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Weighted local progressive-iterative approximation property for triangular Bézier surfaces

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Abstract

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 Bézier surface of degree \(n \le 17\) with uniform parameters. Furthermore, in order to accelerate the convergence rate, we develop a weighted LPIA format for triangular Bézier 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.

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Acknowledgements

The authors thank the anonymous referees for their valuable suggestions and comments. This work is supported by the National Natural Science Foundation of China (Grant Nos 61772025, 61872316), the Natural Science Foundation of Zhejiang Province, China (No. LY19F020004).

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Authors and Affiliations

  1. School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou, 310018, China

    Qianqian Hu, Jiadong Wang & Ruyi Liang

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  1. Qianqian Hu
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  2. Jiadong Wang
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Appendix A. Main symbol table

Appendix A. Main symbol table

\({I}_{n}\)

The set of all 3-composition of the integer n

\(\Gamma\)

The set of the multi-index of the adjusted control points

\(\Phi\)

The set of the multi-index of the fixed control points

\(D_{{n,T}}\)

\(\{ ({i \mathord{\left/ {\vphantom {i n}} \right. \kern-\nulldelimiterspace} n},{j \mathord{\left/ {\vphantom {j n}} \right. \kern-\nulldelimiterspace} n},{k \mathord{\left/ {\vphantom {k n}} \right. \kern-\nulldelimiterspace} n}):(i,j,k) \in {I}_{n} \}\)

\({\mathbf{\xi }}_{{\mathbf{i}}}\)

The uniformly distributed parameter of the control points

\({\mathbf{R}}_{{\mathbf{i}}}\)

The initial data points

\({\mathbf{R}}_{{\mathbf{s}}}^{k}\)

The adjusted control points after the k-th iteration

\({\mathbf{R}}_{{\mathbf{h}}}^{k}\)

The fixed control points after the k-th iteration

\(\Delta _{{\mathbf{s}}}^{k}\)

The adjusting vectors after the k-th iteration

\(\Delta _{{\mathbf{h}}}^{k}\)

The difference vectors after the k-th iteration

B

The collocation matrix with uniformly distributed parameters

\({\mathbf{R}}^{k} ( \cdot )\)

The iterative B-B surface after the k-th iteration with LPIA method

\({\mathbf{R}}_{\omega }^{k} ( \cdot )\)

The iterative B-B surface after the k-th iteration with weighted LPIA method

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Hu, Q., Wang, J. & Liang, R. Weighted local progressive-iterative approximation property for triangular Bézier surfaces. Vis Comput 38, 3819–3830 (2022). https://doi.org/10.1007/s00371-021-02223-1

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