

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Gauss-Seidel Progressive and Iterative Approximation for Least Squares Fitting

Yusuf¹,

Jiang²,

Lin³

2021

J Comput-Aided Design Comput Graphics

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Progressive-iterative approximation (PIA) is an efficient method for data fitting that attracts the attention of many researchers and has a wide range of applications. However, the convergence rate of LSPIA is prolonged. In this study, we design a fast PIA format based on the Gauss-Seidel iterative method named Gauss-Seidel progressive and iterative approximation for least squares curve and surface fitting (GS-LSPIA). Firstly, the control points of the fitting curve (surface) are selected from the given data points.Then, the chord length method is used to assign the parameters of the given data points. GS-LSPIA generates a series of fitting curves (surfaces) by refining the control points iteratively, and the limit of the generated curve (surface) is the least square fitting result to the given data points. Several experimental results presented in this paper demonstrate that, to achieve the same accuracy for GS-LSPIA and LSPIA, GS-LSPIA required fewer steps and shorter running time compared with LSPIA. Thus, the proposed GS-LSPIA is efficient and has a faster convergence rate compared with the LSPIA algorithm.

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Progressive iterative approximation for regularized least square bivariate B-spline surface fitting

Cited by 31 publications

References 21 publications

Stationary splitting iterative methods for the matrix equation AXB=C

Stationary splitting iterative methods for the matrix equation AXB=C

On a progressive and iterative approximation method with memory for least square fitting

Gauss-Seidel Progressive and Iterative Approximation for Least Squares Fitting

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