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Accelerated local progressive-iterative approximation methods for curve and surface fitting

VISUAL COMPUTER 2025年第8期41卷 5979-5993页

作者： Yao, Zhenmin Hu, Qianqian Zhejiang Gongshang Univ Sch Stat & Math Hangzhou 310018 Peoples R China

progressive-iterative approximation is an efficient tool for data interpolation and approximation in CAD systems. For greater flexibility, a local progressive-iterative approximation method was proposed in [Lin HW, Local progressive-iterative approximation format for blending curves and patches, Computer Aided Geometric Design 27 (2010) 322-339] by adjusting only a subset of the control points. To speed up the convergence rate, this paper presents a family of hybrid local progressive-iterative approximation methods based on a high-order convergent iterative method for computing the inverse matrix. We demonstrate that the proposed accelerated local iterative procedure is convergent and the limit curve interpolates the corresponding adjusted data points. Furthermore, we generalize the proposed iterative format of blending curves to the surface case. The iterative format is represented in matrix form to avoid the Kronecker product computation, which significantly increases computational cost and could potentially produce an ill-conditioned matrix. Several numerical examples, including adaptive fitting, are given to validate that our method outperforms the existing approaches, particularly with larger datasets.

关键词： progressive-iterative approximation Local format Convergence Data fitting Geometric iterative method

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Fairing-PIA: progressive-iterative approximation for fairing curve and surface generation

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VISUAL COMPUTER 2024年第3期40卷 1467-1484页

作者： Jiang, Yini Lin, Hongwei Huang, Weixian Zhejiang Univ Sch Math Sci Hangzhou 310058 Peoples R China ZWSOFT Guangzhou 510623 Peoples R China

The fairing curves and surfaces are used extensively in geometric design, modeling, and industrial manufacturing. However, the majority of conventional fairing approaches, which lack sufficient parameters to improve fairness, are based on energy minimization problems. In this study, we develop a novel progressive-iterative approximation method for the fairing curve and surface generation (fairing-PIA). Fairing-PIA is an iteration method that can generate a series of curves (surfaces) by adjusting the control points of B-spline curves (surfaces). In fairing-PIA, each control point is endowed with an individual weight. Thus, the fairing-PIA has many parameters to optimize the shapes of curves and surfaces. Not only a fairing curve (surface) can be generated globally through fairing-PIA, but also the curve (surface) can be improved locally. Moreover, we prove the convergence of the developed fairing-PIA and show that the conventional energy minimization fairing model is a special case of fairing-PIA. Finally, numerical examples indicate that the proposed method is effective and efficient.

关键词： Curve and surface fairing progressive-iterative approximation Fairing-PIA Energy minimization Geometric iteration method

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Conjugate-Gradient progressive-iterative approximation for Loop and Catmull-Clark Subdivision Surface Interpolation

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Journal of Computer Science & Technology 2022年第2期37卷 487-504页

作者： Yusuf Fatihu Hamza Hong-Wei Lin School of Mathematical Science Zhejiang UniversityHangzhou 310027China State Key Laboratory of CAD Sz CG Zhejiang UniversityHangzhou 310058China

Loop and Catmull-Clark are the most famous approximation subdivision schemes,but their limit surfaces do not interpolate the vertices of the given ***-iterative approximation(PIA)is an efficient method for data interpolation and has a wide range of applications in many fields such as subdivision surface fitting,parametric curve and surface fitting among ***,the convergence rate of classical PIA is *** this paper,we present a new and fast PIA format for constructing interpolation subdivision surface that interpolates the vertices of a mesh with arbitrary *** proposed method,named Conjugate-Gradient progressive-iterative approximation(CG-PIA),is based on the Conjugate-Gradient iterative algorithm and the progressive iterative approximation(PIA)*** method is presented using Loop and Catmull-Clark subdivision ***-PIA preserves the features of the classical PIA method,such as the advantages of both the local and global scheme and resemblance with the given ***,CG-PIA has the following features.1)It has a faster convergence rate compared with the classical PIA and W-PIA.2)CG-PIA avoids the selection of weights compared with W-PIA.3)CG-PIA does not need to modify the subdivision schemes compared with other methods with fairness *** examples for Loop and Catmull-Clark subdivision surfaces are provided in this paper to demonstrate the efficiency and effectiveness of CG-PIA.

关键词： progressive-iterative approximation Loop subdivision Catmull-Clark subdivision conjugate-gradient method

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Weighted local progressive-iterative approximation property for triangular Bezier surfaces

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VISUAL COMPUTER 2022年第11期38卷 3819-3830页

作者： Hu, Qianqian Wang, Jiadong Liang, Ruyi Zhejiang Gongshang Univ Sch Math & Stat Hangzhou 310018 Peoples R China

progressive-iterative approximation (abbr. PIA) is an important and intuitive method for fitting and interpolating scattered data points. The triangular Bernstein basis with uniformly distributed parameters has the PIA property. For the sake of more flexibility, this paper presents a local progressive-iterative approximation (abbr. LPIA) format, which allows only a chosen subset of the initial control points to adjust and shows that the LPIA format is convergent for triangular Bezier surface of degree n <= 17 with uniform parameters. Furthermore, in order to accelerate the convergence rate, we develop a weighted LPIA format for triangular Bezier surfaces and prove that the weighted LPIA format has a faster convergence rate than the LPIA format when an optimal value of the weight is chosen. Finally, some numerical examples are presented to show the effectiveness of the LPIA method and the fast convergence of the weighted LPIA method.

关键词： progressive-iterative approximation Local format Weight Triangular Bezier surface Bernstein basis function Convergence rate

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Generalized and optimal sequence of weights on a progressive-iterative approximation method with memory for least square fitting

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2022年第17期45卷 11013-11030页

作者： Channark, Saknarin Kumam, Poom Martinez-Moreno, Juan Chaipunya, Parin Jirakitpuwapat, Wachirapong King Mongkuts Univ Technol Thonburi KMUTT Fac Sci Dept Math 126 Pracha Uthit Rd Bangkok 10140 Thailand King Mongkuts Univ Technol Thonburi KMUTT Fac Sci Ctr Excellence Theoret & Computat Sci TaCS CoE Sci Lab Bldg126 Pracha Uthit Rd Bangkok 10140 Thailand China Med Univ China Med Univ Hosp Dept Med Res Taichung Taiwan Univ Jaen Fac Expt Sci Dept Math Jaen Spain Natl Sci & Technol Dev Agcy NSTDA Natl Secur & Dual Use Technol Ctr Pathum Thani Thailand

The generalized and optimal sequence of weights on a progressive-iterative approximation method with memory for least square fitting (GOLSPIA) improves the MLSPIA method by extends to the multidimensional data fitting. In addition, weights of the moving average are varied between iterations, using the three optimal sequences of weights derived from the singular values of the collocation matrix. It is proved that a series of data fitting with an appropriate alternative of weights converge to the solution of least square fitting. Moreover, the convergence rate of the new method is faster than that of the MLSPIA method. Some examples and applications in this paper show the efficiency and effectiveness of the GOLSPIA method.

关键词： least square fitting progressive-iterative approximation optimal sequence of weights

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Variational progressive-iterative approximation for RBF-based surface reconstruction

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VISUAL COMPUTER 2021年第9-11期37卷 2485-2497页

作者： Liu, Shengjun Liu, Tao Hu, Ling Shang, Yuanyuan Liu, Xinru Cent South Univ Sch Math & Stat Changsha Peoples R China Hunan First Normal Univ Sch Math & Comp Sci Changsha Peoples R China

RBF-based methods play a very important role in the point cloud reconstruction field. However, solving a linear system is the bottleneck of such methods, especially when there are a large number of points and lead the computing to be time-consuming and unstable. In this paper, we firstly construct a novel implicit progressive-iterative approximation framework based on RBFs, which could elegantly reconstruct curves and surfaces or even higher dimensional data in an approximation or interpolation way, avoiding expensive computational cost on solving linear systems. Then, we further accelerate the proposed method with a strategy inspired from the conjugate gradient algorithm. In our framework, using proper RBFs allows to simply transform the iteration matrix to be symmetrical and positive definite. Such a property contributes to reduce the computational cost greatly and produce high-quality reconstruction results. Plenty of numerical examples on various challenging data are provided to demonstrate our efficiency, effectiveness, and superiority to other methods.

关键词： Implicit curve and surface progressive-iterative approximation Conjugate gradient method

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Implicit Randomized progressive-iterative approximation for Curve and Surface Reconstruction

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COMPUTER-AIDED DESIGN 2022年 152卷

作者： Wang, Huidi China Jiliang Univ Sch Sci Hangzhou 310018 Peoples R China

In this paper, we propose the implicit randomized progressive-iterative approximation (IR-PIA) method for curve and surface reconstruction. By introducing the effective probability criterion, the IR-PIA method selects the working elements of the collocation matrix to adjust the control coefficients. It is proved that the sequence of curves and surfaces generated by the control coefficients converge to the least-norm results in expectation. The IR-PIA method reduces the computation complexity and speeds up the curve and surface reconstruction compared with the implicit progressive-iterative approximation (I-PIA) method (Hamza et al., 2020). Numerical examples show that the IR-PIA method can be more efficient than the I-PIA method (Hamza et al., 2020).(c) 2022 Elsevier Ltd. All rights reserved.

关键词： I-PIA method Curve and surface reconstruction Convergence The implicit randomized progressive-iterative approximation method

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Implicit progressive-iterative approximation for curve and surface reconstruction

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COMPUTER AIDED GEOMETRIC DESIGN 2020年 77卷 101817-101817页

作者： Hamza, Yusuf Fatihu Lin, Hongwei Li, Zihao Zhejiang Univ Sch Math Sci Hangzhou 310027 Peoples R China Zhejiang Univ State Key Lab CAD&CG Hangzhou 310058 Peoples R China

Implicit curve and surface reconstruction attracts the attention of many researchers and gains a wide range of applications, due to its ability to describe objects with complicated geometry and topology. However, extra zero-level sets or spurious sheets arising in the reconstruction process make the reconstruction result challenging to be interpreted and damage the final result. In this paper, we propose an implicit curve and surface reconstruction method based on the progressive-iterative approximation, named implicit progressive-iterative approximation (I-PIA). The proposed method elegantly and naturally eliminates the spurious sheets without requiring any explicit minimization procedure, thus significantly reducing the computing cost and providing high-quality reconstruction results. Numerical examples are presented to demonstrate the efficiency and effectiveness of the proposed method. (C) 2020 Elsevier B.V. All rights reserved.

关键词： Implicit curve and surface Curve and surface fitting progressive-iterative approximation

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Composite iteration for isogeometric collocation method using LSPIA and Schulz iteration

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2025年 181卷 111-132页

作者： Li, Gengchen Lin, Hongwei Gao, Depeng Zhejiang Univ Sch Math Sci Hanghzou 310058 Peoples R China

The isogeometric least-squares collocation method (IGA-L) is an effective numerical technique for solving partial differential equations (PDEs), which utilizes the non-uniform rational B-splines (NURBS) to represent the numerical solution and constructs a system of equations using more collocation points than the number of unknowns. However, on the one hand, the convergence rate of the isogeometric collocation method is lower than that of the isogeometric Galerkin (IGA-G) method;on the other hand, the freedom of the numerical solution cannot be determined in advance to reach specified precision. In this paper, we model the solution of PDEs using IGA-L as a data fitting problem, in which the linear combination of the numerical solution and its derivatives is employed to fit the load function. Moreover, we develop a composite iterative method combining the least-squares progressive-iterative approximation (LSPIA) with the three-order Schulz iteration to solve the data fitting problem. The convergence of composite iterative method is proved, and the error bound is analyzed. Numerical results demonstrate that the convergence rate of the composite iterative method for IGA-L is nearly the same as that of IGA-G. Finally, we propose an incremental fitting algorithm with the composite iterative method, by which the freedom of numerical solution can be determined automatically to reach the specified fitting precision.

关键词： Isogeometric analysis Isogeometric collocation method progressive-iterative approximation Schulz iteration

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Persistence B-spline grids: stable vector representation of persistence diagrams based on data fitting

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MACHINE LEARNING 2024年第3期113卷 1373-1420页

作者： Dong, Zhetong Lin, Hongwei Zhou, Chi Zhang, Ben Li, Gengchen Zhejiang Univ Sch Math Sci Hangzhou 310058 Peoples R China

Many attempts have been made in recent decades to integrate machine learning (ML) and topological data analysis. A prominent problem in applying persistent homology to ML tasks is finding a vector representation of a persistence diagram (PD), which is a summary diagram for representing topological features. From the perspective of data fitting, a stable vector representation, namely, persistence B-spline grid (PBSG), is developed based on the efficient technique of progressive-iterative approximation for least-squares B-spline function fitting. We theoretically prove that the PBSG method is stable with respect to the metric of 1-Wasserstein distance defined on the PD space. The developed method was tested on a synthetic data set, data sets of randomly generated PDs, data of a dynamical system, and 3D CAD models, showing its effectiveness and efficiency.

关键词： Persistent homology Machine learning Persistence diagram B-spline function progressive-iterative approximation

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