Spectral adaptivity for iterative least-squares B-spline curve and surface fitting

Xingxuan Peng; Yutong Li; Xingxuan Peng; Yutong Li

doi:10.3934/math.2026998

AIMS Mathematics

2026, Volume 11, Issue 8: 24808-24838. doi: 10.3934/math.2026998

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Research article

Spectral adaptivity for iterative least-squares B-spline curve and surface fitting

Xingxuan Peng ^,,
Yutong Li

School of Mathematics, Liaoning Normal University, Dalian 116081, China

Received: 09 June 2026 Revised: 29 July 2026 Accepted: 11 August 2026 Published: 13 August 2026
MSC : 65D07, 65D17, 65D10, 65F10, 65F20

For curve and surface fitting, the least-squares progressive iterative approximation (LSPIA) method updates the control points with a constant weight. To accelerate the iteration, this paper proposes a spectrally adaptive LSPIA (denoted by SaLSPIA) method. SaLSPIA updates the control points by an adaptive weight constructed from the adjusting vectors of two consecutive iterations, so that the weight adapts to the spectral scale of the collocation matrix, while retaining the simple control-point update form of LSPIA. We establish the global convergence and an $R$-linear convergence rate of SaLSPIA for both the full-rank and the rank-deficient cases, including convergence to a least-squares fitting curve or surface when the corresponding fitting system is rank deficient. Numerical experiments on curve, surface, and missing-data fitting show that SaLSPIA is efficient and robust.
- least-squares fitting,
- progressive iterative approximation,
- B-spline,
- spectral adaptivity,
- curve and surface reconstruction,
- missing data fitting
Citation: Xingxuan Peng, Yutong Li. Spectral adaptivity for iterative least-squares B-spline curve and surface fitting[J]. AIMS Mathematics, 2026, 11(8): 24808-24838. doi: 10.3934/math.2026998

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Abstract

For curve and surface fitting, the least-squares progressive iterative approximation (LSPIA) method updates the control points with a constant weight. To accelerate the iteration, this paper proposes a spectrally adaptive LSPIA (denoted by SaLSPIA) method. SaLSPIA updates the control points by an adaptive weight constructed from the adjusting vectors of two consecutive iterations, so that the weight adapts to the spectral scale of the collocation matrix, while retaining the simple control-point update form of LSPIA. We establish the global convergence and an $R$-linear convergence rate of SaLSPIA for both the full-rank and the rank-deficient cases, including convergence to a least-squares fitting curve or surface when the corresponding fitting system is rank deficient. Numerical experiments on curve, surface, and missing-data fitting show that SaLSPIA is efficient and robust.

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