Progressive Iterative Approximation for Extended B-Spline Interpolation Surfaces

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.


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][2][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. e 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][6][7][8][9][10][11][12][13][14] and a lot of literatures therein. e emergence of blending bases with shape parameters has enriched the theories and methods of geometric modeling [1,[15][16][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][19][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. e 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.

Extended Cubic Uniform B-Spline Surface.
We begin with the definition of the extended cubic uniform B-spline basis with a shape parameter.
If we want the λ-B-spline surface to interpolate the boundary control points, we have to add several control vertices according to and N s (t; λ 2 ) 3 s�0 be the λ-Bspline bases defined as in (1). Given a set of organized data points p ij i�0,...,n i�0,...,m to be interpolated, we assign a pair of Firstly, the points p ij i�0,...,n i�0,...,m as well as these added points (4) are interpreted as the control points of a λ-Bspline surface. erefore, we can obtain the initial approximate interpolation curve be the adjusting vectors of the control points. en, we can adjust the control points according to Suppose that we have obtained the (k − 1)th, k � 1, 2, . . ., approximate interpolation surface S (k− 1) (u, v); then, the kth approximate interpolation surface can be defined as erefore, we obtain a sequence of approximate interpolation surfaces S (k) (u, v), k � 0, 1, . . . ,. e initial surface S (0) (u, v) is said to have the PIA property if the limit of S (k) (u, v) interpolates the points p ij i�0,...,n i�0,...,m . 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 λ-B-spline basis is normalized and totally positive; therefore, the initial λ-B-spline surface has the PIA property. Let en, equation (9) can be written as where ⊗ is the Kronecker product, I is the identity matrix, and B 1 ∈ R (m+3)×(m+3) and B 2 ∈ R (n+3)×(n+3) are the collocation matrices resulting from the λ-B-spline basis; in detail,

Mathematical Problems in Engineering
We refer to the iterative format (12) as the PIA format. (12) is the iteration matrix. It is well known that the PIA converges if and only if the spectral radius of the iteration matrix I − B 1 ⊗ B 2 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 λ 1 and λ 2 . is indicates that we can minimize the spectral radius by selecting optimal shape parameters λ 1 and λ 2 , and hence the convergence of PIA can be improved. We will discuss the optimal shape parameters λ 1 and λ 2 in the following section.
Finally, we give some notations. e spectral radius of a matrix A is denoted by ρ(A), the eigenvalues of A are denoted by μ i (A), and the smallest eigenvalue of a matrix A is denoted by μ min (A).

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 p ij i�0,...,n i�0,...,m be the points to be interpolated, and let S (k) (u, v) be the kth approximate interpolation λ-B-spline surface. en, the interpolation error of S (k) (u, v) can be defined as where ‖·‖ is the Euclidean norm.
in the following way: at e PIA for λ-B-spline surfaces with different λ 1 and λ 2 is employed to interpolate the points in Examples 1-4. It should be pointed out that the PIA for λ-B-spline surfaces will degenerate into the PIA for the classic bicubic B-spline surfaces if λ 1 � λ 2 � 0.
As an illustration, we show in Figure 1 the spectral radii of PIA's iteration matrices with different shape parameters λ 1 , λ 2 ∈ (0, 1] in Examples 1-4. In Table 1, we list the spectral radii of PIA's iteration matrices. For convenience, the notation (λ 1 , λ 2 ) in Table 1 and the subsequent tables represents the values of the shape parameters λ 1 and λ 2 . We can see from Figure 1 and Table 1 that the spectral radii of iteration matrices are less than 1 for any λ 1 , λ 2 ∈ (− 1, 1] and minimize at λ 1 � λ 2 � 1, and hence the PIA converges for λ 1 , λ 2 ∈ (− 1, 1] and has the fastest convergence rate when λ 1 � λ 2 � 1. ose results coincide with the conclusions in eorem 1. us, for the optimal shape parameters λ 1 and λ 2 , the convergence rate of PIA for λ-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 Mathematical Problems in Engineering of PIA with λ 1 � λ 2 � 1 is the smallest. In Tables 3 and 4 , we list the interpolation errors of Examples 2-4 obtained by implementing the PIA for λ-B-spline surfaces with different λ 1 and λ 2 . We can see that with the same iterations, the interpolation errors obtained by the PIA with λ 1 � λ 2 � 1 are the smallest. Figures 2-9 display the λ-B-spline surfaces with different shape parameters when we employ the PIA to interpolate the data given in Examples 1-4.  Example          All these numerical results shown in these tables and figures indicate that the PIA for λ-B-spline surfaces has the fastest convergence rate when λ 1 � λ 2 � 1.

Conclusion
In this paper, we have exploited the PIA format for λ-Bspline 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. erefore, 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 e data are included within tha article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.