Variational progressive-iterative approximation for RBF-based surface reconstruction

Liu, Bo; Liu, Tao; Hu, Ling; Shang, Yuanyuan; Liu, Xinru

doi:10.1007/s00371-021-02213-3

Cited by 7 publications

(5 citation statements)

References 28 publications

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“…The AA-I-PIA and I-PIA ( [12]) are used to reconstruct the Skeleton, Max Plank, and Armadillo models. Note in [16] that the conjugate gradient (CG) method was used for the surface reconstruction based on radial basis functions. We also compared the B-spline surface reconstruction using the CG method, referred to as CG-PIA.…”

Section: B Numerical Results Of the Aa-i-piamentioning

confidence: 99%

“…We note in [12] that the I-PIA can deal with the problem of surface reconstruction from noisy data. We also note in [13] and [16] that regularization methods can reduce the instability of the reconstruction algorithm caused by data perturbations. The commonly used regularization methods include Tikhonov regularization based on variational principles, mixed gradient Tikhonov regularization, sparsity regularization, regularization based on singular value decomposition, and other improved methods.…”

Section: Regularized Aa-i-pia For Noisy Datamentioning

confidence: 95%

“…For large datasets, the convergence of I-PIA may be relatively slow, affecting the reconstruction algorithm's computational efficiency. Therefore, some work has been done to accelerate the convergence rate of I-PIA or improve the computational efficiency of the I-PIA; see the relaxed implicit randomized algebraic reconstruction technique [13]; randomized implicit progressive-iterative approximation [14]; I-PIA with acceleration factor [15]; accelerated I-PIA based on conjugate gradient method [16]; and so on.…”

Section: Introductionmentioning

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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Section: B Numerical Results Of the Aa-i-piamentioning

confidence: 99%

Section: Regularized Aa-i-pia For Noisy Datamentioning

confidence: 95%

Section: Introductionmentioning

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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“…Dedicated node positioning algorithms for meshless discretization support variable nodal density [7] that can also be applied to parametric surfaces, e.g., a dendrite envelope. In the context of surface reconstruction from a cloud of points, important advances have been just recently made with a two dimensional algorithm using splines [8] and radial basis functions based three-dimensional algorithms [9,10].…”

Section: Introductionmentioning

confidence: 99%

A sharp-interface mesoscopic model for dendritic growth

Jančič

Založnik

Kosec

2023

IOP Conf. Ser.: Mater. Sci. Eng.

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The grain envelope model (GEM) describes the growth of envelopes of dendritic crystal grains during solidification. Numerically the growing envelopes are usually tracked using an interface capturing method employing a phase field equation on a fixed grid. Such an approach describes the envelope as a diffuse interface, which can lead to numerical artefacts that are possibly detrimental. In this work, we present a sharp-interface formulation of the GEM that eliminates such artefacts and can thus track the envelope with high accuracy. The new formulation uses an adaptive meshless discretization method to solve the diffusion in the liquid around the grains. We use the ability of the meshless method to operate on scattered nodes to accurately describe the interface, i.e., the envelope. The proposed algorithm combines parametric surface reconstruction, meshless discretization of parametric surfaces, global solution construction procedure and partial differential operator approximation using monomials as basis functions. The approach is demonstrated on a two-dimensional h-adaptive solution of diffusive growth of dendrites and assessed by comparison to a conventional diffuse-interface fixed-grid GEM.

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“…In view of the similarities between PIAs and geometric interpolation methods in the iterative process, they are collectively referred to as the geometric iterative methods (GIMs). Through diversified development, PIA with constraints [11,12] and implicit PIA [13,14] are also widely studied.…”

Section: Introductionmentioning

confidence: 99%

Progressive Iterative Approximation of Non-Uniform Cubic B-Spline Curves and Surfaces via Successive Over-Relaxation Iteration

Shou

Fang

2022

Mathematics

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Geometric iteration (GI) is one of the most efficient curve- or surface-fitting techniques in recent years, which is famous for its remarkable geometric significance. In essence, GI can be thought of as the sum of iterative methods for solving systems of linear equations, such as progressive iterative approximation (PIA) which relies on the theory of Richardson iteration. Thus, when the curve- or surface-fitting error is at a desired level, we want to have as few iterations as possible to improve efficiency when dealing with large data sets. Based on the idea of successive over-relaxation (SOR) iteration, we formulate a faster PIA curve and surface interpolation scheme using classical non-uniform cubic B-splines, named SOR-PIA. The genetic algorithm is utilized to estimate the best approximate relaxation factor of SOR-PIA. Similar to standard PIA, SOR-PIA can also be regarded as a process in which the control points move in one direction, but it can greatly reduce the number of iterations in the iterative process with the same fitting accuracy. By comparing with the standard PIA and WPIA algorithms, the effectiveness of the SOR-PIA iterative interpolation algorithm can be verified.

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Variational progressive-iterative approximation for RBF-based surface reconstruction

Cited by 7 publications

References 28 publications

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

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

A sharp-interface mesoscopic model for dendritic growth

Progressive Iterative Approximation of Non-Uniform Cubic B-Spline Curves and Surfaces via Successive Over-Relaxation Iteration

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