Jacobi–PIA algorithm for bi-cubic B-spline interpolation surfaces

Liu, Chengzhi; Li, Juncheng; Hu, Lijuan

doi:10.1016/j.gmod.2022.101134

Cited by 7 publications

(4 citation statements)

References 15 publications

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“…In [22] and [23], similar analyses were conducted on selecting µ, leading to the assertion of the following lemma.…”

Section: Convergence Analysis Of Aa-i-piamentioning

confidence: 99%

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Fast Surface Reconstruction Technique Based on Anderson Accelerated I-PIA Method

Dai,

Yi,

Liu

2023

IEEE Access

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For curve and surface reconstruction, the implicit progressive-iterative approximation (I-PIA) is widely used in many applications because it is efficient in data fitting and handling noise and missing data. By introducing Anderson extrapolation to the I-PIA, in this paper we exploit an accelerated method, namely AA-I-PIA, for the I-PIA of the B-spline function. For noisy point cloud data, we have exploited the regularized I-PIA and its accelerated version. We have shown that the AA-I-PIA method converges for appropriate parameters. Numerical results show that regardless of missing or noisy data, the (regularized) AA-I-PIA outperforms the classical I-PIA method regarding the number of iterations and elapsed CPU time.

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“…In [22] and [23], similar analyses were conducted on selecting µ, leading to the assertion of the following lemma.…”

Section: Convergence Analysis Of Aa-i-piamentioning

confidence: 99%

“…1: Compute r (0) = B T (b−Bc (0) ) and c (1) = (1−τ µ)c (0) + µr (0) . 2: m = 1. if m p = 0, compute c (m+1) according to (23).…”

Section: Algorithmmentioning

confidence: 99%

Fast Surface Reconstruction Technique Based on Anderson Accelerated I-PIA Method

Dai,

Yi,

Liu

2023

IEEE Access

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“…Clearly, the selection of τ k will affect the convergence of iterative sequence { Ũ(k) } ∞ k=0 . If τ k equals to zero, the solution obtained by the CG method is the exact one for the system (14). In this case, the IPPIA format is reduced into the PPIA format.…”

Section: The Inexact Solvermentioning

confidence: 99%

“…This is because the collocation matrices resulting from some normalized totally positive (NTP) bases are usually ill-conditioned; see [10,11]. Therefore, much more attention was paid to accelerating the convergence rate of PIA; see the weighted progressive iteration approximation (WPIA) (also named modified Richardson method) in [12,13]; the QR-PIA in [11]; the Jacobi-PIA in [14]; the Chebyshev semi-iterative method in [15]; the progressive iterative approximation with different weights (DWPIA) in [16]; the GS-PIA in [17]; the PIA with mutually different weights in [18]; the Schulz iterative method in [19]; and a lot literature therein.…”

Section: Introductionmentioning

confidence: 99%

Progressive Iterative Approximation with Preconditioners

Liu

2020

Mathematics

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The progressive iterative approximation (PIA) plays an important role in curve and surface fitting. By using the diagonally compensated reduction of the collocation matrix, we propose the preconditioned progressive iterative approximation (PPIA) to improve the convergence rate of PIA. For most of the normalized totally positive bases, we show that the presented PPIA can accelerate the convergence rate significantly in comparison with the weighted progressive iteration approximation (WPIA) and the progressive iterative approximation with different weights (DWPIA). Furthermore, we propose an inexact variant of the PPIA (IPPIA) to reduce the computational complexity of the PPIA. We introduce the inexact solver of the preconditioning system by employing some state-of-the-art iterative methods. Numerical results show that both the PPIA and the IPPIA converge faster than the WPIA and DWPIA, while the elapsed CPU times of the PPIA and IPPIA are less than those of the WPIA and DWPIA.

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Distributed least-squares progressive iterative approximation for blending curves and surfaces

Yao,

2024

Computer-Aided Design

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Jacobi–PIA algorithm for bi-cubic B-spline interpolation surfaces

Cited by 7 publications

References 15 publications

Fast Surface Reconstruction Technique Based on Anderson Accelerated I-PIA Method

Fast Surface Reconstruction Technique Based on Anderson Accelerated I-PIA Method

Progressive Iterative Approximation with Preconditioners

Distributed least-squares progressive iterative approximation for blending curves and surfaces

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