Mathematical Problems in Engineering
Volume 2021 (2021), Article ID 5556771, 10 pages
https://doi.org/10.1155/2021/5556771

Research Article

Progressive Iterative Approximation for Extended B-Spline Interpolation Surfaces

Yeqing Yi,¹ Zixuan Tang,² and Chengzhi Liu²

¹School of Information, Hunan University of Humanities, Science and Technology, Loudi 417000, China
²School of Mathematics and Finance, Hunan University of Humanities, Science and Technology, Loudi 417000, China

Correspondence should be addressed to Chengzhi Liu; it-rocket@163.com

Received 8 February 2021; Revised 23 March 2021; Accepted 13 April 2021; Published 27 April 2021

Academic Editor: Giovanni Garcea

Copyright © 2021 Yeqing Yi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In order to improve the computational efficiency of data interpolation, we study the progressive iterative approximation (PIA) for tensor product extended cubic uniform B-spline surfaces. By solving the optimal shape parameters, we can minimize the spectral radius of PIA’s iteration matrix, and hence the convergence rate of PIA is accelerated. Stated numerical examples show that the optimal shape parameters make the PIA have the fastest convergence rate.

1. Introduction

Data interpolation plays important roles in scientific research and engineering applications. How to solve interpolation curves/surfaces efficiently has been one of the most popular topics in computer-aided geometric design (see [1–3]). Oftentimes, one has to solve a linear system to obtain the interpolation curves or surfaces. Efficient and accurate algorithms are required to guarantee the computational efficiency. For small-scale systems, direct methods are typically the preferred choices. However, for large-scale systems, it becomes necessary to employ iterative methods to obtain the solutions. In recent years, an iterative method, namely, progressive iterative approximation (PIA), has attracted a lot of attention and has become a very hot research area. The PIA stands out because it has the advantages of clear geometric meaning, stable convergence, simple iterative format, local modification, and so on. Furthermore, it avoids to solve a linear system directly. For more details about PIA, we refer the readers to read a recent survey [4].

Despite the fact that the PIA offers many advantages, there is a disadvantage, that is, slow rate of convergence. To overcome this limitation and further improve the computational efficiency, a great deal of acceleration techniques have been conducted. Examples of such approaches include [5–14] and a lot of literatures therein.

The emergence of blending bases with shape parameters has enriched the theories and methods of geometric modeling [1, 15–17]. Due to the flexibly in shape adjustment, splines with shape parameters have drawn much attention for decades and a large number of splines with shape parameters were exploited (see, for example, [18–20]). Very often, the aim of shape parameters is to adjust the shapes of splines, while in [21], the introduction of shape parameters is to speed up the convergence rate of PIA. In that paper, the eigenvalues of the collocation matrix were expressed explicitly, and hence the optimal shape parameters were solved to make the PIA have the fastest convergence rate. Based on this conclusion, we further study the PIA format for tensor product extended cubic uniform B-spline surfaces, which is an extension of the PIA for the classic bicubic uniform B-spline curves. By solving the optimal shape parameters, the convergence rate of PIA is accelerated, and thus the computational efficiency of data interpolation can be improved.

The rest of this paper is organized as follows. After recapping the definition of the extended cubic uniform B-spline surfaces with shape parameters, we exploit the PIA format for extended cubic uniform B-spline surfaces in Section 2. In Section 3, we study the optimal shape parameters to make the PIA have the fastest convergence rate. Some numerical examples are given to illustrate the acceleration effect in Section 4. Finally, we give some concluding remarks in Section 5.

2. PIA for Extended Cubic Uniform B-Spline Surface

2.1. Extended Cubic Uniform B-Spline Surface

We begin with the definition of the extended cubic uniform B-spline basis with a shape parameter.

Definition 1. (see [22]). For , the extended cubic uniform B-spline basis (abbr. -B-spline basis) is

where is the so-called shape parameter.
The -B-spline basis has the properties of non-negativity and symmetry, and it will degenerate into the classic cubic B-spline basis if [22].

Definition 2. (see [22]). Given knot vectors and such that and . Let be the control points, and Let and be the -B-spline bases defined as in (1). Then, for , and , , we can define extended cubic uniform B-spline patches with shape parameters and as

All these patches comprise an entire extended cubic uniform B-spline surface (abbr. -B-spline surface):

Due to the degeneracy property of the -B-spline basis, it is easy to verify that the -B-spline surface will degenerate into the classic bicubic B-spline surface if .
If we want the -B-spline surface to interpolate the boundary control points, we have to add several control vertices according to

2.2. PIA

Let and be the -B-spline bases defined as in (1). Given a set of organized data points to be interpolated, we assign a pair of parameters to the th point , where and .

Firstly, the points as well as these added points (4) are interpreted as the control points of a -B-spline surface. Therefore, we can obtain the initial approximate interpolation curve

where .

Secondly, let

be the adjusting vectors of the control points. Then, we can adjust the control points according to

Suppose that we have obtained the th, , approximate interpolation surface ; then, the th approximate interpolation surface can be defined as

where

Therefore, we obtain a sequence of approximate interpolation surfaces . The initial surface is said to have the PIA property if the limit of interpolates the points . It was shown in [23] that tensor product surfaces generated by normalized and totally positive bases have the PIA property. We note in [22] that the -B-spline basis is normalized and totally positive; therefore, the initial -B-spline surface has the PIA property.

Let

Then, equation (9) can be written as

where is the Kronecker product, is the identity matrix, and and are the collocation matrices resulting from the -B-spline basis; in detail,

We refer to the iterative format (12) as the PIA format. The matrix in (12) is the iteration matrix. It is well known that the PIA converges if and only if the spectral radius of the iteration matrix is less than 1. Moreover, the smaller the spectral radius is, the faster the PIA converges. According to (13), the spectral radius is a function of the shape parameters and . This indicates that we can minimize the spectral radius by selecting optimal shape parameters and , and hence the convergence of PIA can be improved. We will discuss the optimal shape parameters and in the following section.

Finally, we give some notations. The spectral radius of a matrix is denoted by , the eigenvalues of are denoted by , and the smallest eigenvalue of a matrix is denoted by .

3. Optimal Shape Parameters

In order to make the PIA have the fastest convergence rate, we have to solve the optimal shape parameters and that minimize the spectral radius of PIA’s iteration matrix, i.e.,

Lemma 1 (see [24]). Suppose that and are square matrices of size and , respectively. Let and be the eigenvalues of and , respectively. Then, the eigenvalues of are .

Lemma 2 (see [21]). Let be an collocation matrix resulting from the -B-spline basis. Then, the eigenvalues of are

By direct deduction, we have the following corollary.

Corollary 1. Let be an collocation matrix resulting from the -B-spline basis. Then,

(1)The eigenvalues of are distributed at the interval .(2)Given , the smallest eigenvalue of is .(3) decreases as increases and reaches minimum when .

Theorem 1. Let and be the collocation matrices defined as in (13). For fixed , the spectral radius of the iteration matrix of PIA is

The PIA has the fastest convergence rate when , and in such case, the spectral radius is

Proof. According to Corollary 1, for ; , we have , so is the product of and , i.e., . Combined with Lemma¹, we have

From Corollary 1, and minimize at and , respectively. By substituting into (16), the result (17) follows straightforwardly.

4. Numerical Examples

In this section, several numerical examples are presented to assess the effectiveness of the optimal shape parameters. All experiments were performed by Matlab R2012b.

Let be the points to be interpolated, and let be the th approximate interpolation -B-spline surface. Then, the interpolation error of can be defined as

where is the Euclidean norm.

Example 1. Consider the data interpolation of points sampled from the peaks function

in the following way:

Example 2. Consider the data interpolation of 16 points: , , , , , , , , , , , , , , , .

Example 3. Consider the data interpolation of points sampled from the function

at .

Example 4. Consider the data interpolation of points sampled from the function

at .
The PIA for -B-spline surfaces with different and is employed to interpolate the points in Examples 1–4. It should be pointed out that the PIA for -B-spline surfaces will degenerate into the PIA for the classic bicubic B-spline surfaces if .
As an illustration, we show in Figure 1 the spectral radii of PIA’s iteration matrices with different shape parameters in Examples 1–4. In Table 1, we list the spectral radii of PIA’s iteration matrices. For convenience, the notation in Table 1 and the subsequent tables represents the values of the shape parameters and . We can see from Figure 1 and Table 1 that the spectral radii of iteration matrices are less than 1 for any and minimize at , and hence the PIA converges for and has the fastest convergence rate when . Those results coincide with the conclusions in Theorem 1. Thus, for the optimal shape parameters and , the convergence rate of PIA for -B-spline surfaces would achieve a great acceleration compared with that for the classic bicubic B-spline surfaces.
Given interpolation errors, we list in Table 2 the number of required iterations when we test Example 1 with different shape parameters. It is evident from Table 2 that under the requirement of the same precision, the number of iterations of PIA with is the smallest. In Tables 3 and 4 , we list the interpolation errors of Examples 2–4 obtained by implementing the PIA for -B-spline surfaces with different and . We can see that with the same iterations, the interpolation errors obtained by the PIA with are the smallest. Figures 2–9 display the -B-spline surfaces with different shape parameters when we employ the PIA to interpolate the data given in Examples 1–4.
All these numerical results shown in these tables and figures indicate that the PIA for -B-spline surfaces has the fastest convergence rate when .

(a)

(b)

(c)

(d)

Figure 1: Spectral radii of PIA’s iteration matrices with different and in Examples 1–4. (a) Example 1. (b) Example 2. (c) Example 3. (d) Example 4.

Table 1: Spectral radii of PIA’s iteration matrices with different in Examples 1–4.

Example

Example 1
0.8400
0.8000
0.7714
0.7500
0.7500
0.6875
0.6875
0.6429
0.6094

Example 2
0.9518
0.9232
0.9028
0.8875
0.8777
0.8209
0.8209
0.7733
0.7376

Example 3
0.9599
0.9332
0.9142
0.8999
0.8888
0.8332
0.8332
0.7856
0.7499

Example 4
0.9600
0.9333
0.9142
0.9000
0.8888
0.8333
0.8333
0.7857
0.7500

Table 2: Number of required PIA iterations for fixed interpolation error in Example 1.

1e − 03
43
30
26
24
21
16
17
12
12

1e − 04
76
49
40
36
33
24
24
19
16

1e − 05
114
72
58
50
47
32
33
26
22

1e − 06
152
96
77
66
63
42
43
33
28

1e − 07
191
122
97
84
78
52
54
41
35

1e − 08
232
147
118
102
94
63
65
49
42

1e − 09
273
174
139
120
110
73
76
57
49

1e − 10
317
200
160
139
126
84
86
66
56

Table 3: Interpolation errors of PIA in Example 2 with different .

0
2.20e − 01
2.00e − 01
1.86e − 01
1.75e − 01
1.81e − 01
1.56e − 01
1.56e − 01
1.33e − 01

1
1.52e − 02
1.22e − 02
1.10e − 02
1.06e − 02
8.87e − 03
6.73e − 03
6.73e − 03
3.78e − 03

2
7.33e − 03
5.89e − 03
4.61e − 03
3.55e − 03
5.00e − 03
3.35e − 03
3.35e − 03
2.53e − 03

3
5.20e − 03
3.80e − 03
2.86e − 03
2.21e − 03
2.82e − 03
1.67e − 03
1.67e − 03
1.02e − 03

4
2.38e − 03
1.61e − 03
1.15e − 03
8.64e − 04
1.07e − 03
5.69e − 04
5.69e − 04
2.93e − 04

5
9.60e − 04
6.00e − 04
4.09e − 04
2.97e − 04
3.64e − 04
1.72e − 04
1.72e − 04
7.52e − 05

10
6.54e − 06
2.80e − 06
1.46e − 06
8.65e − 07
1.08e − 06
2.85e − 07
2.85e − 07
5.84e − 08

15
3.98e − 08
1.16e − 08
4.52e − 09
2.15e − 09
2.89e − 09
4.19e − 10
4.19e − 10
4.14e − 11

20
2.41e − 10
4.78e − 11
1.39e − 11
5.23e − 12
7.69e − 12
6.10e − 13
6.10e − 13
2.89e − 14

Table 4: Interpolation errors of PIA in Examples 3 and 4 with different .

Example 3
Example 4

0
1.48e − 03
1.24e − 03
9.27e − 04
1.35e − 03
1.12e − 03
8.42e − 04

1
4.87e − 05
3.39e − 05
1.91e − 05
4.71e − 05
3.27e − 05
1.84e − 05

2
1.94e − 05
1.12e − 05
4.75e − 06
1.88e − 05
1.09e − 05
4.58e − 06

3
9.67e − 06
4.67e − 06
1.48e − 06
9.37e − 06
4.52e − 06
1.43e − 06

4
5.41e − 06
2.18e − 06
5.17e − 07
5.24e − 06
2.11e − 06
5.00e − 07

5
3.70e − 06
1.24e − 06
2.21e − 07
3.59e − 06
1.20e − 06
2.14e − 07

10
6.22e − 07
8.39e − 08
3.55e − 09
6.04e − 07
8.13e − 08
3.43e − 09

20
3.37e − 08
7.35e − 10
1.76e − 12
3.27e − 08
7.11e − 10
1.69e − 12

50
2.09e − 11
1.21e − 13
3.15e − 16
1.67e − 11
7.89e − 14
1.99e − 15

(a)

(b)

(c)

Figure 2: Interpolation surfaces obtained by PIA with at the th iteration in Example 1. (a) k = 0. (b) k = 1. (c) k = 10.

(a)

(b)

(c)

Figure 3: Interpolation surfaces obtained by PIA with at the th iteration in Example 1. (a) k = 0. (b) k = 1. (c) k = 10.

(a)

(b)

(c)

Figure 4: Interpolation surfaces obtain by PIA with at the th iteration in Example 2. (a) k = 0. (b) k = 1. (c) k = 5.

(a)

(b)

(c)

Figure 5: Interpolation surfaces obtain by PIA with at the th iteration in Example 2. (a) k = 0. (b) k = 1. (c) k = 5.

(a)

(b)

Figure 6: Interpolation surfaces obtained by PIA with at the th iteration in Example 3. (a) k = 0. (b) k = 2.

(a)

(b)

Figure 7: Interpolation surfaces obtained by PIA with at the th iteration in Example 3. (a) k = 0. (b) k = 2.

(a)

(b)

Figure 8: Interpolation surfaces obtained by PIA with at the th iteration in Example 4. (a) k = 0. (b) k = 2.

(a)

(b)

Figure 9: Interpolation surfaces obtained by PIA with at the th iteration in Example 4. (a) k = 0. (b) k = 2.

5. Conclusion

In this paper, we have exploited the PIA format for -B-spline surfaces. Due to the introduction of shape parameters, we can make the PIA have the fastest convergence rate by solving the optimal shape parameters, while the amount of calculation does not increase. Therefore, it inherits the merits of the PIA for the classic bicubic B-spline surfaces, e.g., simple iterative scheme, stable convergence, clear geometric meaning, local modification, etc. More importantly, the computational efficiency of data interpolation is improved by accelerating the convergence rate.

Data Availability

The data are included within tha article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by the Natural Science Foundation of Hunan Province (grant no. 2020JJ5267) and Scientific Research Funds of Hunan Provincial Education Department (grant nos. CX20201192 and 19B301).

References

Y. Zhu and X. Han, “C² rational quartic interpolation spline with local shape preserving property,” Applied Mathematics Letters, vol. 46, pp. 57–63, 2015.

X. Han, “Shape-preserving piecewise rational interpolation with higher order continuity,” Applied Mathematics and Computation, vol. 337, pp. 1–13, 2018.

M. Abbas, A. A. Majid, and J. M. Ali, “Positivity-preserving rational bi-cubic spline interpolation for 3D positive data,” Applied Mathematics and Computation, vol. 234, pp. 460–476, 2014.

H. Lin, T. Maekawa, and C. Deng, “Survey on geometric iterative methods and their applications,” Computer-Aided Design, vol. 95, pp. 40–51, 2017.

L. Lu, “Weighted progressive iteration approximation and convergence analysis,” Computer Aided Geometric Design, vol. 27, no. 2, pp. 129–137, 2010.

J. M. Carnicer, J. Delgado, and J. Peña, “Richardson method and totally nonnegative linear systems,” Linear Algebra and Its Applications, vol. 11, 2010.

J. M. Carnicer, J. Delgado, and J. M. Peña, “On the progressive iteration approximation property and alternative iterations,” Computer Aided Geometric Design, vol. 28, no. 9, pp. 523–526, 2011.

S. Deng and G. Wang, “Numerical analysis of the progressive iterative approximation method,” Computer Aided Geometric Design, vol. 7, pp. 879–884, 2012.

C. Liu, X. Han, and J. Li, “The Chebyshev accelerating method for progressive iterative approximation,” Communications in Information and Systems, vol. 17, no. 1, pp. 25–43, 2017.

L. Zhang, J. Tan, X. Ge, and G. Zheng, “Generalized B-splines' geometric iterative fitting method with mutually different weights,” Journal of Computational and Applied Mathematics, vol. 329, pp. 331–343, 2018.

A. Ebrahimi and G. B. Loghmani, “A composite iterative procedure with fast convergence rate for the progressive-iteration approximation of curves,” Journal of Computational and Applied Mathematics, vol. 359, pp. 1–15, 2019.

C. Liu and Z. Liu, “Progressive iterative approximation with preconditioners,” Mathematics, vol. 8, no. 9, p. 1503, 2020.

C. Liu, X. Han, and J. Li, “Preconditioned progressive iterative approximation for triangular Bézier patches and its application,” Journal of Computational and Applied Mathematics, vol. 366, no. 112389, 2020.

C. Liu, Z. Liu, and X. Han, “Preconditioned progressive iterative approximation for tensor product Bézier patches,” Mathematics and Computers in Simulation, vol. 185, pp. 372–383, 2021.

U. Bashir and M. Abbas, “A class of quasi-quintic trigonometric Bézier curve with two shape parameters,” Science Asia, vol. 39, no. 2, pp. 11–15, 2013.

U. Bashir, M. Abbas, and J. Ali, “The and ational quadratic trigonometric Bézier curve with two shape parameters with applications,” Applied Mathematics and Computation, vol. 219, no. 20, pp. 183–197, 2013.

A. Ghaffar, M. Iqbal, M. Bari et al., “Construction and application of nine-tic B-spline tensor product SS,” Mathematics, vol. 7, no. 8, p. 675, 2019.

M. Usman, M. Abbas, and K. T. Miura, “Some engineering applications of new trigonometric cubic Bézier-like curves to free-form complex curve modeling,” Journal of Advanced Mechanical Design Systems and Manufacturing, vol. 14, no. 4, 2020.

M. Abbas, N. Ramli, A. A. Majid, and J. M. Ali, “The representation of circular arc by using rational cubic timmer curve,” Mathematical Problems in Engineering, vol. 2014, pp. 1–6, 2014.

A. Majeed, M. Abbas, K. T. Miura, M. Kamran, and T. Nazir, “Surface modeling from 2D contours with an application to craniofacial fracture construction,” Mathematics, vol. 8, no. 8, p. 1246, 2020.

Y. Yi, L. Hu, C. Liu et al., “Progressive iterative approximation for extended cubic uniform B-splines with shape parameters,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 10, 2020.

L. Yan and X. Han, “The extended cubic uniform B-spline curve based on totally positive basis,” Journal of Graphics, vol. 37, pp. 329–336, 2016.

H.-W. Lin, H.-J. Bao, and G.-J. Wang, “Totally positive bases and progressive iteration approximation,” Computers & Mathematics with Applications, vol. 50, no. 3-4, pp. 575–586, 2005.

R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, UK, 1991.