1. Introduction
Scattered data interpolation and approximation are still active research topics in computer-aided design (CAD) and geometric modeling [
1,
2,
3,
4,
5,
6,
7,
8,
9]. This is because engineers and scientists often face the problem of how to produce smooth curves and surfaces for the raw data obtained from experiments or observations. This is where scattered data interpolation can be used to assist them. To construct smooth curves and surfaces, some mathematical formulations are required. This can be achieved using functions which are well-established, such as the Bézier, B-spline, and radial basis functions (RBFs). All these methods are guaranteed to produce smooth curves and surfaces.
The formulation problem in scattered data interpolation can be described as follows:
Given functional data
construct a smooth
surface
such that
To solve the above problem, there are many methods that can be used, such as meshless methods (e.g., radial basis functions (RBFs) and many types of Shepard’s families). However, some meshless schemes are global. Fasshauer [
10] gave details on many meshless methods to solve the problems arising in scattered data interpolation and approximation, as well as partial differential equations. Beyond that, another approach that can be used to solve the problem is the triangulation of the given data points. Then, the Bézier or spline triangular can be used to construct a piecewise smooth surface with some degree of smoothness, such as
or
. The Shepard triangular can also be used to produce a continuous surface from irregular scattered data. For instance, Cavoretto et al. [
6], Dell’Accio and Di Tommaso [
11], and Dell’Accio et al. [
12,
13] have discussed the application of the Shepard triangular for surface reconstruction. However, their schemes require more computation time in order to produce the interpolated surfaces.
Crivellaro et al. [
14] applied RBFs to reconstruct 3D scattered data via new algorithms, which involves an adaptive multi-level interpolation approach based on implicit surface representation. The least squares approximation is used to remove the noise that appears in the scattered data. Chen and Cao [
15] employed neural network operators of a logistic function through translations and dilation. Meanwhile, Bracco et al. [
2] considered scattered data fitting using hierarchical splines where the local solutions are represented in variable degrees of the polynomial spline. Zhou and Li [
16] studied scattered noise data by extending the weighted least squares method via triangulating the data points. Zhou and Li [
17] discussed the scattered data interpolation for noisy data by using bivariate splines defined on triangulation. Qian et al. [
18] also considered scattered data interpolation by using a new recursive algorithm based on the non-tensor product of bivariate splines. Liu [
19] constructed local multilevel scattered data interpolation by proposing a new idea (i.e., nested scattered data sets), and scaled the compactly supported RBFs. Borne and Wende [
3] also considered the meshless scheme based on definite RBFs for scattered data interpolation. In their study, they applied the domain decomposition methods to produce a symmetric-saddle point system. Joldes et al. [
20] modified the moving least squares (MLS) methods by integrating the polynomial bases to solve the scattered data interpolation problem. Brodlie et al. [
5] discussed the constrained surface interpolation by using the Shepard interpolant. The solution to the problem is obtained by solving some optimization. Lai and Meile [
21] discussed scattered data interpolation by using nonnegative bivariate triangular splines to preserve the shape of the scattered data. Schumaker and Speleers [
22] considered the nonnegativity preservation of scattered data by using macro-element spline spaces including Clough–Tocher macro-elements. Furthermore, they also give general results for range-restricted interpolation. Karim et al. [
23] discussed the spatial interpolation for rainfall data by employing cubic Bézier triangular patches to interpolate the scattered data. Karim et al. [
24] have constructed a new type of cubic Bézier-like triangular patches for scattered data interpolation. Karim and Saaban [
25] constructed the terrain data using cubic Ball triangular patches [
23]. In this study, they show that the scattered data interpolation scheme by Said and Rahmat [
26] is not
everywhere. Thus, a new condition for
continuity is derived. The final surface is
and provides a smooth surface. Feng and Zhang [
27] proposed piecewise bivariate Hermite interpolations based on triangulation. They applied the scheme for large scattered data sets to produce high-accuracy surface reconstruction. Sun et al. [
28] constructed bivariate rational interpolation defined on a triangular domain for scattered data lying on a parallel line. They only considered a few data sets, and it was not tested for large data sets. By using a rational spline, the computation time increases. Bozzini et al. [
4] proposed a polyharmonic spline to approximate the noisy scattered data.
The main motivation of the present study is described in the following paragraphs. In triangulation-based approaches to scattered data interpolation, cubic Bézier triangular or quintic Bézier triangular patches are the common methods. The quartic Bézier triangular has received less attention due to the need to solve optimization problems in order to calculate the Bézier ordinates. This approach increases the computation time. There are four steps in constructing a surface using a triangulation method: (a) triangulate the domain by using Delaunay triangulation; (b) specify the first partial derivative at the data points (sites); (c) assign the control points or ordinates for each triangular patch; and finally (d) the surface is constructed via a convex combination scheme. Goodman and Said [
29] constructed a suitable
triangular interpolant for scattered data interpolation using a convex combination scheme between three local schemes. Their work is different from that of Foley and Opitz [
30]. However, both studies developed a
cubic triangular convex combination scheme. Said and Rahmat [
26] constructed a scattered data surface using cubic Ball triangular patches [
31,
32] with the same approach as in Goodman and Said [
29]. Based on the numerical results, their scheme gave the same results as cubic Bézier triangular patches. The main advantages of cubic Ball triangular patches are that the required computation is 7% less when compared with the work of Goodman and Said [
29]. This is what has been claimed by References [
26,
29]. However, in the work of Karim and Saaban [
25], it was proved that Said and Rahmat [
26] is not
continuous everywhere, and Karim and Saaban [
25] found that the [
26] scheme produced the same surface for scattered data interpolation when the inner coefficient was calculated by using Reference [
29]. Hussain and Hussain [
33] proposed the rational cubic Bézier triangular for positivity-preserving scattered data interpolation. They claimed that their proposed scheme is
positive everywhere. However, from their results, it is possible that their scheme may not be positive everywhere. Chan and Ong [
7] considered range-restricted interpolation using a cubic Bézier triangular comprising three local schemes. All the schemes were implemented by estimating the partial derivatives at the respective knots using the method proposed by Goodman et al. [
34].
Other than the use of cubic Ball and cubic Bézier triangular patches for scattered data interpolation, there are some studies that have utilized quartic Bézier triangular and rational quartic Bézier triangular patches for scattered data interpolation. For instance, Saaban et al. [
35] constructed
(or
) scattered data interpolation based on the quartic Bézier triangular. Piah et al. [
36] considered
range-restricted positivity-preserving scattered data interpolation by using the quartic Bézier triangular. They employed an optimization method (i.e., the minimized sum of squares) to calculate the inner Bézier points proposed in Saaban et al. [
35]. Hussain et al. [
37] extended this idea to construct convexity-preserving scattered data interpolation. Hussain et al. [
38] constructed a new scattered data interpolation scheme by using the rational quartic Bézier triangular. They applied it to positivity-preserving interpolation. However, to achieve
continuity, we still need to solve some optimization problems. This is the main weakness of quartic Bézier triangular patches when applied to scattered data interpolation. Some good surveys on scattered data interpolation can be found in [
39,
40,
41,
42,
43].
The present study aims to answer the following research questions:
a. Can we construct a scattered data interpolation scheme by using quartic triangular patches but without an optimization method?
b. How can we produce a surface (everywhere)?
c. Is the proposed scheme better than some existing schemes in terms of CPU time, coefficient of determination (R2), and maximum error?
To answer these research questions, we will use the quartic triangular basis initiated by Zhu and Han [
44]. The main advantage of using this quartic basis is that it only requires ten control points to construct one triangular patch. This is the same as the number of control points in the cubic Bézier triangular patch. Thus, in order to construct
scattered data interpolation using the quartic spline triangular, we can employ the Foley and Opitz [
30] cubic precision scheme to calculate the inner ordinates. With this, the optimization problem required in a quartic triangular basis will be avoided. Hence, this will show that the proposed scheme is local. Furthermore, the proposed scheme is different from the works of Lai and Meile [
21] and Schumaker and Spellers [
22], even though all schemes required triangulation of the given data in the first step.
Some contributions from the present study are described below:
1. The proposed scattered data interpolation scheme produces a
surface without any optimization method like Piah et al. [
36], Saaban et al. [
35] and Hussain et al. [
37,
38].
2. The proposed scheme is local; meanwhile, the schemes presented in Piah et al. [
36], Saaban et al. [
35] and Hussain et al. [
37,
38] are global.
3. Based on the CPU time needed to construct the surface, the proposed scheme is faster than quartic Bézier triangular patches. Thus, the reconstruction of scattered surfaces from large data sets can be performed in less time.
4. Furthermore, the proposed positivity-preserving scattered data interpolation is capable of producing a better interpolated surface than quartic Bézier triangular patches. This lies in contrast to scattered data schemes by Ali et al. [
1], Draman et al. [
9] and Karim et al. [
24], which are not positivity-preserving interpolations.
This paper is organized as follows: In
Section 2 we give a review of the triangular basis initiated by Zhu and Han [
44], and the derivation of the quartic triangular basis with ten control points. Some graphical results are presented, as well as the construction of a local scheme comprising convex combination between three local schemes. The numerical results and the discussion are given in
Section 3 with various numerical and graphical results, including a comparison with some existing schemes. Error analysis is also investigated in this section. The construction of the positive scattered data interpolant is discussed in
Section 4. Meanwhile, numerical results for positivity-preserving scattered data interpolation are shown in
Section 5. Conclusions and future recommendations are given in the final section.
3. Results and Discussion for Scattered Data Interpolation
We tested the proposed scheme using two well-known test functions
and
:
We implemented the proposed scheme using MATLAB 2017 version on Intel® Core ™ i5-8250U 1.60 GHz. MATLAB coding was developed based on Algorithm 1. About 25 MATLAB functions were used to obtain all the results.
We chose 36 data point samples in the domain
as shown in
Table 1.
Figure 8 shows the Delaunay triangulation for the data.
Figure 6 shows examples of surface interpolation for both functions. Comparing
Figure 9 and
Figure 10, the surfaces produced by the proposed scheme visually look smoother than the surfaces obtained from the quartic Bézier triangular of Saaban et al. [
35] and Piah et al. [
36].
Figure 10 shows an example of scattered data interpolation using quartic Bézier triangular patches.
To validate the proposed scheme, we calculated the maximum error (Max Error) and coefficient of determination (COD; i.e., R2) for both functions and compared them with those obtained for quartic Bézier triangular for three different numbers of points i.e., 100, 65, and 36 for both functions and . Functions 1 and 2 represent and , respectively.
Table 2 shows the error analysis for both tested functions by using (a) quartic Zhu and Han and (b) quartic Bézier triangular. Meanwhile,
Table 3 shows CPU time in seconds. From
Table 2, we can see that the proposed quartic triangular patches for scattered data interpolation gave smaller Max Error values than the quartic Bézier triangular. Additionally, the proposed scheme gave higher R
2 values for all numbers of data points (100, 65, and 36). From
Table 3, the proposed scheme required less CPU time than the quartic Bézier. For instance, for 100 data points, the proposed scheme only required 0.71 s for data from function
and 0.42 s for data from function
compared with the quartic Bézier which requires 5.6 s and 3.57 s for 100 data points from functions
and
, respectively. Thus, the proposed scheme in this study gave very good results, and was better at treating scattered data than using the quartic Bézier triangular proposed by Piah et al. [
36], Saaban et al. [
35], and Hussain et al. [
37,
38]. We conclude that the proposed scheme required less CPU time than the quartic Bézier triangular. This reduction of CPU time consumption is an advantage when the goal is to construct a surface with thousands of data points or big data.
This can be seen clearly from
Table 3. With the largest number of data points, the CPU time for the proposed scheme was approximately 12.5% that of the CPU time required for the quartic Bézier triangular based scheme. This is very significant, especially when the user wants to render and reconstruct surfaces obtained from very dense data sets. Many studies in scattered data interpolation usually involve the use of Shepard-type interpolants such as Shepard triangular schemes for scattered data interpolation [
6,
11,
12,
13]. We also implemented the Shepard triangular to the same data sets as listed in
Table 1.
Table 4 and
Table 5 show the error analysis for the schemes of Cavoretto et al. [
6], Dell’Accio et al. [
12,
13], and Dell’Accio and Di Tommaso [
11]. Based on CPU time, for all tested data sets, the proposed scheme was faster than the schemes in [
6,
11,
12,
13], except for the case with 100 data points for the data from function
Considering the Max Error, the proposed scheme was better than all four schemes except for the case with 100 data points from function
Therefore, we can conclude that the proposed scheme is better than quartic triangular patch and the Shepard triangular based schemes [
6,
11,
12,
13].
4. Positivity-Preserving Scattered Data Interpolation
In this section, we apply the proposed scheme discussed in the previous section to preserve the positivity of scattered data sets. To do this, first we derive the sufficient condition for the positivity of the quartic triangular spline defined in (5). Finally, the rational corrected scheme defined by (19) will be used to construct a positive surface with C1 continuity.
To derive the sufficient condition for the positivity of the quartic spline triangular patch, we adopted a similar approach to Saaban et al. [
35] and Piah et al. [
36]. Assume that the quartic ordinates at the vertices are strictly positive such
. Let
therefore
(see
Figure 11). Meanwhile, let the other ordinates have the same value, that is
where
. Thus, Equation (5) becomes
From (23) we can observe that when
then
. Meanwhile as
increases,
decreases. We want to find the value of
when the minimum value of
. By taking first partial derivatives, we will obtain the following:
The minimum value of occurs when
and
or equivalently.
From Equation (25) we have
and
Since
, we obtain the following relations:
Substituting this value into (23) we obtain the minimum value of i.e.,
which can be simplified to
Now,
when
Then Equation (27) can be written as . This equation can be solved by using regula-falsi method with suitable choice of initial guess.
Since
and
then
and
(see
Figure 12). Thus, the curve is convex on that region. Let
and
, then the following holds
Figure 12 shows the example of the relative locations of
and
and
.
Now we establish the main theorem for positivity preservation using the proposed scheme.
Theorem 3. Consider the quartic triangular patchwith vertexsuch that. If the remaining quartic triangular ordinates are equal towhereis a unique solution to (28), thenfor alland.
Some observations from Theorem 3 can be made as follows:
Let
. Then, we have
Therefore, as
, then
. Hence,
and therefore
. Thus, the ordinate values are unbounded compared with the work of Chan and Ong [
7] in which the Bézier ordinates are bounded by a lower bound −1/3.
Remark 1. The sufficient condition for the positivity of the quartic triangular patch developed in this study is the same as the sufficient condition for the quartic Bézier triangular patch developed in Saaban et al. [
35]
. The main difference is that the proposed quartic polynomial only requires ten control points (or ordinates) as compared to the quartic Bézier triangular which requires 15 control points and involves some optimization problems as shown in Saaban et al. [
35]
and Hussain et al. [
37,
38]
. Therefore, the proposed positivity preservation using a quartic triangular patch requires less computation time than some established schemes for scattered data interpolation. The final construction of the positive scattered surface is described below:
1. Input positive scattered data points;
2. Triangulate the scattered data using Delaunay triangulation;
3. Assign the first partial derivative at the respective data sites and adjust if necessary, to provide the positivity preservation;
4. The triangular surface is constructed via convex combination between three local schemes;
5. Repeat Steps 1 through 4 for other positive scattered data sets.
5. Numerical Results and Discussion for Positivity-Preserving Scattered Data Interpolation
After we derive the sufficient condition for the positivity of quartic triangular patch, the final
scattered data scheme for positivity preservation can be written as follows:
with
We test the proposed scheme by using four well-known test functions given below:
The positive test functions
F1,
F2,
F3, and
F4 were evaluated on 36, 33, 26, and 100 node points respectively (
Table 6,
Table 7,
Table 8 and
Table 9) where all function values were greater than or equal to zero. The nodes of 36 and 33 points were defined on a rectangular domain (
Figure 13a,b), while the 26- and 100-point nodes were defined on a sparse non-rectangular domain (
Figure 13c,d).
Table 8 and
Table 9 show examples of irregular scattered data sets.
For test function
as the data in
Table 6, the interpolated surface did not preserve the positivity of the original surface for the
C1 Zhu and Han quartic (from Theorem 1), as shown in
Figure 14a with calculated
. Observe that these surfaces cross the
xy-plane at a number of places. After applying positivity-preserving methods from Theorem 3, the result is shown in
Figure 14b, where the interpolated surfaces lie above or on the
xy-plane
.
For test function
as the data in
Table 7, the interpolated surface did not preserve the positivity of the original surface for the
C1 Zhu and Han quartic as shown in
Figure 15a, with calculated
. These surfaces cross the
xy-plane at a number of places. Using the proposed positivity-preserving methods, the interpolated surface lies above or on the
xy-plane, as shown in
Figure 15b, with calculated
.
For the third test function defined on a sparse non-rectangular domain (data in
Table 8), the interpolated surface did not preserve the positivity, as shown in
Figure 16a where the surface crosses below the
xy-plane with
and the positivity-preserving interpolated surface using the proposed scheme is shown in
Figure 16b where the surface lies above or on the
xy-plane, with calculated.
The interpolated surface of the Zhu and Han C
1 quartic without positivity preservation is given in
Figure 17a, with calculated
, while the positivity-preserved surface lying above the
xy-plane is illustrated in
Figure 17b with calculated
.
We also calculated the CPU time (in seconds), maximum error, and coefficient of determination (R
2) for the positivity-preserving scattered data interpolation as shown in
Table 10 and
Table 11. Once again, the proposed scheme was superior to the quartic Bézier triangular patch. For positivity preservation in scattered data interpolation with dense data sets (i.e., 100 data points with 1697 points of evaluation), the proposed scheme only required 0.5168 s, compared with the quartic Bézier which required 18.5996 s. This is about 36 times faster than the times obtained by the schemes of Saaban et al. [
35] and Piah et al. [
36]. Roughly, the proposed scheme only required about 2.78% of the CPU times of schemes [
35,
36]. This is very significant when we want to visualize thousands of scattered data points.
Our final example is devoted to the coronavirus disease 2019 (COVID-19) cases at Selangor State and Klang Valley in Malaysia until 15 April 2020. There were 5072 positive cases in Malaysia on 15 April 2020. Selangor and Klang Valley alone had about 2296 positive cases. This represents 45.27% of all COVID-19 cases in Malaysia.
Table 12 shows the number of positive COVID-19 cases in 14 districts of Selangor and Klang Valley, including Putrajaya [
45].
Figure 18 and
Figure 19 show the example of surface interpolation for COVID-19 scattered data listed in
Table 12.
Figure 18 shows the interpolated surface without positivity preservation.
Figure 19 shows the interpolated surface after we applied the positivity-preserving scheme. Clearly,
Figure 19 is suitable for the relevant agency to visualize the number of COVID-19 cases. Then, they can prepare any contingency plan for the spread of COVID-19. They could also try to minimize the spread of COVID-19. This is very crucial, since at the time of writing there are no available vaccines to cure the patients.