# Progressive Iterative Approximation for Extended B-Spline Interpolation Surfaces

Authors: Yeqing Yi; Zixuan Tang; Chengzhi Liu
Journal: Mathematical Problems in Engineering
 (2021)

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## 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.


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## Body

## 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 Section2. 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 0≤t≤1, the extended cubic uniform B-spline basis (abbr. λ-B-spline basis) is(1)N0t;λ=16+2λ1−λt1−t3,N1t;λ=2+λ3+λ1−λt1−t3+2+λ3+λ3+λ−λt1−t2t+13+λ3+λt1−tt2+16+2λ1−λ+λtt3,N2t;λ=16+2λ1−λt1−t3+13+λ3+λ−λt1−t2t+2+λ3+λ3+λt1−tt2+2+λ3+λ1−λ+λtt3,N3t;λ=16+2λ1−λ+λtt3,where λ∈−1,1 is the so-called shape parameter.
Theλ-B-spline basis has the properties of non-negativity and symmetry, and it will degenerate into the classic cubic B-spline basis if λ=0 [22].Definition 2.
(see [22]). Given knot vectors u1,u2,…,um−1 and v1,v2,…,vn−1 such that u1<u2<…<um−1 and v1<v2<…<vun−1. Let pij∈ℝ3i=0,1,…,m;j=0,1,…,n be the control points, and Let Nlt;λ1l=03 and Nst;λ2s=03 be the λ-B-spline bases defined as in (1). Then, for u∈ui,ui+1, i=1,2,…,m−2 and v∈vj,vj+1, j=1,2,…,n−2, we can define m−2×n−2 extended cubic uniform B-spline patches with shape parameters λ1 and λ2 as(2)Siju,v=∑l=03∑s=03pi+l−1,j+s−1Nlu−uiui+1−ui;λ1Nsv−vjvj+1−vj;λ2.
All these patches comprise an entire extended cubic uniform B-spline surface (abbr.λ-B-spline surface):(3)Su,v=Siju,v,i=1,2,…,m−2;j=1,2,…,n−2.
Due to the degeneracy property of theλ-B-spline basis, it is easy to verify that the λ-B-spline surface will degenerate into the classic bicubic B-spline surface if λ1=λ2=0.
If we want theλ-B-spline surface to interpolate the boundary control points, we have to add several control vertices according to(4)p−1,−1=2p0,0−p1,1,p−1,n+1=2p0,n−p1,n−1,pm+1,−1=2pm,0−pm−1,1,pm+1,n+1=2pm,n−pm−1,n−1,pi,−1=2pi,0−pi,1,pi,n+1=2pi,n−pi,n−1,i=0,1,…,n,p−1,j=2p0,j−p1,j,pm+1,j=2pm,j−pm−1,j,j=0,1,…,m.


### 2.2. PIA

LetNlt;λ1l=03 and Nst;λ2s=03 be the λ-B-spline bases defined as in (1). Given a set of organized data points piji=0,…,mi=0,…,n to be interpolated, we assign a pair of parameters ui,vj to the i,jth point pij, where u0<u1<…<um and v0<v1<…<vn.Firstly, the pointspiji=0,…,mi=0,…,n as well as these added points (4) are interpreted as the control points of a λ-B-spline surface. Therefore, we can obtain the initial approximate interpolation curve(5)S0u,v=Sij0u,v=∑l=03∑s=03pi+l−1,j+s−10Nlu−uiui+1−ui;λ1Nsv−vjvj+1−vj;λ2,i=0,1,…,m−1;j=0,1,…,n−1,where pij0=pij,i=−1,0,…,m+1;j=−1,0,…,n+1.Secondly, let(6)δij0=pij−S0ui,vj,i=−1,0,…,m+1;j=−1,0,…,n+1be the adjusting vectors of the control points. Then, we can adjust the control points according to(7)pij1=pij0+δij0,i=−1,0,…,m+1;j=−1,0,…,n+1.Suppose that we have obtained thek−1th, k=1,2,…, approximate interpolation surface Sk−1u,v; then, the kth approximate interpolation surface can be defined as(8)Sku,v=Sijku,v=∑l=03∑s=03pi+l−1,j+s−1kNlu−uiui+1−ui;λ1Nsv−vjvj+1−vj;λ2,i=0,1,…,m−1;j=0,1,…,n−1,where(9)pijk=pijk−1+δijk−1,δijk−1=pij−Sk−1ui,vj,i=−1,0,…,m+1;j=−1,0,…,n+1.Therefore, we obtain a sequence of approximate interpolation surfacesSku,v,k=0,1,…,. The initial surface S0u,v is said to have the PIA property if the limit of Sku,v interpolates the points piji=0,…,mi=0,…,n. 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(10)P=p−1,−1,p−1,0,…,p−1,n+1,p0,−1,…,p0,n+1,…,pm+1,−1,…,pm+1,n+1T,(11)Pk=p−1,−1k,p−1,0k,…,p−1,n+1k,p0,−1k,…,p0,n+1k,…,pm+1,−1k,…,pm+1,n+1kT.Then, equation (9) can be written as(12)Pk+1=I−B1⊗B2Pk+P,where ⊗ is the Kronecker product, I is the identity matrix, and B1∈ℝm+3×m+3 and B2∈ℝn+3×n+3 are the collocation matrices resulting from the λ-B-spline basis; in detail,(13)B1=1016+2λ12+λ13+λ116+2λ1⋱⋱⋱16+2λ12+λ13+λ116+2λ101,B2=1016+2λ22+λ23+λ216+2λ2⋱⋱⋱16+2λ22+λ23+λ216+2λ201.We refer to the iterative format (12) as the PIA format. The matrix I−B1⊗B2 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 I−B1⊗B2 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 λ1 and λ2. This indicates that we can minimize the spectral radius by selecting optimal shape parameters λ1 and λ2, and hence the convergence of PIA can be improved. We will discuss the optimal shape parameters λ1 and λ2 in the following section.Finally, we give some notations. The spectral radius of a matrixA is denoted by ρA, the eigenvalues of A are denoted by μiA, and the smallest eigenvalue of a matrix A is denoted by μminA.


## 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 0≤t≤1, the extended cubic uniform B-spline basis (abbr. λ-B-spline basis) is(1)N0t;λ=16+2λ1−λt1−t3,N1t;λ=2+λ3+λ1−λt1−t3+2+λ3+λ3+λ−λt1−t2t+13+λ3+λt1−tt2+16+2λ1−λ+λtt3,N2t;λ=16+2λ1−λt1−t3+13+λ3+λ−λt1−t2t+2+λ3+λ3+λt1−tt2+2+λ3+λ1−λ+λtt3,N3t;λ=16+2λ1−λ+λtt3,where λ∈−1,1 is the so-called shape parameter.
Theλ-B-spline basis has the properties of non-negativity and symmetry, and it will degenerate into the classic cubic B-spline basis if λ=0 [22].Definition 2.
(see [22]). Given knot vectors u1,u2,…,um−1 and v1,v2,…,vn−1 such that u1<u2<…<um−1 and v1<v2<…<vun−1. Let pij∈ℝ3i=0,1,…,m;j=0,1,…,n be the control points, and Let Nlt;λ1l=03 and Nst;λ2s=03 be the λ-B-spline bases defined as in (1). Then, for u∈ui,ui+1, i=1,2,…,m−2 and v∈vj,vj+1, j=1,2,…,n−2, we can define m−2×n−2 extended cubic uniform B-spline patches with shape parameters λ1 and λ2 as(2)Siju,v=∑l=03∑s=03pi+l−1,j+s−1Nlu−uiui+1−ui;λ1Nsv−vjvj+1−vj;λ2.
All these patches comprise an entire extended cubic uniform B-spline surface (abbr.λ-B-spline surface):(3)Su,v=Siju,v,i=1,2,…,m−2;j=1,2,…,n−2.
Due to the degeneracy property of theλ-B-spline basis, it is easy to verify that the λ-B-spline surface will degenerate into the classic bicubic B-spline surface if λ1=λ2=0.
If we want theλ-B-spline surface to interpolate the boundary control points, we have to add several control vertices according to(4)p−1,−1=2p0,0−p1,1,p−1,n+1=2p0,n−p1,n−1,pm+1,−1=2pm,0−pm−1,1,pm+1,n+1=2pm,n−pm−1,n−1,pi,−1=2pi,0−pi,1,pi,n+1=2pi,n−pi,n−1,i=0,1,…,n,p−1,j=2p0,j−p1,j,pm+1,j=2pm,j−pm−1,j,j=0,1,…,m.


## 2.2. PIA

LetNlt;λ1l=03 and Nst;λ2s=03 be the λ-B-spline bases defined as in (1). Given a set of organized data points piji=0,…,mi=0,…,n to be interpolated, we assign a pair of parameters ui,vj to the i,jth point pij, where u0<u1<…<um and v0<v1<…<vn.Firstly, the pointspiji=0,…,mi=0,…,n as well as these added points (4) are interpreted as the control points of a λ-B-spline surface. Therefore, we can obtain the initial approximate interpolation curve(5)S0u,v=Sij0u,v=∑l=03∑s=03pi+l−1,j+s−10Nlu−uiui+1−ui;λ1Nsv−vjvj+1−vj;λ2,i=0,1,…,m−1;j=0,1,…,n−1,where pij0=pij,i=−1,0,…,m+1;j=−1,0,…,n+1.Secondly, let(6)δij0=pij−S0ui,vj,i=−1,0,…,m+1;j=−1,0,…,n+1be the adjusting vectors of the control points. Then, we can adjust the control points according to(7)pij1=pij0+δij0,i=−1,0,…,m+1;j=−1,0,…,n+1.Suppose that we have obtained thek−1th, k=1,2,…, approximate interpolation surface Sk−1u,v; then, the kth approximate interpolation surface can be defined as(8)Sku,v=Sijku,v=∑l=03∑s=03pi+l−1,j+s−1kNlu−uiui+1−ui;λ1Nsv−vjvj+1−vj;λ2,i=0,1,…,m−1;j=0,1,…,n−1,where(9)pijk=pijk−1+δijk−1,δijk−1=pij−Sk−1ui,vj,i=−1,0,…,m+1;j=−1,0,…,n+1.Therefore, we obtain a sequence of approximate interpolation surfacesSku,v,k=0,1,…,. The initial surface S0u,v is said to have the PIA property if the limit of Sku,v interpolates the points piji=0,…,mi=0,…,n. 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(10)P=p−1,−1,p−1,0,…,p−1,n+1,p0,−1,…,p0,n+1,…,pm+1,−1,…,pm+1,n+1T,(11)Pk=p−1,−1k,p−1,0k,…,p−1,n+1k,p0,−1k,…,p0,n+1k,…,pm+1,−1k,…,pm+1,n+1kT.Then, equation (9) can be written as(12)Pk+1=I−B1⊗B2Pk+P,where ⊗ is the Kronecker product, I is the identity matrix, and B1∈ℝm+3×m+3 and B2∈ℝn+3×n+3 are the collocation matrices resulting from the λ-B-spline basis; in detail,(13)B1=1016+2λ12+λ13+λ116+2λ1⋱⋱⋱16+2λ12+λ13+λ116+2λ101,B2=1016+2λ22+λ23+λ216+2λ2⋱⋱⋱16+2λ22+λ23+λ216+2λ201.We refer to the iterative format (12) as the PIA format. The matrix I−B1⊗B2 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 I−B1⊗B2 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 λ1 and λ2. This indicates that we can minimize the spectral radius by selecting optimal shape parameters λ1 and λ2, and hence the convergence of PIA can be improved. We will discuss the optimal shape parameters λ1 and λ2 in the following section.Finally, we give some notations. The spectral radius of a matrixA is denoted by ρA, the eigenvalues of A are denoted by μiA, and the smallest eigenvalue of a matrix A is denoted by μminA.


## 3. Optimal Shape Parameters

In order to make the PIA have the fastest convergence rate, we have to solve the optimal shape parametersλ1 and λ2 that minimize the spectral radius of PIA’s iteration matrix, i.e.,(14)minλ1,λ2∈−1,1ρI−B1⊗B2.Lemma 1 (see [24]).
Suppose thatA and B are square matrices of size m and n, respectively. Let μiAi=1m and μjBj=1n be the eigenvalues of A and B, respectively. Then, the eigenvalues of A⊗B are μiAμjB,i=1,2,…,m;j=1,2,…,n.Lemma 2 (see [21]).
LetB be an n×n collocation matrix resulting from the λ-B-spline basis. Then, the eigenvalues of B are(15)μiB=2+λ3+λ+13+λcosin−1π,i=0,1,…,n−2.1,i=n−1.
By direct deduction, we have the following corollary.Corollary 1.
LetB be an n×n collocation matrix resulting from the λ-B-spline basis. Then,(1)
The eigenvalues ofB are distributed at the interval 0,1.(2)
Givenλ∈0,1, the smallest eigenvalue of B is μminB=2+λ/3+λ+1/3+λcosn−2/n−1π.(3)
μminB decreases as λ increases and reaches minimum when λ=1.Theorem 1.
LetB1∈ℝm+3×m+3 and B2∈ℝn+3×n+3 be the collocation matrices defined as in (13). For fixed λ1,λ2∈−1,1, the spectral radius of the iteration matrix of PIA is(16)ρI−B1⊗B2=1−2+λ1+cosm+1/m+2π2+λ2+cosn+1/n+2π3+λ13+λ2.
The PIA has the fastest convergence rate whenλ1=λ2=1, and in such case, the spectral radius is(17)ρI−B1⊗B2=1−3+cosm+1/m+2π3+cosn+1/n+2π16.Proof.
According to Corollary1, for i=0,1,…,m+2; j=0,1,…,n+2, we have 0<μiB1,μjB2≤1, so is the product of μiB1 and μjB2, i.e., 0<μiB1μjB2≤1. Combined with Lemma1, we have(18)ρI−B1⊗B2=1−μminB1⊗B2=1−μminB1μminB2=1−2+λ1+cosm+1/m+2π2+λ2+cosn+1/n+2π3+λ13+λ2.
From Corollary1, μminB1 and μminB2 minimize at λ1=1 and λ2=1, respectively. By substituting λ1=λ2=1 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.Letpiji=0,…,mi=0,…,n be the points to be interpolated, and let Sku,v be the kth approximate interpolation λ-B-spline surface. Then, the interpolation error of Sku,v can be defined as(19)εk=max0≤i≤m,0≤j≤nδijk=max0≤i≤m,0≤j≤npij−Skui,vj,where · is the Euclidean norm.Example 1.
Consider the data interpolation of11×11 points piji=0,1,…,10j=0,1,…,10 sampled from the peaks function(20)fx,y=31−x2e−x2−y+12−1015x−x3−y5e−x2−y2−13e−x+12−y2,x,y∈−3,3⊗−4,4,in the following way:(21)pij=−3+35i,−4+45j,f−3+35i,−4+45j,i;j=0,1,…,10.Example 2.
Consider the data interpolation of 16 points:1,1,1, 2,1,2, 3,1,2, 4,1,1, 1,2,2, 2,2,2.5, 3,2,2.5, 4,2,2, 2,3,2, 1,3,2, 2,3,2.5, 4,3,2, 1,4,1, 2,4,2, 3,4,2, 4,4,1.Example 3.
Consider the data interpolation of90×100 points piji=0,1,…,89j=0,1,…,99 sampled from the function(22)fx,y=34e−9x−22+9y−22/4+34e−9x+12/49−9y+1/10+12e−9x−72+9y−32/4−15e−9x−42−9y−72,at xi,yj:xi=i/89,i=0,1,…,89;yj=j/99,j=0,1,…,99.Example 4.
Consider the data interpolation of160×160 points piji=0,1,…,10j=0,1,…,10 sampled from the function(23)fx,y=sinx2+y2/x2+y2,at xi,yj:xi=−8+16i/159,yj=−8+16j/159,i;j=0,1,…,159.
The PIA forλ-B-spline surfaces with different λ1 and λ2 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 λ1=λ2=0.
As an illustration, we show in Figure1 the spectral radii of PIA’s iteration matrices with different shape parameters λ1,λ2∈0,1 in Examples 1–4. In Table 1, we list the spectral radii of PIA’s iteration matrices. For convenience, the notation λ1,λ2 in Table 1 and the subsequent tables represents the values of the shape parameters λ1 and λ2 . We can see from Figure 1 and Table 1 that the spectral radii of iteration matrices are less than 1 for any λ1,λ2∈−1,1 and minimize at λ1=λ2=1, and hence the PIA converges for λ1,λ2∈−1,1 and has the fastest convergence rate when λ1=λ2=1. Those results coincide with the conclusions in Theorem 1. Thus, for the optimal shape parameters λ1 and λ2, 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 Table2 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 λ1=λ2=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 λ1 and λ2. We can see that with the same iterations, the interpolation errors obtained by the PIA with λ1=λ2=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.
All these numerical results shown in these tables and figures indicate that the PIA forλ-B-spline surfaces has the fastest convergence rate when λ1=λ2=1.Figure 1
Spectral radii of PIA’s iteration matrices with differentλ1 and λ2 in Examples 1–4. (a) Example 1. (b) Example 2. (c) Example 3. (d) Example 4.
(a)(b)(c)(d)Table 1
Spectral radii of PIA’s iteration matrices with differentλ1,λ2 in Examples 1–4.
ExampleρI−B1⊗B2−1/2,−1/2−1/2,0−1/2,1/2−1/2,10,00,11,01/2,11,1Example10.84000.80000.77140.75000.75000.68750.68750.64290.6094Example20.95180.92320.90280.88750.87770.82090.82090.77330.7376Example30.95990.93320.91420.89990.88880.83320.83320.78560.7499Example40.96000.93330.91420.90000.88880.83330.83330.78570.7500Table 2
Number of required PIA iterations for fixed interpolation error in Example1.
ε−1/2,−1/2−1/2,0−1/2,1/2−1/2,10,00,11,01/2,11,11e − 034330262421161712121e − 047649403633242419161e − 0511472585047323326221e − 0615296776663424333281e − 07191122978478525441351e − 0823214711810294636549421e − 09273174139120110737657491e − 1031720016013912684866656Table 3
Interpolation errors of PIA in Example2 with different λ1,λ2.
k−1/2,−1/2−1/2,0−1/2,1/2−1/2,10,00,11,01/2,102.20e − 012.00e − 011.86e − 011.75e − 011.81e − 011.56e − 011.56e − 011.33e − 0111.52e − 021.22e − 021.10e − 021.06e − 028.87e − 036.73e − 036.73e − 033.78e − 0327.33e − 035.89e − 034.61e − 033.55e − 035.00e − 033.35e − 033.35e − 032.53e − 0335.20e − 033.80e − 032.86e − 032.21e − 032.82e − 031.67e − 031.67e − 031.02e − 0342.38e − 031.61e − 031.15e − 038.64e − 041.07e − 035.69e − 045.69e − 042.93e − 0459.60e − 046.00e − 044.09e − 042.97e − 043.64e − 041.72e − 041.72e − 047.52e − 05106.54e − 062.80e − 061.46e − 068.65e − 071.08e − 062.85e − 072.85e − 075.84e − 08153.98e − 081.16e − 084.52e − 092.15e − 092.89e − 094.19e − 104.19e − 104.14e − 11202.41e − 104.78e − 111.39e − 115.23e − 127.69e − 126.10e − 136.10e − 132.89e − 14Table 4
Interpolation errors of PIA in Examples3 and 4 with different λ1,λ2.
kExample3Example4−1/2,−1/20,01,1−1/2,−1/20,01,101.48e − 031.24e − 039.27e − 041.35e − 031.12e − 038.42e − 0414.87e − 053.39e − 051.91e − 054.71e − 053.27e − 051.84e − 0521.94e − 051.12e − 054.75e − 061.88e − 051.09e − 054.58e − 0639.67e − 064.67e − 061.48e − 069.37e − 064.52e − 061.43e − 0645.41e − 062.18e − 065.17e − 075.24e − 062.11e − 065.00e − 0753.70e − 061.24e − 062.21e − 073.59e − 061.20e − 062.14e − 07106.22e − 078.39e − 083.55e − 096.04e − 078.13e − 083.43e − 09203.37e − 087.35e − 101.76e − 123.27e − 087.11e − 101.69e − 12502.09e − 111.21e − 133.15e − 161.67e − 117.89e − 141.99e − 15Figure 2
Interpolation surfaces obtained by PIA withλ1=λ2=0 at the kth iteration in Example 1. (a) k = 0. (b) k = 1. (c) k = 10.
(a)(b)(c)Figure 3
Interpolation surfaces obtained by PIA withλ1=λ2=1 at the kth iteration in Example 1. (a) k = 0. (b) k = 1. (c) k = 10.
(a)(b)(c)Figure 4
Interpolation surfaces obtain by PIA withλ1=λ2=0 at the kth iteration in Example 2. (a) k = 0. (b) k = 1. (c) k = 5.
(a)(b)(c)Figure 5
Interpolation surfaces obtain by PIA withλ1=λ2=1 at the kth iteration in Example 2. (a) k = 0. (b) k = 1. (c) k = 5.
(a)(b)(c)Figure 6
Interpolation surfaces obtained by PIA withλ1=λ2=0 at the kth iteration in Example 3. (a) k = 0. (b) k = 2.
(a)(b)Figure 7
Interpolation surfaces obtained by PIA withλ1=λ2=1 at the kth iteration in Example 3. (a) k = 0. (b) k = 2.
(a)(b)Figure 8
Interpolation surfaces obtained by PIA withλ1=λ2=0 at the kth iteration in Example 4. (a) k = 0. (b) k = 2.
(a)(b)Figure 9
Interpolation surfaces obtained by PIA withλ1=λ2=1 at the kth iteration in Example 4. (a) k = 0. (b) k = 2.
(a)(b)


## 5. Conclusion

In this paper, we have exploited the PIA format forλ-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.



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*Source: 5556771-2021-04-27.xml*