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<p>Surface fitting of discrete data points is a critical challenge in geometric modeling design and reverse engineering applications. Although progressive-iterative approximation for least-squares fitting (LSPIA) provides an effective solution, it is inherently constrained by slow convergence rates. To address this limitation, this paper proposes a novel randomized Gauss–Seidel LSPIA (RGS-LSPIA) method for efficient surface fitting. By introducing a randomized selection strategy for control point adjustments in each iteration, our approach significantly reduces computational cost and enhances efficiency. Moreover, the proposed algorithm decomposes the surface approximation problem into two curve approximation subproblems, thereby avoiding the computation of the Kronecker product of matrices. This mitigates the issue of numerical instability caused by ill-conditioned matrix properties and significantly improves computational efficiency. Additionally, a detailed convergence analysis is also provided. Numerical examples demonstrate that the proposed method outperforms existing methods, particularly in processing large-scale data sets, achieving both faster convergence and superior computational efficiency.</p>

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Efficient surface fitting via randomized Gauss–Seidel LSPIA: a novel iterative approach

  • 期刊论文

  • 发表日期：

    2026年2月16日

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  • Zhenmin Yao,
  • Qianqian Hu

摘要

Surface fitting of discrete data points is a critical challenge in geometric modeling design and reverse engineering applications. Although progressive-iterative approximation for least-squares fitting (LSPIA) provides an effective solution, it is inherently constrained by slow convergence rates. To address this limitation, this paper proposes a novel randomized Gauss–Seidel LSPIA (RGS-LSPIA) method for efficient surface fitting. By introducing a randomized selection strategy for control point adjustments in each iteration, our approach significantly reduces computational cost and enhances efficiency. Moreover, the proposed algorithm decomposes the surface approximation problem into two curve approximation subproblems, thereby avoiding the computation of the Kronecker product of matrices. This mitigates the issue of numerical instability caused by ill-conditioned matrix properties and significantly improves computational efficiency. Additionally, a detailed convergence analysis is also provided. Numerical examples demonstrate that the proposed method outperforms existing methods, particularly in processing large-scale data sets, achieving both faster convergence and superior computational efficiency.