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Papers by lin eagle

ArXiv, 2021

In this paper, we propose a stochastic geometric iterative method to approximate the high-resolut... more In this paper, we propose a stochastic geometric iterative method to approximate the high-resolution 3D models by finite Loop subdivision surfaces. Given an input mesh as the fitting target, the initial control mesh is generated using the mesh simplification algorithm. Then, our method adjusts the control mesh iteratively to make its finite Loop subdivision surface approximates the input mesh. In each geometric iteration, we randomly select part of points on the subdivision surface to calculate the difference vectors and distribute the vectors to the control points. Finally, the control points are updated by adding the weighted average of these difference vectors. We prove the convergence of our method and verify it by demonstrating error curves in the experiment. In addition, compared with an existing geometric iterative method, our method has a faster fitting speed and higher fitting precision.

ArXiv, 2019

A porous scaffold is a three-dimensional network structure composed of a large number of pores, a... more A porous scaffold is a three-dimensional network structure composed of a large number of pores, and triply periodic minimal surfaces (TPMSs) are one of conventional tools for designing porous scaffolds. However, discontinuity, incompleteness, and high storage space requirements are the three main shortcomings of TPMSs for porous scaffold design. In this study, we developed an effective method for heterogeneous porous scaffold generation to overcome the abovementioned shortcomings of TPMSs. The input of the proposed method is a trivariate B-spline solid (TBSS) with a cubic parameter domain. The proposed method first constructs a threshold distribution field (TDF) in the cubic parameter domain, and then produces a continuous and complete TPMS within it. Moreover, by mapping the TPMS in the parametric domain to the TBSS, a continuous and complete porous scaffold is generated in the TBSS. In addition, if the TBSS does not satisfy engineering requirements, the TDF can be locally modified...

Journal of Computer-Aided Design & Computer Graphics, 2021

Progressive-iterative approximation (PIA) is an efficient method for data fitting that attracts t... more Progressive-iterative approximation (PIA) is an efficient method for data fitting that attracts the attention of many researchers and has a wide range of applications. However, the convergence rate of LSPIA is prolonged. In this study, we design a fast PIA format based on the Gauss-Seidel iterative method named Gauss-Seidel progressive and iterative approximation for least squares curve and surface fitting (GS-LSPIA). Firstly, the control points of the fitting curve (surface) are selected from the given data points. Then, the chord length method is used to assign the parameters of the given data points. GS-LSPIA generates a series of fitting curves (surfaces) by refining the control points iteratively, and the limit of the generated curve (surface) is the least square fitting result to the given data points. Several experimental results presented in this paper demonstrate that, to achieve the same accuracy for GS-LSPIA and LSPIA, GS-LSPIA required fewer steps and shorter running time compared with LSPIA. Thus, the proposed GS-LSPIA is efficient and has a faster convergence rate compared with the LSPIA algorithm.

The advent of isogeometric analysis has prompted a need for methods to generate Trivariate B-spli... more The advent of isogeometric analysis has prompted a need for methods to generate Trivariate B-spline Solids (TBS) with positive Jacobian. However, it is difficult to guarantee a positive Jacobian of a TBS since the geometric pre-condition for ensuring the positive Jacobian is very complicated. In this paper, we propose a method for generating TBSs with guaranteed positive Jacobian. For the study, we used a tetrahedral (tet) mesh model and segmented it into sub-volumes using the pillow operation. Then, to reduce the difficulty in ensuring a positive Jacobian, we separately fitted the boundary curves and surfaces and the sub-volumes using a geometric iterative fitting algorithm. Finally, the smoothness between adjacent TBSs is improved. The experimental examples presented in this paper demonstrate the effectiveness and efficiency of the developed algorithm.

Journal of Systems Science and Complexity, 2020

In this paper, we present the isogeometric least-squares collocation (IGA-L) method, which determ... more In this paper, we present the isogeometric least-squares collocation (IGA-L) method, which determines the numerical solution by making the approximate differential operator fit the real differential operator in a least-squares sense. The number of collocation points employed in IGA-L can be larger than that of the unknowns. Theoretical analysis and numerical examples presented in this paper show the superiority of IGA-L over state-of-the-art collocation methods. First, a small increase in the number of collocation points in IGA-L leads to a large improvement in the accuracy of its numerical solution. Second, IGA-L method is more flexible and more stable, because the number of collocation points in IGA-L is variable. Third, IGA-L is convergent in some cases of singular parameterization. Moreover, the consistency and convergence analysis are also developed in this paper.

Computers & Graphics, 2020

Persistence barcode is a topological summary for persistent homology to exhibit topological featu... more Persistence barcode is a topological summary for persistent homology to exhibit topological features with different persistence. Persistence rank function (PRF), derived from persistence barcode, organizes persistence Betti numbers in the form of an integervalued function. To obtain topological patterns of objects such as point clouds represented by finite-dimensional vectors for machine learning classification tasks, the vectorizing representations of barcodes is generated via decomposing PRF on a system of Haar basis. Theoretically, the generated vectorizing representation is proved to have 1-Wasserstein stability. In practice, to reduce training time and achieve better results, a technique of dimensionality reduction through out-of-sample mapping in supervised manifold learning is used to generate a low-dimensional vector. Experiments demonstrate that the representation is effective for capturing the topological patterns of data sets. Moreover, the classification of porous structures has become an essential problem in the fields such as material science in recent decades. The proposed method is successfully applied to distinguish porous structures on a novel data set of porous models.

Computer Aided Geometric Design, 2020

Implicit curve and surface reconstruction attracts the attention of many researchers and gains a ... more Implicit curve and surface reconstruction attracts the attention of many researchers and gains a wide range of applications, due to its ability to describe objects with complicated geometry and topology. However, extra zero-level sets or spurious sheets arise in the reconstruction process makes the reconstruction result challenging to be interpreted and damage the final result. In this paper, we proposed an implicit curve and surface reconstruction method based on the progressive-iterative approximation method, named implicit progressive-iterative approximation (I-PIA). The proposed method elegantly eliminates the spurious sheets naturally without requiring any explicit minimization procedure, thus reducing the computational cost greatly and providing high-quality reconstruction results. Numerical examples are provided to demonstrate the efficiency and effectiveness of the proposed method.

Applied Mathematics-A Journal of Chinese Universities, 2019

In isogeometric analysis, it is frequently required to handle the geometric models enclosed by fo... more In isogeometric analysis, it is frequently required to handle the geometric models enclosed by foursided or non-four-sided boundary patches, such as trimmed surfaces. In this paper, we develop a Gregory solid based method to parameterize those models. First, we extend the Gregory patch representation to the trivariate Gregory solid representation. Second, the trivariate Gregory solid representation is employed to interpolate the boundary patches of a geometric model, thus generating the polyhedral volume parametrization. To improve the regularity of the polyhedral volume parametrization, we formulate the construction of the trivariate Gregory solid as a sparse optimization problem, where the optimization objective function is a linear combination of some terms, including a sparse term aiming to reduce the negative Jacobian area of the Gregory solid. Then, the alternating direction method of multipliers (ADMM) is used to solve the sparse optimization problem. Lots of experimental examples illustrated in this paper demonstrate the effectiveness and efficiency of the developed method.

Science China Information Sciences, 2018

Computer-Aided Design, 2018

Computers & Mathematics with Applications, 2014

Computer-Aided Design, 2015

h i g h l i g h t s • An iterative algorithm is developed to fill a triangular mesh with an all-h... more h i g h l i g h t s • An iterative algorithm is developed to fill a triangular mesh with an all-hex mesh. • The Jacobian values of the all-hex mesh are guaranteed to be positive. • The convergence of the iterative algorithm is proved.

Science in China Series F, 2004

In this paper, based on the idea of profit and loss modification, we present the iterative non-un... more In this paper, based on the idea of profit and loss modification, we present the iterative non-uniform B-spline curve and surface to settle a key problem in computer aided geometric design and reverse engineering, that is, constructing the curve (surface) fitting (interpolating) a given ordered point set without solving a linear system. We start with a piece of initial non-uniform B-spline curve (surface) which takes the given point set as its control point set. Then by adjusting its control points gradually with iterative formula, we can get a group of non-uniform B-spline curves (surfaces) with gradually higher precision. In this paper, using modern matrix theory, we strictly prove that the limit curve (surface) of the iteration interpolates the given point set. The non-uniform B-spline curves (surfaces) generated with the iteration have many advantages, such as satisfying the NURBS standard, having explicit expression, gaining locality, and convexity preserving, etc.

Computer Aided Geometric Design, 2015

The Visual Computer, 2005

Curve reconstruction that generates a piece of centric curve from a piece of planar stripshaped p... more Curve reconstruction that generates a piece of centric curve from a piece of planar stripshaped point cloud is a fundamental problem in reverse engineering. In this paper, we present a new curvereconstruction algorithm based on an interval B-spline curve. The algorithm constructs a rectangle sequence approximating the point cloud using a new data clustering technique, which facilitates the determination of curve order implied in the shape of the point cloud. A quasicentric point sequence and two pieces of boundary point sequences are then computed, based on which a piece of interval B-spline curve representing the geometric shape of the point cloud is constructed. Its centric curve is the final reconstructed curve. The whole algorithm is intuitive, simple, and efficient, as demonstrated by experimental results.

2007 10th IEEE International Conference on Computer-Aided Design and Computer Graphics, 2007

We develop a novel method for feature point detection and employ it to generate outline font from... more We develop a novel method for feature point detection and employ it to generate outline font from historical document of Chinese calligraphy. The feature points at a character contour subdivide the contour into segments. Each segment can be then fitted by a parametric curve to obtain the outline font. Some experimental results are also presented in the paper.

2009 IEEE International Conference on Shape Modeling and Applications, 2009

The coarse bounding cage of a dense mesh plays important roles in computer graphics, computer vis... more The coarse bounding cage of a dense mesh plays important roles in computer graphics, computer vision, and geometric design. Specifically, in volume-based deformation, a coarse bounding cage is required to manipulate the dense mesh model it enclosed; in subdivision surface fitting, the fitting starts from a coarse cage bounding the fitted dense mesh or point set; and so on. However, the generation of a coarse bounding cage is mainly by interactive ways, which are very tedious and time-consuming. In this paper, we develop a fully automatic method to generate a coarse cage bounding a dense mesh model. The automatically generated coarse bounding cage can keep the topological structure and major geometric features of the original mesh model, which is validated by theoretical analysis and experimental data presented in this paper. Further more, we employ the automatically generated coarse bounding cage in some applications, such as deformation, and subdivision fitting, producing good results.

T-spline overcomes the topological constraints of the control net of NURBS model successfully. Ho... more T-spline overcomes the topological constraints of the control net of NURBS model successfully. However, the introduction of T-junctions, L-junctions and the isolated vertices in the T-mesh makes its topological structure very flexible. As a result, not only the T-mesh is hard to be represented, but the computation and local refinement of T-spline are difficult to be implemented as well. This hinders the studies and applications of T-splines in practice. In this paper, we develop the extended T-mesh, which can be represented in an obj-like format file, and converted into the face-edge-vertex data structure conveniently. With such data structure, the computation of T-splines can be made much easier. Furthermore, we develop a new local refinement algorithm, by virtue of the extended T-mesh. The new algorithm is easy to be implemented, by separating the local refinement into two procedures, the mesh refinement, and blending function refinement.

2011 12th International Conference on Computer-Aided Design and Computer Graphics, 2011

Progressive-iterative approximation presents an intuitive way to generate a sequence of curves or... more Progressive-iterative approximation presents an intuitive way to generate a sequence of curves or patches, whose limit interpolates the given data points. It has been shown that the blending curves and tensor product blending patches with normalized totally positive basis have the progressiveiterative approximation property. In this paper, we prove that, the quadratic, cubic, and quartic non-uniform triangular Bernstein-Bézier patches also have the progressive-iterative approximation property. Since the most often empolyed in geometric design are the low degree curves or patches, especially the cubic curves and patches, the result shown in this paper has practical significance for geometric design.

2011 12th International Conference on Computer-Aided Design and Computer Graphics, 2011

Fairing curve and surface generation is an important topic in geometric design. However, the conv... more Fairing curve and surface generation is an important topic in geometric design. However, the conventional method for generating the fairing curve and surface, which fit the giving data points, is hard to control the fitting precision, because it is a minimization problem where the objective function is the weighted sum of a fitting term and a fairness term. In this paper, we develop the variational progressive-iterative approximation (abbr. variational PIA) method for fitting a data point sequence. While the variational PIA is easy to control the fitting precision, the generated fitting curve or surface is the most fairing one in some scope. Lots of comparisons show that the fairness results of the variational PIA are comparable to that of the conventional method.

ArXiv, 2021

In this paper, we propose a stochastic geometric iterative method to approximate the high-resolut... more In this paper, we propose a stochastic geometric iterative method to approximate the high-resolution 3D models by finite Loop subdivision surfaces. Given an input mesh as the fitting target, the initial control mesh is generated using the mesh simplification algorithm. Then, our method adjusts the control mesh iteratively to make its finite Loop subdivision surface approximates the input mesh. In each geometric iteration, we randomly select part of points on the subdivision surface to calculate the difference vectors and distribute the vectors to the control points. Finally, the control points are updated by adding the weighted average of these difference vectors. We prove the convergence of our method and verify it by demonstrating error curves in the experiment. In addition, compared with an existing geometric iterative method, our method has a faster fitting speed and higher fitting precision.

ArXiv, 2019

A porous scaffold is a three-dimensional network structure composed of a large number of pores, a... more A porous scaffold is a three-dimensional network structure composed of a large number of pores, and triply periodic minimal surfaces (TPMSs) are one of conventional tools for designing porous scaffolds. However, discontinuity, incompleteness, and high storage space requirements are the three main shortcomings of TPMSs for porous scaffold design. In this study, we developed an effective method for heterogeneous porous scaffold generation to overcome the abovementioned shortcomings of TPMSs. The input of the proposed method is a trivariate B-spline solid (TBSS) with a cubic parameter domain. The proposed method first constructs a threshold distribution field (TDF) in the cubic parameter domain, and then produces a continuous and complete TPMS within it. Moreover, by mapping the TPMS in the parametric domain to the TBSS, a continuous and complete porous scaffold is generated in the TBSS. In addition, if the TBSS does not satisfy engineering requirements, the TDF can be locally modified...

Journal of Computer-Aided Design & Computer Graphics, 2021

Progressive-iterative approximation (PIA) is an efficient method for data fitting that attracts t... more Progressive-iterative approximation (PIA) is an efficient method for data fitting that attracts the attention of many researchers and has a wide range of applications. However, the convergence rate of LSPIA is prolonged. In this study, we design a fast PIA format based on the Gauss-Seidel iterative method named Gauss-Seidel progressive and iterative approximation for least squares curve and surface fitting (GS-LSPIA). Firstly, the control points of the fitting curve (surface) are selected from the given data points. Then, the chord length method is used to assign the parameters of the given data points. GS-LSPIA generates a series of fitting curves (surfaces) by refining the control points iteratively, and the limit of the generated curve (surface) is the least square fitting result to the given data points. Several experimental results presented in this paper demonstrate that, to achieve the same accuracy for GS-LSPIA and LSPIA, GS-LSPIA required fewer steps and shorter running time compared with LSPIA. Thus, the proposed GS-LSPIA is efficient and has a faster convergence rate compared with the LSPIA algorithm.

The advent of isogeometric analysis has prompted a need for methods to generate Trivariate B-spli... more The advent of isogeometric analysis has prompted a need for methods to generate Trivariate B-spline Solids (TBS) with positive Jacobian. However, it is difficult to guarantee a positive Jacobian of a TBS since the geometric pre-condition for ensuring the positive Jacobian is very complicated. In this paper, we propose a method for generating TBSs with guaranteed positive Jacobian. For the study, we used a tetrahedral (tet) mesh model and segmented it into sub-volumes using the pillow operation. Then, to reduce the difficulty in ensuring a positive Jacobian, we separately fitted the boundary curves and surfaces and the sub-volumes using a geometric iterative fitting algorithm. Finally, the smoothness between adjacent TBSs is improved. The experimental examples presented in this paper demonstrate the effectiveness and efficiency of the developed algorithm.

Journal of Systems Science and Complexity, 2020

In this paper, we present the isogeometric least-squares collocation (IGA-L) method, which determ... more In this paper, we present the isogeometric least-squares collocation (IGA-L) method, which determines the numerical solution by making the approximate differential operator fit the real differential operator in a least-squares sense. The number of collocation points employed in IGA-L can be larger than that of the unknowns. Theoretical analysis and numerical examples presented in this paper show the superiority of IGA-L over state-of-the-art collocation methods. First, a small increase in the number of collocation points in IGA-L leads to a large improvement in the accuracy of its numerical solution. Second, IGA-L method is more flexible and more stable, because the number of collocation points in IGA-L is variable. Third, IGA-L is convergent in some cases of singular parameterization. Moreover, the consistency and convergence analysis are also developed in this paper.

Computers & Graphics, 2020

Persistence barcode is a topological summary for persistent homology to exhibit topological featu... more Persistence barcode is a topological summary for persistent homology to exhibit topological features with different persistence. Persistence rank function (PRF), derived from persistence barcode, organizes persistence Betti numbers in the form of an integervalued function. To obtain topological patterns of objects such as point clouds represented by finite-dimensional vectors for machine learning classification tasks, the vectorizing representations of barcodes is generated via decomposing PRF on a system of Haar basis. Theoretically, the generated vectorizing representation is proved to have 1-Wasserstein stability. In practice, to reduce training time and achieve better results, a technique of dimensionality reduction through out-of-sample mapping in supervised manifold learning is used to generate a low-dimensional vector. Experiments demonstrate that the representation is effective for capturing the topological patterns of data sets. Moreover, the classification of porous structures has become an essential problem in the fields such as material science in recent decades. The proposed method is successfully applied to distinguish porous structures on a novel data set of porous models.

Computer Aided Geometric Design, 2020

Implicit curve and surface reconstruction attracts the attention of many researchers and gains a ... more Implicit curve and surface reconstruction attracts the attention of many researchers and gains a wide range of applications, due to its ability to describe objects with complicated geometry and topology. However, extra zero-level sets or spurious sheets arise in the reconstruction process makes the reconstruction result challenging to be interpreted and damage the final result. In this paper, we proposed an implicit curve and surface reconstruction method based on the progressive-iterative approximation method, named implicit progressive-iterative approximation (I-PIA). The proposed method elegantly eliminates the spurious sheets naturally without requiring any explicit minimization procedure, thus reducing the computational cost greatly and providing high-quality reconstruction results. Numerical examples are provided to demonstrate the efficiency and effectiveness of the proposed method.

Applied Mathematics-A Journal of Chinese Universities, 2019

In isogeometric analysis, it is frequently required to handle the geometric models enclosed by fo... more In isogeometric analysis, it is frequently required to handle the geometric models enclosed by foursided or non-four-sided boundary patches, such as trimmed surfaces. In this paper, we develop a Gregory solid based method to parameterize those models. First, we extend the Gregory patch representation to the trivariate Gregory solid representation. Second, the trivariate Gregory solid representation is employed to interpolate the boundary patches of a geometric model, thus generating the polyhedral volume parametrization. To improve the regularity of the polyhedral volume parametrization, we formulate the construction of the trivariate Gregory solid as a sparse optimization problem, where the optimization objective function is a linear combination of some terms, including a sparse term aiming to reduce the negative Jacobian area of the Gregory solid. Then, the alternating direction method of multipliers (ADMM) is used to solve the sparse optimization problem. Lots of experimental examples illustrated in this paper demonstrate the effectiveness and efficiency of the developed method.

Science China Information Sciences, 2018

Computer-Aided Design, 2018

Computers & Mathematics with Applications, 2014

Computer-Aided Design, 2015

h i g h l i g h t s • An iterative algorithm is developed to fill a triangular mesh with an all-h... more h i g h l i g h t s • An iterative algorithm is developed to fill a triangular mesh with an all-hex mesh. • The Jacobian values of the all-hex mesh are guaranteed to be positive. • The convergence of the iterative algorithm is proved.

Science in China Series F, 2004

In this paper, based on the idea of profit and loss modification, we present the iterative non-un... more In this paper, based on the idea of profit and loss modification, we present the iterative non-uniform B-spline curve and surface to settle a key problem in computer aided geometric design and reverse engineering, that is, constructing the curve (surface) fitting (interpolating) a given ordered point set without solving a linear system. We start with a piece of initial non-uniform B-spline curve (surface) which takes the given point set as its control point set. Then by adjusting its control points gradually with iterative formula, we can get a group of non-uniform B-spline curves (surfaces) with gradually higher precision. In this paper, using modern matrix theory, we strictly prove that the limit curve (surface) of the iteration interpolates the given point set. The non-uniform B-spline curves (surfaces) generated with the iteration have many advantages, such as satisfying the NURBS standard, having explicit expression, gaining locality, and convexity preserving, etc.

Computer Aided Geometric Design, 2015

The Visual Computer, 2005

Curve reconstruction that generates a piece of centric curve from a piece of planar stripshaped p... more Curve reconstruction that generates a piece of centric curve from a piece of planar stripshaped point cloud is a fundamental problem in reverse engineering. In this paper, we present a new curvereconstruction algorithm based on an interval B-spline curve. The algorithm constructs a rectangle sequence approximating the point cloud using a new data clustering technique, which facilitates the determination of curve order implied in the shape of the point cloud. A quasicentric point sequence and two pieces of boundary point sequences are then computed, based on which a piece of interval B-spline curve representing the geometric shape of the point cloud is constructed. Its centric curve is the final reconstructed curve. The whole algorithm is intuitive, simple, and efficient, as demonstrated by experimental results.

2007 10th IEEE International Conference on Computer-Aided Design and Computer Graphics, 2007

We develop a novel method for feature point detection and employ it to generate outline font from... more We develop a novel method for feature point detection and employ it to generate outline font from historical document of Chinese calligraphy. The feature points at a character contour subdivide the contour into segments. Each segment can be then fitted by a parametric curve to obtain the outline font. Some experimental results are also presented in the paper.

2009 IEEE International Conference on Shape Modeling and Applications, 2009

The coarse bounding cage of a dense mesh plays important roles in computer graphics, computer vis... more The coarse bounding cage of a dense mesh plays important roles in computer graphics, computer vision, and geometric design. Specifically, in volume-based deformation, a coarse bounding cage is required to manipulate the dense mesh model it enclosed; in subdivision surface fitting, the fitting starts from a coarse cage bounding the fitted dense mesh or point set; and so on. However, the generation of a coarse bounding cage is mainly by interactive ways, which are very tedious and time-consuming. In this paper, we develop a fully automatic method to generate a coarse cage bounding a dense mesh model. The automatically generated coarse bounding cage can keep the topological structure and major geometric features of the original mesh model, which is validated by theoretical analysis and experimental data presented in this paper. Further more, we employ the automatically generated coarse bounding cage in some applications, such as deformation, and subdivision fitting, producing good results.

T-spline overcomes the topological constraints of the control net of NURBS model successfully. Ho... more T-spline overcomes the topological constraints of the control net of NURBS model successfully. However, the introduction of T-junctions, L-junctions and the isolated vertices in the T-mesh makes its topological structure very flexible. As a result, not only the T-mesh is hard to be represented, but the computation and local refinement of T-spline are difficult to be implemented as well. This hinders the studies and applications of T-splines in practice. In this paper, we develop the extended T-mesh, which can be represented in an obj-like format file, and converted into the face-edge-vertex data structure conveniently. With such data structure, the computation of T-splines can be made much easier. Furthermore, we develop a new local refinement algorithm, by virtue of the extended T-mesh. The new algorithm is easy to be implemented, by separating the local refinement into two procedures, the mesh refinement, and blending function refinement.

2011 12th International Conference on Computer-Aided Design and Computer Graphics, 2011

Progressive-iterative approximation presents an intuitive way to generate a sequence of curves or... more Progressive-iterative approximation presents an intuitive way to generate a sequence of curves or patches, whose limit interpolates the given data points. It has been shown that the blending curves and tensor product blending patches with normalized totally positive basis have the progressiveiterative approximation property. In this paper, we prove that, the quadratic, cubic, and quartic non-uniform triangular Bernstein-Bézier patches also have the progressive-iterative approximation property. Since the most often empolyed in geometric design are the low degree curves or patches, especially the cubic curves and patches, the result shown in this paper has practical significance for geometric design.

2011 12th International Conference on Computer-Aided Design and Computer Graphics, 2011

Fairing curve and surface generation is an important topic in geometric design. However, the conv... more Fairing curve and surface generation is an important topic in geometric design. However, the conventional method for generating the fairing curve and surface, which fit the giving data points, is hard to control the fitting precision, because it is a minimization problem where the objective function is the weighted sum of a fitting term and a fairness term. In this paper, we develop the variational progressive-iterative approximation (abbr. variational PIA) method for fitting a data point sequence. While the variational PIA is easy to control the fitting precision, the generated fitting curve or surface is the most fairing one in some scope. Lots of comparisons show that the fairness results of the variational PIA are comparable to that of the conventional method.