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 introd...
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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.
Volumetric modeling is an important topic for material modeling and isogeometric simulation. In this paper, two kinds of interpolatory Catmull-Clark volumetric subdivision approaches over unstructured hexahedral meshe...
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Volumetric modeling is an important topic for material modeling and isogeometric simulation. In this paper, two kinds of interpolatory Catmull-Clark volumetric subdivision approaches over unstructured hexahedral meshes are proposed based on the limit point formula of Catmull-Clark subdivision volume. The basic idea of the first method is to construct a new control lattice, whose limit volume by the Catmull-Clark subdivision scheme interpolates vertices of the original hexahedral mesh. The new control lattice is derived by the local push-back operation from one Catmull-Clark subdivision step with modified geometric rules. This interpolating method is simple and efficient, and several shape parameters are involved in adjusting the shape of the limit volume. The second method is based on progressive-iterative approximation using limit point formula. At each iteration step, we progressively modify vertices of an original hexahedral mesh to generate a new control lattice whose limit volume interpolates all vertices in the original hexahedral mesh. The convergence proof of the iterative process is also given. The interpolatory subdivision volume has C-2-smoothness at the regular region except around extraordinary vertices and edges. Furthermore, the proposed interpolatory volumetric subdivision methods can be used not only for geometry interpolation, but also for material attribute interpolation in the field of volumetric material modeling. The application of the proposed volumetric subdivision approaches on isogeometric analysis is also given with several examples. (C) 2020 Elsevier B.V. All rights reserved.
作者：
Lin, HongweiZhejiang Univ
Dept Math Inst Comp Image & Graph Hangzhou 310058 Zhejiang Peoples R China Zhejiang Univ
State Key Lab CAD & CG Hangzhou 310058 Zhejiang Peoples R China
In this paper, we develop the adaptive data fitting algorithms by virtue of the local property of the progressive-iterative approximation (abbr. PIA), which generates the fitting curve (patch) by adjusting the control...
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In this paper, we develop the adaptive data fitting algorithms by virtue of the local property of the progressive-iterative approximation (abbr. PIA), which generates the fitting curve (patch) by adjusting the control points of a blending curve (patch) iteratively. In the adaptive data fitting algorithms, the control points are classified into two classes, namely, active and fixed control points, and only the active control points need to be adjusted in each iteration, thus saving computation greatly. Lots of examples and experimental data are presented to demonstrate the efficiency of the adaptive data fitting algorithm. Since the PIA method can be made parallel easily, the adaptive data fitting algorithm developed in this paper has important applications in parallel large scale data fitting. (C) 2012 Elsevier B.V. All rights reserved.
Geometric iterative methods (GIM), including the progressive-iterative approximation (PIA) and the geometric interpolation/approximation method, are a class of iterative methods for fitting curves and surfaces with cl...
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Geometric iterative methods (GIM), including the progressive-iterative approximation (PIA) and the geometric interpolation/approximation method, are a class of iterative methods for fitting curves and surfaces with clear geometric meanings. In this paper, we provide an overview of the interpolatory and approximate geometric iteration methods, present the local properties and accelerating techniques, and show their convergence. Moreover, because it is easy to integrate geometric constraints in the iterative procedure, GIM has been widely applied in geometric design and related areas. We survey the successful applications of geometric iterative methods, including applications in geometric design, data fitting, reverse engineering, mesh and NURBS solid generation. (c) 2017 Elsevier Ltd. All rights reserved.
Just by adjusting the control points iteratively, progressive-iterative approximation presents an intuitive and straightforward way to fit data points. It generates a curve or patch sequence with finer and finer preci...
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Just by adjusting the control points iteratively, progressive-iterative approximation presents an intuitive and straightforward way to fit data points. It generates a curve or patch sequence with finer and finer precision, and the limit of the sequence interpolates the data points. The progressive-iterative approximation brings more flexibility for shape controlling in data fitting. In this paper, we design a local progressive-iterative approximation format, and show that the local format is convergent for the blending curve with normalized totally positive basis, and the bi-cubic B-spline patch, which is the most commonly used patch in geometric design. Moreover, a special adjustment manner is designed to make the local progressive-iterative approximation format is convergent for a generic blending patch with normalized totally positive basis. The local progressive-iterative approximation format adjusts only a part of the control points of a blending curve or patch, and the limit curve or patch interpolates the corresponding data points. Based on the local format, data points can be fit adaptively. (C) 2010 Elsevier B.V. All rights reserved.
The geometric interpolation algorithm is proposed by Maekawa et al. in [Maekawa T, Matsumoto Y, Namiki K. Interpolation by geometric algorithm. Computer-Aided Design 2007:39:313-23]. Without solving a system of equati...
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The geometric interpolation algorithm is proposed by Maekawa et al. in [Maekawa T, Matsumoto Y, Namiki K. Interpolation by geometric algorithm. Computer-Aided Design 2007:39:313-23]. Without solving a system of equations, the algorithm generates a curve (surface) sequence, of which the limit curve (surface) interpolates the given data points. However, the convergence of the algorithm is a conjecture in the reference above, and demonstrated by lots of empirical examples. In this paper, we prove the conjecture given in the reference in theory, that is, the geometric interpolation algorithm is convergent for a blending curve (surface) with normalized totally positive basis, under the condition that the minimal eigenvalue lambda(min)(D(k)) of the collocation matrix D(k) of the totally positive basis in each iteration satisfies lambda(min)(D(k)) >= alpha > 0. As a consequence, the geometric interpolation algorithm is convergent for Bezier, B-spline, rational Bezier, and NURBS curve (surface) if they satisfy the condition aforementioned, since Bernstein basis and B-spline basis are both normalized totally positive. (C) 2010 Elsevier Ltd. All rights reserved.
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