Progressive-iterative approximation method

Progressive-iterative approximation method is an iterative method of data fitting with geometric meanings.[1] Given the data points to be fitted, the method obtains a series of fitting curves (surfaces) by iteratively updating the control points, and the limit curve (surface) can interpolate or approximate the given data points.[2] It avoids solving a linear system of equations directly and allows flexibility in adding constraints during the iterative process.[3] Therefore, it has been widely used in geometric design and related fields.[2]

The study of the iterative method with geometric meaning can be traced back to the work of scholars such as Prof. Dongxu Qi and Prof. Carl de Boor in the 1970s.[4][5] In 1975, Qi et al. developed and proved the "profit and loss" algorithm for uniform cubic B-spline curves,[4] and in 1979, de Boor independently proposed this algorithm.[5] In 2004, Hongwei Lin and coauthors proved that non-uniform cubic B-spline curves and surfaces have the "profit and loss" property.[3] Later, in 2005, Lin et al. proved that the curves and surfaces with normalized and totally positive basis all have this property and named it progressive iterative approximation (PIA).[1] In 2007, Maekawa et al. changed the algebraic distance in PIA to geometric distance and named it geometric interpolation (GI).[6] In 2008, Cheng et al. extended it to subdivision surfaces and named the method progressive interpolation (PI).[7] Since the iteration steps of the PIA, GI, and PI algorithms are similar and all have geometric meanings, we collectively referred to them as geometric iterative methods (GIM).[2]

PIA is now extended to several common curves and surfaces in the geometric design field,[8] including NURBS curves and surfaces,[9] T-spline surfaces,[10] implicit curves and surfaces,[11] etc.

Iteration Methods

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Generally, progressive-iterative approximation can be divided into interpolation and approximation schemes.[2] In interpolation algorithms, the number of control points is equal to that of the data points; in approximation algorithms, the number of control points can be less than that of the data points. Specifically, there are some representative iteration methods, such as local-PIA,[12] implicit-PIA,[11] fairing-PIA,[13] and isogeometric least-squares progressive-iterative approximation (IG-LSPIA),[14] which is specialized for solving the isogeometric analysis problem.[15]

Interpolation scheme: PIA[1][3][9][16]

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Interpolation scheme of PIA: Top left: The data points and the initial control polygon (Here, the initial control points are taken as the data points). Top right: The initial curve and the difference vectors. Bottom left: New control polygon is generated by adding the difference vectors to the old control points. Bottom right: The new control polygon and new curve (in purple).

To facilitate the description of the PIA iteration format for different forms of curves and surfaces, we write B-spline curves and surfaces, NURBS curves and surfaces, B-spline solids, T-spline surfaces, and triangular Bernstein–Bézier (B–B) surfaces uniformly in the following form:[17]

 

For example:

  • If   is a B-spline curve,then   is a scalar,   is a B-spline basis function,and   denotes the control point;[8]
  • If   is a B-spline patch with   control points,then   ,where   and   are B-spline basis functions;[8]
  • If   is a trivariate B-spline solid with   control points, then  , , where  ,   and   are B-spline basis functions.[18]

Given an ordered data set  ,with parameters   satisfying  , the initial fitting curve (surface) is[1]

 ,

where the initial control points of the initial fitting curve (surface)   can be randomly selected. Suppose that after the  th iteration, the  th fitting curve (surface)  is generated by

 

To construct the  st curve (surface),we first calculate the difference vectors

 

and then update the control points by

 

leading to the  st fitting curve (surface):

 

In this way, we obtain a sequence of curves (surfaces)

 

It has been proved that this sequence of curves (surfaces) converges to a limit curve (surface) that interpolates the give data points,[1][9] i.e.,

 

Approximation scheme: LSPIA[19][10]

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Approximation scheme: LSPIA: Top left: The data points (blue circles) and initial control polygon. Top right: Difference vectors for data points and difference vectors for control points (DVC). Bottom: The new control polygon is generated by adding the DVCs to the old control points.

For the B-spline curve and surface fitting problem, Deng and Lin proposed a least-squares progressive–iterative approximation(LSPIA),[19] which allows the number of control points to be less than that of the data points and is more suitable for large-scale data fitting problems.[10]

Assume that the number of data points is  ,and the number of control points is  . Following the notations in the Section above, the  th fitting curve (surface) generated after the  th iteration is  , i.e.,

 

To generate the  st fitting curve (surface),we compute the following difference vectors in turn:[10][19]

Difference vectors for data points

 

and,

Difference vectors for control points

 

where   is the index set of the data points in the  th group,whose parameters fall in the local support of the  th basis function, i.e.,  .   are weights that guarantee the convergence of the algorithm, usually taken as  .

Finally, the control points of the  st curve (surface) are updated by   leading to the  st fitting curve (surface)  . In this way,we obtain a sequence of curve (surface),and the limit curve (surface) converges to the least-squares fitting result to the given data points.[10][19]

Local-PIA[12]

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Local PIA: If only one control point is adjusted, the Bézier curve just interpolates the data point (in red) corresponding to the adjusted control point.

In the local-PIA, the control points are divided into active and fixed control points, whose subscripts are denoted as   and  , respectively.[12] Assume that, the  th fitting curve (surface) is  ,where the fixed control point satisfy

 

Then,on the one hand, the iterative formula of the difference vector   corresponding to the fixed control points is

 

On the other hand, the iterative formula of the difference vector   corresponding to the active control points is

 

Arranging the above difference vectors into a one-dimensional sequence,

 

the local iteration format in matrix form is,

 

where,   is the iteration matrix,

 

  and   are the identity matrices and

 

The above local iteration format converges and can be extended to blending surfaces[12] and subdivision surfaces.[20]

Implicit-PIA[11]

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The progressive iterative approximation format for implicit curve and surface reconstruction is presented in the following. Given an ordered point cloud   and a unit normal vector   on the data points, we want to reconstruct an implicit curve (surface) from the given point cloud. To avoid trivial solution, some offset points   are added to the point cloud.[11] They are offset by a distance   along the unit normal vector of each point

 

Assume that   is the value of the implicit function at the offset point

 

Let the implicit curve after the  th iteration be

 

where   is the control point.

Define the difference vector of data points as[11]

 

Next, calculate the difference vector of control coefficients

 

where   is the convergence coefficient. As a result, the new control coefficients are

 

leading to the new algebraic B-spline curve

 

The above procedure is carried out iteratively to generate a sequence of algebraic B-spline functions  . The sequence converges to a minimization problem with constraints when the initial control coefficients  .[11]

Assume that the implicit surface generated after the  th iteration is

 

the iteration format is similar to that of the curve case.[11][21]

Fairing-PIA[13]

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To develop fairing-PIA, we first define the functionals as follows:[13]

 

where   represents the  th derivative of the basis function  ,[8] (e.g. B-spline basis function).

Let the curve after the  th iteration be

 

To construct the new curve  ,we first calculate the  st difference vectors for data points,[13]

 

Then, the fitting difference vectors and the fairing vectors for control points are calculated by[13]

   

Finally, the control points of the  st curve are produced by[13]

 

where   is a normalization weight, and   is a smoothing weight corresponding to the  th control point. The smoothing weights can be employed to adjust the smoothness individually, thus bringing great flexibility for smoothness.[13] The larger the smoothing weight is, the smoother the generated curve is. The new curve is obtained as follows

 

In this way, we obtain a sequence of curves  . The sequence converges to the solution of the conventional fairing method based on energy minimization when all smoothing weights are equal (  ).[13] Similarly, the fairing-PIA can be extended to the surface case.

IG-LSPIA[14]

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Given a boundary value problem[15]

 

where   is the unknown solution,  and   are the differential operator and boundary operator, respectively.   and   are the continuous functions. In the isogeometric analysis method, NURBS basis functions[8] are used as shape functions to solve the numerical solution of this boundary value problem.[15] The same basis functions are applied to represent the numerical solution   and the geometric mapping  :

 

where   denotes the NURBS basis function,  is the control coefficient. After substituting the collocation points[22]   into the strong form of PDE,we obtain a discretized problem[22]

 

where   and   denote the subscripts of internal and boundary collocation points, respectively.

Arranging the control coefficients   of the numerical solution   into an  -dimensional column vector, i.e.,  , the discretized problem can be reformulated in matrix form

 

where   is the collocation matrix,and   is the load vector.

Assume that the discretized load values are data points   to be fitted. Given the initial guess of the control coefficients  ),we obtain an initial blending function[14]

 

where   ,represents the combination of different order derivatives of the NURBS basis functions determined using the operators   and  

 

where   and   indicate the interior and boundary of the parameter domain, respectively. Each   corresponds to the  th control coefficient. Assume that   and   are the index sets of the internal and boundary control coefficients, respectively. Without loss of generality, we further assume that the boundary control coefficients have been obtained using strong or weak imposition and are fixed, i.e.,

 

The  th blending function, generated after the  th iteration of IG-LSPIA,[14] is assumed to be as follows:

 

Then, the difference vectors for collocation points (DCP) in the  st iteration are obtained using

 

Moreover, group all load values whose parameters fall in the local support of the  th derivatives function, i.e.,  , into the  th group corresponding to the  th control coefficient, and denote the index set of the  th group of load values as  . Lastly, the differences for control coefficients (DCC) can be constructed as follows:[14]

 

where   is a normalization weight to guarantee the convergence of the algorithm.

Thus, the new control coefficients are updated via the following formula,

 

Consequently, the  st blending function is generated as follows:

 

The above iteration process is performed until the desired fitting precision is reached and a sequence of blending functions is obtained

 

The IG-LSPIA converges to the solution of a constrained least-squares collocation problem.[14]

Proof of convergence

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Non-singular case

  • If  , the PIA iterative format in matrix form is[1]

 

where,

 

The convergence of the PIA is related to the properties of the collocation matrix. If the spectral radius of iteration matrix   is less than  ,then the PIA is convergent. It has been shown that the PIA methods for Bézier curves and surface, B-spline curves and surfaces, NURBS curves and surfaces, Triangular Bernstein-Bézier surface, and subdivision surfaces (Loop, Catmull-Clark, Doo-Sabin) are convergent.[2]

  • If  ,the LSPIA in matrix form is[10][19]

 

When the matrix   is nonsingular,the following results can be obtained.[23]

Lemma If   ,where   is the largest eigenvalue of the matrix  ,then the eigenvalues of   are real numbers and satisfy  .

Proof Since   is nonsingular,and  ,then  . Moreover,

 

In summary, .

Theorem If   ,LSPIA is convergent, and converges to the least-squares fitting result to the given data points.[10][19]

Proof From the matrix form of iterative format, we obtain the following :

 

According to above Lemma, the spectral radius of the matrix   satisfies

 

Thus,the spectral radius of the iteration matrix satisfies

 

When  

 

As a result,

 

i.e.,  ,which is equivalent to the normal equation of the fitting problem. Hence, the LSPIA algorithm converges to the least squares result for a given sequence of points.

Singular case

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Lin et al. showed that LSPIA converges even when the iteration matrix is singular.[17]

Acceleration algorithms and others

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  • Precondition:Liu et al. proposed a preconditioned PIA for Bézier surfaces via the diagonally compensated reduction method, effectively improving the accuracy and efficiency of the classical algorithm.[24]
  • Iteration matrix inverse approximation: Sajavičius improves the LSPIA based on the matrix approximate inverse method. In each iteration step, the approximate inverse of the coefficient matrix of the least-squares fitting problem is first computed and then used as the weight to adjust the control points.[25]
  • Optimal weight: Lu initially presented a weighted progressive-iterative approximation (WPIA) that introduces the optimal weight of difference vectors for control points to accelerate the convergence.[26] Moreover, Zhang et al. proposed a weighted local PIA format for tensor Bézier surfaces.[27] Li et al. assigned initial weights to each data point, and the weights of the interpolated points are determined adaptively during the iterative process.[28]
  • Acceleration with memory: In 2020, Huang et al. proposed a progressive-iterative approximation method with memory for least square fitting (MLSPIA), which has a similar format to the momentum method. MLSPIA generates a series of fitting curves (surfaces) with three weights by iteratively adjusting the control points. With appropriate parameter selection, these curves (surfaces) converge to the least squares fit results for a given data point and are more efficient than LSPIA.[29]
  • Stochastic descent strategy:Rios and Jüttle explored the relationship between LSPIA and gradient descent method and proposed a stochastic LSPIA algorithm with parameter correction.[30]

Applications

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Since PIA has obvious geometric meaning, constraints can be easily integrated in the iterations. Currently, PIA has been widely applied in many fields, such as data fitting, reverse engineering, geometric design, mesh generation, data compression, fairing curve and surface generation, and isogeometric analysis.

Data fitting

  • Adaptive data fitting:The control points are divided into active control points and fixed control points. In each round of iteration, if the fitting error of a data point reaches a given precision, its corresponding control point is fixed and not updated. The above iterative process is repeated until all control points are fixed . The algorithm performs well on large-scale data fitting by adaptively reducing the number of active control points.[31]
  • Large-scale data fitting:By combining T-spline with PIA, an incremental fitting algorithm suitable for fitting large-scale data sets is proposed. During the incremental iteration, each new round of iterations reuses information from the last round of iterations to save computation. While the convergence speed of the traditional point-by-point iterative algorithm decreases as the number of control points increases, in PIA the computation of each iteration step is unrelated to the number of control points, thus exhibiting a powerful data fitting capability.[10]
  • Local fitting:Based on the local property of PIA,a series of local PIA formats have been proposed.[12][32]

Implicit reconstruction

For implicit curve and surface reconstruction, the PIA avoids the additional zero level set and regularization term, which greatly improves the speed of the reconstruction algorithm.[11]

Offset curve approximation

Firstly, the data points are sampled on the original curve. Then, the initial polynomial approximation curve or rational approximation curve of the offset curve is generated from these sampled points. Finally, the offset curve is approximated iteratively using the PIA method.[33]

Mesh generation

Input a triangular mesh model, the algorithm first constructs the initial hexahedral mesh, and extracts the quadrilateral mesh of the surface as the initial boundary mesh. During the iterations, the movement of each mesh vertex is constrained to ensure the validity of the mesh. Finally, the hexahedral model is fitted to the given input model. The algorithm can guarantee the validity of the generated hexahedral mesh, i.e., the Jacobi value at each mesh vertex is greater than zero.[34]

Data compression

First, the image data are converted into a one-dimensional sequence by Hilbert scan; then, these data points are fitted by LSPIA to generate a Hilbert curve; finally, the Hilbert curve is sampled, and the compressed image can be reconstructed. This method can well preserve the neighborhood information of pixels.[35]

Fairing curve and surface generation

Given a data point set, we first define the fairing functional, and calculate the fitting difference vector and the fairing vector of the control point; then, adjust the control points with fairing weights. According to the above steps, the fairing curve and surface can be generated iteratively. Due to the sufficient fairing parameters, the method can achieve global or local fairing. It is also flexible to adjust knot vectors, fairing weights, or data parameterization after each round of iteration. The traditional energy-minimization method is a special case of this method, i.e., when the smooth weights are all the same.[13]

Isogeometric analysis

The discretized load values are regarded as the set of data points, and the combination of the basis functions and their derivative functions is used as the blending function for fitting. The method automatically adjusts the degrees of freedom of the numerical solution of the partial differential equation according to the fitting result of the blending function to the load values. In addition, the average iteration time per step is only related to the number of data points (i.e., collocation points) and unrelated to the number of control coefficients.[14]

References

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  •   This article incorporates text by zju_cagd available under the CC BY 4.0 license.
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