# Scattered Data Interpolation Using Quartic Triangular Patch for Shape-Preserving Interpolation and Comparison with Mesh-Free Methods

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{2}) and smaller maximum error (Max Error), requires about 12.5% of the CPU time of the quartic Bézier triangular, and is on par with Shepard triangular-based schemes. Therefore, the proposed scheme is significant for use in visualizing large and irregular scattered data sets. Finally, we tested the proposed positivity-preserving interpolation scheme to visualize coronavirus disease 2019 (COVID-19) cases in Malaysia.

## 1. Introduction

^{2}), and maximum error?

## 2. Materials and Methods

#### 2.1. Review of the Cubic Triangular Bases of Zhu And Han

**Definition 1.**

#### 2.2. Quartic Zhu and Han Triangular Patches

#### 2.3. Scattered Data Interpolation Using Quartic Zhu and Han Triangular Patches

^{1}continuity along all edges, the following equations must be satisfied:

**Theorem 1.**

Algorithm 1 (Scattered Data Interpolation) |

Step 1: Input scattered data points;Step 2: Estimate the partial derivative at the data points by using [25];Step 3: Triangulate the domain of the data points;Step 4: Calculate the boundary control points using Equations (7)–(12);Step 5: Calculate inner control points for the local scheme, ${b}_{111}^{i}$, $i=1,2,3$ by using the cubic precision method as in Foley and Opitz [30];Step 6: Construct the interpolated surface using the convex combination method of three local schemes defined by (6);Step 7: Calculate CPU time (in seconds), R^{2}, and maximum error. Repeat steps 1 through 6 for the other scattered data sets. |

**Theorem 2.**

^{1}quartic Bézier triangular patches using minimized sum of squares of principal curvatures [35].

_{t}and the total number of interior edges be n

_{e}. S(x,y) which will minimize the functional I(S(x,y)) leads to the optimization problem

**x**

^{T}Q

**x**+

**ex**+ h, subject to the C

^{1}continuity constraint:

_{t}× 6n

_{t}sparse matrix,

**e**is a 1 × 6n

_{t}row vector,

**x**is a 6n

_{t}× 1 column vector consisting of unknown ordinates $\left({b}_{220}^{m}{b}_{211}^{m}{b}_{121}^{m}{b}_{202}^{m}{b}_{112}^{m}{b}_{022}^{m}\right),m=1,\dots ,{n}_{t}$, h is a real constant, A is a 3n

_{e}× 6n

_{t}($3{n}_{e}\le 6{n}_{t}$) coefficient matrix,

**x**is a 6n

_{t}× 1 unknown column vector consisting of the remaining ordinates b

_{220}, b

_{202}, b

_{022}, b

_{211}, b

_{121}and b

_{112}to be determined for the entire triangular mesh, and

**b**is a 3n

_{e}× 1 constant column vector. The optimization problem stated in (18) was solved using the optimization toolbox in MATLAB 2017 on Intel

^{®}Core ™ i5-8250U 1.60 GHz.

^{1}continuity constraint, which results in a global method for scattered data interpolation. Meanwhile, by using the proposed scheme in this study, the resulting surface is local.

## 3. Results and Discussion for Scattered Data Interpolation

^{®}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.

^{2}) 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 ${F}_{1}\left(x,y\right)$ and ${F}_{2}\left(x,y\right)$. Functions 1 and 2 represent ${F}_{1}\left(x,y\right)$ and ${F}_{2}\left(x,y\right)$, respectively.

^{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 ${F}_{1}\left(x,y\right)$ and 0.42 s for data from function ${F}_{2}\left(x,y\right),$ compared with the quartic Bézier which requires 5.6 s and 3.57 s for 100 data points from functions ${F}_{1}\left(x,y\right)$ and ${F}_{2}\left(x,y\right)$, 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.

## 4. Positivity-Preserving Scattered Data Interpolation

^{1}continuity.

**Theorem 3.**

**Remark 1.**

## 5. Numerical Results and Discussion for Positivity-Preserving Scattered Data Interpolation

_{1}, F

_{2}, F

_{3}, and F

_{4}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.

^{1}Zhu and Han quartic (from Theorem 1), as shown in Figure 14a with calculated $\underset{\left(x,y\right)\in D}{min{F}_{1}\left(x,y\right)}=-0.039975$. 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 $\underset{\left(x,y\right)\in D}{min{F}_{1}\left(x,y\right)}=0$.

^{1}Zhu and Han quartic as shown in Figure 15a, with calculated $\underset{\left(x,y\right)\in D}{min{F}_{2}\left(x,y\right)}=-0.039975$. 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 $\underset{\left(x,y\right)\in D}{min{F}_{2}\left(x,y\right)}=0.0072657$.

^{1}quartic without positivity preservation is given in Figure 17a, with calculated $\underset{\left(x,y\right)\in D}{min{F}_{4}\left(x,y\right)}=-0.67634$, while the positivity-preserved surface lying above the xy-plane is illustrated in Figure 17b with calculated $\underset{\left(x,y\right)\in D}{min{F}_{4}\left(x,y\right)}=0.000074928$.

^{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.

## 6. Conclusions

^{2}, and requiring only 12.5% of the CPU time needed by the quartic Bézier triangular scheme. This is very significant, especially when the goal is to reconstruct surfaces from large scattered data sets. Furthermore, based on a comparison against the Shepard triangular for scattered data, the proposed scheme was also superior to the schemes of Cavoretto et al. [6], Dell’Accio et al. [12,13] and Dell’Accio and Di Tommaso [11]. Finally, we constructed a positive interpolant based on the proposed quartic triangular spline to preserve the positivity of scattered data. Numerical results suggest that the proposed scheme is better than existing schemes, especially in terms of CPU time—our proposed scheme requires less computation time than positivity schemes proposed by Piah et al. [36] and Saaban et al. [35]. Finally, we implemented our proposed positivity-preserving interpolation to visualize COVID-19 cases in Selangor State and Klang Valley, Malaysia. The resulting surfaces were smooth and positive everywhere. Future works will focus on the construction of a quintic Zhu and Han spline for scattered data interpolation with quintic precision as well as shape-preserving interpolation (e.g., positivity-preserving and range-restricted interpolation). This can be achieved by extending the main idea from Karim et al. [46]. Another potential study could be a comparison between the use of a CPU and a graphical processing unit (GPU) for large scattered data sets. Finally, the proposed scheme can also be applied to visualize large sets of scattered data, such as from geophysical data, medical imaging, and total COVID-19 cases around the world.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Ali, F.A.M.; Karim, S.A.A.; Bin Saaban, A.; Hasan, M.K.; Ghaffar, A.; Nisar, K.S.; Baleanu, D. Construction of Cubic Timmer Triangular Patches and its Application in Scattered Data Interpolation. Mathematics
**2020**, 8, 159. [Google Scholar] [CrossRef] [Green Version] - Bracco, C.; Gianelli, C.; Sestini, A. Adaptive scattered data fitting by extension of local approximations to hierachical splines. Comput. Aided Geom. Des.
**2017**, 52–53, 90–105. [Google Scholar] [CrossRef] [Green Version] - Borne, S.L.; Wende, M. Domain decomposition methods in scattered data interpolation with conditionally positive definite radial basis functions. Comput. Math. Appl.
**2018**, 77, 1178–1196. [Google Scholar] [CrossRef] - Bozzini, M.; Lenarduzzi, L.; Rossini, M. Polyharmonic splines: An approximation method for noisy scattered data of extra-large size. Appl. Math. Comput.
**2010**, 216, 317–331. [Google Scholar] [CrossRef] - Brodlie, K.W.; Asim, M.R.; Unsworth, K. Constrained Visualization Using the Shepard Interpolation Family. Comput. Graph. Forum
**2005**, 24, 809–820. [Google Scholar] [CrossRef] - Cavoretto, R.; De Rossi, A.; Dell’Accio, F.; Di Tommaso, F. Fast computation of triangular Shepard interpolants. J. Comput. Appl. Math.
**2019**, 354, 457–470. [Google Scholar] [CrossRef] - Chan, E.; Ong, B. Range restricted scattered data interpolation using convex combination of cubic Bézier triangles. J. Comput. Appl. Math.
**2001**, 136, 135–147. [Google Scholar] [CrossRef] [Green Version] - Chang, L.; Said, H. A C
^{2}triangular patch for the interpolation of functional scattered data. Comput. Des.**1997**, 29, 407–412. [Google Scholar] [CrossRef] - Draman, N.N.C.; Karim, S.A.A.; Hashim, I. Scattered Data Interpolation Using Rational Quartic Triangular Patches With Three Parameters. IEEE Access
**2020**, 8, 44239–44262. [Google Scholar] [CrossRef] - Fasshauer, G.E. Meshfree Approximation Methods with Matlab; World Scientific Publishing Co Pte Ltd.: Singapore, 2007. [Google Scholar]
- Dell’Accio, F.; Di Tommaso, F. On the hexagonal Shepard method. Appl. Numer. Math.
**2020**, 150, 51–64. [Google Scholar] [CrossRef] - Dell’Accio, F.; Di Tommaso, F.; Nouisser, O.; Zerroudi, B. Increasing the approximation order of the triangular Shepard method. Appl. Numer. Math.
**2018**, 126, 78–91. [Google Scholar] [CrossRef] - Dell’Accio, F.; Di Tommaso, F.; Hormann, K. On the approximation order of triangular Shepard interpolation. IMA J. Numer. Anal.
**2015**, 36, 359–379. [Google Scholar] [CrossRef] - Crivellaro, A.; Perotto, S.; Zonca, S. Reconstruction of 3D scattered data via radial basis functions by efficient and robust techniques. Appl. Numer. Math.
**2017**, 113, 93–108. [Google Scholar] [CrossRef] - Chen, Z.; Cao, F. Scattered data approximation by neural networks operators. Neurocomputing
**2016**, 190, 237–242. [Google Scholar] [CrossRef] - Zhou, T.; Li, Z. Scattered Noisy Data Fitting Using Bivariate Splines. Procedia Eng.
**2011**, 15, 1942–1946. [Google Scholar] [CrossRef] [Green Version] - Zhou, T.; Li, Z. Scattered noisy Hermite data fitting using an extension of the weighted least squares method. Comput. Math. Appl.
**2013**, 65, 1967–1977. [Google Scholar] [CrossRef] - Qian, J.; Wang, F.; Zhu, C. Scattered data interpolation based upon bivariate recursive polynomials. J. Comput. Appl. Math.
**2018**, 329, 223–243. [Google Scholar] [CrossRef] - Liu, Z. Local multilevel scattered data interpolation. Eng. Anal. Bound. Elements
**2018**, 92, 101–107. [Google Scholar] [CrossRef] - Joldes, G.; Chowdhury, H.A.; Wittek, A.; Doyle, B.J.; Miller, K. Modified moving least squares with polynomial bases for scattered data approximation. Appl. Math. Comput.
**2015**, 266, 893–902. [Google Scholar] [CrossRef] [Green Version] - Lai, M.-J.; Meile, C. Scattered data interpolation with nonnegative preservation using bivariate splines and its application. Comput. Aided Geom. Des.
**2015**, 34, 37–49. [Google Scholar] [CrossRef] - Schumaker, L.L.; Speleers, H. Nonnegativity preserving macro-element interpolation of scattered data. Comput. Aided Geom. Des.
**2010**, 27, 245–261. [Google Scholar] [CrossRef] [Green Version] - Karim, S.A.A.; Saaban, A.; Hasan, M.K.; Sulaiman, J.; Hashim, I. Interpolation using cubic Bézier triangular patches. Int. J. Adv. Sci. Eng. Inf. Technol.
**2018**, 8, 1746–1752. [Google Scholar] [CrossRef] - Karim, S.A.A.; Saaban, A.; Skala, V.; Ghaffar, A.; Nisar, K.S.; Baleanu, D. Construction of new cubic Bézier-like triangular patches with application in scattered data interpolation. Available online: https://link.springer.com/content/pdf/10.1186/s13662-020-02598-w.pdf (accessed on 29 March 2020).
- Karim, S.A.B.A.; Saaban, A. Visualization Terrain Data Using Cubic Ball Triangular Patches. MATEC Web Conf.
**2018**, 225, 06023. [Google Scholar] [CrossRef] - Said, H.B.; Rahmat, R.W. A Cubic Ball Triangular Patch for Scattered Data Interpolation. J. Phys. Sci.
**1995**, 5, 89–101. [Google Scholar] - Feng, R.; Zhang, Y. Piecewise Bivariate Hermite Interpolations for Large Sets of Scattered Data. J. Appl. Math.
**2013**, 2013, 1–10. [Google Scholar] [CrossRef] - Sun, Q.; Bao, F.; Zhang, Y.; Duan, Q. A bivariate rational interpolation based on scattered data on parallel lines. J. Vis. Commun. Image Represent.
**2013**, 24, 75–80. [Google Scholar] [CrossRef] - Goodman, T.N.T.; Said, H.B. A C
^{1}- triangular interpolation suitable for scattered data interpolation. Commun. Appl. Numer. Methods**1991**, 7, 479–485. [Google Scholar] [CrossRef] - Foley, T.A.; Opitz, K. Hybrid cubic Bézier triangle patches. In Mathematical Methods in Computer Aided Geometric Design II; Lyche, T., Schumaker, L.L., Eds.; Academic Press: Cambridge, MA, USA, 1992; pp. 275–286. [Google Scholar]
- Goodman, T.; Said, H. Shape preserving properties of the generalised ball basis. Comput. Aided Geom. Des.
**1991**, 8, 115–121. [Google Scholar] [CrossRef] - Goodman, T.; Said, H. Properties of generalized Ball curves and surfaces. Comput. Aided Des.
**1991**, 23, 554–560. [Google Scholar] [CrossRef] - Hussain, M.Z.; Hussain, M. C
^{1}positivity preserving scattered data interpolation using rational Bernstein-Bézier triangular patch. J. Appl. Math. Comput.**2009**, 35, 281–293. [Google Scholar] [CrossRef] - Goodman, T.; Said, H.; Chang, L. Local derivative estimation for scattered data interpolation. Appl. Math. Comput.
**1995**, 68, 41–50. [Google Scholar] [CrossRef] - Saaban, A.; Piah, A.R.M.; Majid, A.A.; Chang, L.H.T. G
^{1}scattered data interpolation with minimized sum of squares of principle curvatures. In Proceedings of the International Conference on Computer Graphics, Imaging and Visualization (CGIV’05), Beijing, China, 26–29 July 2005; pp. 385–390. [Google Scholar] - Piah, A.R.M.; Saaban, A.; Majid, A.A. Range restricted positivity-preserving scattered data interpolation. Malays. J. Fundam. Appl. Sci.
**2014**, 2, 63–75. [Google Scholar] [CrossRef] [Green Version] - Hussain, M.; Majid, A.A.; Hussain, M.Z. Convexity-preserving Bernstein–Bézier quartic scheme. Egypt. Inform. J.
**2014**, 15, 89–95. [Google Scholar] [CrossRef] - Hussain, M.; Hussain, M.Z.; Buttar, M. C
^{1}Positive Bernstein-Bézier Rational Quartic Interpolation. Int. J. Math. Models Methods Appl. Sci.**2014**, 8, 9–21. [Google Scholar] - Farin, G. Curves and Surfaces for CAGD: A Practicle Guide, 5th ed.; Palmer, C., Ed.; Morgan Kaufmann: San Diego, CA, USA, 2001. [Google Scholar]
- Franke, R. Scattered Data Interpolation: Tests of Some Method. Math. Comput.
**1982**, 38, 181. [Google Scholar] [CrossRef] - Franke, R.; Nielson, G.M. Scattered data interpolation of large Sets of scattered data. Int. J. Numer. Methods Eng.
**1980**, 15, 1691–1704. [Google Scholar] [CrossRef] - Franke, R.; Nielson, G.M. Scattered Data Interpolation and Applications: A Tutorial and Survey; Scattered data interpolation and applications: A tutorial and survey. In Geometric Modelling: Methods and Applications; Hagen, H., Roller, D., Eds.; Springer Science and Business Media LLC: Berlin/Heidelberg, Germany, 1991; pp. 131–160. [Google Scholar]
- Lodha, S.; Franke, R. Scattered Data Techniques for Surfaces. In Proceedings of the Scientific Visualization Conference (dagstuhl ’97), Dagstuhl, Germany, 9–13 June 1997; pp. 189–230. [Google Scholar]
- Zhu, Y.; Han, X. A class of αβγ-Bernstein–Bézier basis functions over triangular domain. Appl. Math. Comput.
**2013**, 220, 446–454. [Google Scholar] [CrossRef] - Press statement by the Director-General of Health Malaysia, Ministry of Health Malaysia. Available online: https://kpkesihatan.com (accessed on 15 April 2020).
- Karim, S.A.A.; Saaban, A.; Skala, V. Range-Restricted Surface Interpolation Using Rational Bi-Cubic Spline Functions with 12 Parameters. IEEE Access
**2019**, 7, 104992–105007. [Google Scholar] [CrossRef]

**Figure 3.**One patch (Zhu and Han [44]).

**Figure 8.**Delaunay triangulation of data in Table 1.

**Figure 9.**Surface reconstruction using the proposed scheme. (

**a**) For ${F}_{1}\left(x,y\right)$; (

**b**) For ${F}_{2}\left(x,y\right).$

**Figure 10.**Surface reconstruction using quartic Bézier triangular. (

**a**) For ${F}_{1}\left(x,y\right);$ (

**b**) For ${F}_{2}\left(x,y\right).$

**Figure 13.**Triangulation domain using Delaunay triangulation: (

**a**) 36 node points; (

**b**) 63 node points; (

**c**) 26 node points; (

**d**) 100 node points.

**Figure 14.**C

^{1}quartic Zhu and Han interpolated surface (data in Table 6): (

**a**) without positivity preserved; (

**b**) with positivity preserved from Theorem 3.

**Figure 15.**C

^{1}quartic Zhu and Han interpolated surface (data Table 7): (

**a**) without positivity preserved; (

**b**) with positivity preserved from Theorem 3.

**Figure 16.**C

^{1}quartic Zhu and Han interpolated surface (data Table 8): (

**a**) without positivity preserved; (

**b**) with positivity preserved from Theorem 3.

**Figure 17.**C

^{1}quartic Zhu and Han interpolated surface (data Table 9): (

**a**) without positivity preserved; (

**b**) with positivity preserved from Theorem 3.

x | y | F_{1}(x,y) | F_{2}(x,y) | x | y | F_{1}(x,y) | F_{2}(x,y) |

0 | 0 | 0.7664 | 1.3333 | 0.80 | 0.85 | 0.0823 | 1.2431 |

0.50 | 0 | 0.4349 | 1.3833 | 0.85 | 0.65 | 0.1412 | 1.2043 |

1.00 | 0 | 0.1076 | 1.2833 | 1.00 | 0.50 | 0.1610 | 1.2199 |

0.15 | 0.15 | 1.1370 | 1.3382 | 1.00 | 1.00 | 0.0359 | 1.2712 |

0.70 | 0.15 | 0.4304 | 1.3020 | 0.50 | 1.00 | 0.1460 | 1.3346 |

0.50 | 0.20 | 0.5345 | 1.3128 | 0.10 | 0.85 | 0.2935 | 1.2363 |

0.25 | 0.30 | 1.0726 | 1.2423 | 0 | 1.00 | 0.2703 | 1.3029 |

0.40 | 0.30 | 0.7134 | 1.2421 | 0.25 | 0 | 0.8189 | 1.4069 |

0.75 | 0.40 | 0.5903 | 1.2139 | 0.75 | 0 | 0.2521 | 1.3150 |

0.85 | 0.25 | 0.5088 | 1.2607 | 0.25 | 1.00 | 0.2222 | 1.3496 |

0.55 | 0.45 | 0.3823 | 1.1613 | 0 | 0.25 | 0.8026 | 1.2683 |

0 | 0.50 | 0.4818 | 1.1747 | 0.75 | 1.00 | 0.0810 | 1.2913 |

0.20 | 0.45 | 0.6458 | 1.1412 | 0 | 0.75 | 0.3395 | 1.1987 |

0.45 | 0.55 | 0.2946 | 1.1037 | 1.00 | 0.25 | 0.2302 | 1.2573 |

0.60 | 0.65 | 0.1920 | 1.1552 | 1.00 | 0.75 | 0.0504 | 1.2295 |

0.25 | 0.70 | 0.2930 | 1.1240 | 0.19 | 0.19 | 1.2118 | 1.3229 |

0.40 | 0.80 | 0.0515 | 1.1887 | 0.32 | 0.75 | 0.2029 | 1.1477 |

0.65 | 0.75 | 0.1372 | 1.1961 | 0.79 | 0.46 | 0.4777 | 1.2041 |

Num. of Data Points | Function | Max Error | R^{2} | ||
---|---|---|---|---|---|

The Proposed Scheme | Quartic Bézier [35] | The Proposed Scheme | Quartic Bézier [35] | ||

100 | 1 | 3.436 × 10^{∧−2} | 3.598 × 10^{∧−2} | 0.99936 | 0.99934 |

2 | 4.500 × 10^{∧−2} | 7.61 × 10^{∧−2} | 0.99977 | 0.99967 | |

65 | 1 | 6.410 × 10^{∧−2} | 6.586 × 10^{∧−2} | 0.99720 | 0.99733 |

2 | 1.732 × 10^{∧−2} | 1.562 × 10^{∧−2} | 0.99796 | 0.99793 | |

36 | 1 | 9.740 × 10^{∧−2} | 9.973 × 10^{∧−2} | 0.99211 | 0.99256 |

2 | 2.675 × 10^{∧−2} | 2.762 × 10^{∧−2} | 0.99332 | 0.99208 |

Num. of Data Points | Function | CPU Time (in Seconds) | |
---|---|---|---|

The Proposed Scheme | Quartic Bézier [35] | ||

100 | 1 | 0.7097807844 | 5.6002481345 |

2 | 0.4234289196 | 3.5703151686 | |

65 | 1 | 0.2741610887 | 1.5474957467 |

2 | 0.2363209917 | 1.3271791002 | |

36 | 1 | 0.1298699059 | 0.5886074910 |

2 | 0.1163838547 | 0.4703613961 |

Num. of Data Points | Function | Max Err | ||||
---|---|---|---|---|---|---|

Dell’Accio et al. [12] | Dell’Accio and Di Tommaso [11] | Dell’Accio et al., [12] and Cavoretto et al. [6] | Dell’Accio et al. [12] | Dell’Accio et al. [13] and Cavoretto et al. [6] | ||

100 | 1 | 5.2990 × 10^{∧}^{−2} | 8.6648 × 10^{∧}^{−2} | 1.0970 × 10^{∧}^{−1} | 6.2438 × 10^{∧}^{−2} | 5.3936 × 10^{∧2} |

2 | 1.8617 × 10^{∧}^{−2} | 5.0590 × 10^{∧}^{−2} | 3.2842 × 10^{∧}^{−2} | 1.5449 × 10^{∧}^{−2} | 1.9619 × 10^{∧}^{−2} | |

65 | 1 | 1.0147 × 10^{∧}^{−1} | 1.1864 × 10^{∧}^{−1} | 1.1221 × 10^{∧}^{−1} | 7.6266 × 10^{∧}^{−2} | 7.1704 × 10^{∧}^{−2} |

2 | 6.4329 × 10^{∧}^{−2} | 3.7704 × 10^{∧}^{−2} | 3.5962 × 10^{∧}^{−2} | 2.7322 × 10^{∧}^{−2} | 2.8894 × 10^{∧}^{−2} | |

36 | 1 | 1.2822 × 10^{∧}^{−1} | 1.6219 × 10^{∧}^{−1} | 1.3564 × 10^{∧}^{−1} | 1.1371 × 10^{∧}^{−1} | 9.8914 × 10^{∧}^{−2} |

2 | 7.9686 × 10^{∧}^{−2} | 5.6713 × 10^{∧}^{−2} | 5.3611 × 10^{∧}^{−2} | 5.1806 × 10^{∧}^{−2} | 4.6253×10^{∧}^{−2} |

Num. of Data Points | Function | CPU Time (Second) | ||||
---|---|---|---|---|---|---|

Dell’Accio et al. [12] | Dell’Accio and Di Tommaso [11] | Dell’Accio et al., [12] and Cavoretto et al. [6] | Dell’Accio et al. [12] | Dell’Accio et al. [13] and Cavoretto et al. [6] | ||

100 | 1 | 0.490380 | 1.894685 | 0.480296 | 0.423177 | 0.401825 |

2 | 0.502923 | 1.881019 | 0.501286 | 0.428949 | 0.417658 | |

65 | 1 | 0.486381 | 1.882420 | 0.478175 | 0.424583 | 0.424086 |

2 | 0.484984 | 1.849458 | 0.459517 | 0.424870 | 0.397832 | |

36 | 1 | 0.437090 | 1.707668 | 0.445523 | 0.415162 | 0.445566 |

2 | 0.448242 | 1.686242 | 0.448589 | 0.417875 | 0.421869 |

x | y | F_{1}(x,y) | x | y | F_{1}(x,y) | x | y | F_{1}(x,y) |
---|---|---|---|---|---|---|---|---|

0 | 0 | 0 | 0.35 | 0 | 0 | 1.4 | 0.8 | 0 |

0.2 | 0.2 | 0 | 0.8 | 0 | 0 | 1.65 | 0.75 | 0 |

0.5 | 0.2 | 0 | 0.1 | 0.85 | 1 | 2 | 1 | 0 |

0.4 | 0.4 | 0 | 0 | 0.25 | 0.5 | 1.25 | 0 | 0 |

0.75 | 0.35 | 0 | 0.8 | 1 | 0.4 | 1.7 | 0 | 0 |

0 | 0.5 | 1 | 2 | 0 | 0 | 1.25 | 1 | 0 |

0.25 | 0.5 | 0.5 | 1.4 | 0.3 | 0.0272 | 1.7 | 1 | 0 |

0.25 | 0.75 | 1 | 1.75 | 0.45 | 0 | 2 | 0.35 | 0 |

0.55 | 0.75 | 0.4 | 1.2 | 0.45 | 0 | 2 | 0.7 | 0 |

0.7 | 0.6 | 0 | 1.45 | 0.5 | 0.9045 | 1.05 | 0.2 | 0 |

0.5 | 1 | 1 | 1.6 | 0.3 | 0.0272 | 1 | 0.5 | 0 |

0 | 1 | 1 | 1.25 | 0.7 | 0 | 0.95 | 0.8 | 0 |

x | y | F_{2}(x,y) | x | y | F_{2}(x,y) | x | y | F_{2}(x,y) |
---|---|---|---|---|---|---|---|---|

0 | 0 | 0.2586 | 0 | 0.50 | 0.5960 | 0.50 | 1.00 | 0.8762 |

0.50 | 0 | 0.6429 | 0.25 | 0.45 | 0.6264 | 0.10 | 0.85 | 0.7316 |

1.00 | 0 | 0.9174 | 0.45 | 0.55 | 0.7981 | 0 | 1.00 | 0.7547 |

0.25 | 0.20 | 0.0056 | 0.60 | 0.65 | 0.8336 | 0.25 | 0 | 0.2629 |

0.70 | 0.15 | 0.6012 | 0.25 | 0.70 | 0.7862 | 0.75 | 0 | 0.7739 |

0.50 | 0.20 | 0.6329 | 0.40 | 0.80 | 0.9941 | 0.25 | 1.00 | 0.8026 |

0.30 | 0.30 | 0.4199 | 0.65 | 0.75 | 0.8825 | 0 | 0.25 | 0.2792 |

0.45 | 0.35 | 0.6618 | 0.80 | 0.85 | 0.9427 | 0.75 | 1.00 | 0.9440 |

0.75 | 0.40 | 0.4361 | 0.85 | 0.65 | 0.8838 | 0 | 0.75 | 0.6865 |

0.85 | 0.25 | 0.5164 | 1.00 | 0.50 | 0.8640 | 1.00 | 0.25 | 0.7948 |

0.55 | 0.45 | 0.6724 | 1.00 | 1.00 | 0.9891 | 1.00 | 0.75 | 0.9746 |

x | y | F_{3}(x,y) | x | y | F_{3}(x,y) | x | y | F_{3}(x,y) |
---|---|---|---|---|---|---|---|---|

0.9375 | −0.4063 | 0.7997 | 0.0469 | −0.7656 | 0.3436 | −0.5156 | −0.1094 | 0.0708 |

−0.1719 | 1.0000 | 1.0009 | −0.7813 | −0.8906 | 1.0017 | 0.4844 | 0.1406 | 0.0554 |

−0.8906 | −0.0938 | 0.6292 | 0.0625 | 0.3750 | 0.0198 | −0.4531 | 0.1563 | 0.0427 |

−0.0625 | −0.6719 | 0.2038 | −0.7656 | −0.2969 | 0.3513 | 0.7031 | 0.3281 | 0.2560 |

−0.8750 | −0.6250 | 0.7388 | 0.0938 | 0.1250 | 0.0003 | −0.4219 | 0.4688 | 0.0800 |

0 | 0.7969 | 0.4033 | −0.6875 | 0.3750 | 0.2432 | 0.9063 | −0.5938 | 0.7990 |

−0.8438 | −0.5313 | 0.5866 | 0.1094 | 0.3281 | 0.0117 | −0.2344 | 0.1406 | 0.0034 |

0.0313 | 0.5313 | 0.0797 | −0.5625 | −0.6563 | 0.2856 | 0.9688 | 0.7188 | 1.1476 |

−0.8438 | 0.1563 | 0.5075 | 0.1563 | 0.4531 | 0.0427 |

x | y | F_{4} (x,y) | x | y | F_{4} (x,y) | x | y | F_{4} (x,y) |
---|---|---|---|---|---|---|---|---|

0.0096 | 0.3083 | 0.119425 | 0.3307 | 0.5159 | 1.155816 | 0.6677 | 0.6764 | 0.442125 |

0.0216 | 0.245 | 0.029055 | 0.3379 | 0.9426 | 0.268844 | 0.6814 | 0.8444 | 0.195332 |

0.0298 | 0.8614 | 0.00111 | 0.3439 | 0.48 | 1.248258 | 0.6888 | 0.3273 | 0.365476 |

0.0417 | 0.0978 | 0.000258 | 0.353 | 0.1783 | 0.345127 | 0.6941 | 0.1894 | 0.158965 |

0.047 | 0.3648 | 0.300748 | 0.3636 | 0.1147 | 0.395081 | 0.7062 | 0.0646 | 0.119387 |

0.0563 | 0.7156 | 0.073456 | 0.3766 | 0.8226 | 0.473071 | 0.7161 | 0.018 | 0.096822 |

0.0647 | 0.5311 | 0.714724 | 0.3822 | 0.2271 | 0.526808 | 0.7317 | 0.8905 | 0.068664 |

0.074 | 0.9756 | 0.000124 | 0.387 | 0.4074 | 1.274589 | 0.7371 | 0.4161 | 0.619367 |

0.0874 | 0.1781 | 0.004419 | 0.3973 | 0.8875 | 0.590813 | 0.7462 | 0.4689 | 0.797375 |

0.0935 | 0.5453 | 0.677296 | 0.4171 | 0.7632 | 0.749339 | 0.7567 | 0.2175 | 0.051462 |

0.1032 | 0.1604 | 0.00273 | 0.4256 | 0.9973 | 0.758236 | 0.77 | 0.5734 | 0.613977 |

0.111 | 0.7837 | 0.013932 | 0.4299 | 0.496 | 2.117705 | 0.7879 | 0.8853 | 0.016309 |

0.1181 | 0.9982 | 0.000684 | 0.4373 | 0.341 | 1.207499 | 0.7944 | 0.8018 | 0.021117 |

0.1252 | 0.6911 | 0.121795 | 0.4705 | 0.2498 | 1.021601 | 0.8164 | 0.6389 | 0.294451 |

0.1327 | 0.105 | 0.001483 | 0.4737 | 0.6409 | 1.512454 | 0.8193 | 0.8931 | 0.006444 |

0.144 | 0.8185 | 0.00648 | 0.4879 | 0.1059 | 0.99334 | 0.8368 | 0.1001 | 0.003695 |

0.1565 | 0.7086 | 0.088121 | 0.494 | 0.5412 | 2.374907 | 0.8501 | 0.279 | 0.067559 |

0.1651 | 0.4457 | 0.653239 | 0.5055 | 0.009 | 0.998497 | 0.8588 | 0.9083 | 0.001782 |

0.1786 | 0.1178 | 0.006221 | 0.5163 | 0.8784 | 0.987962 | 0.8646 | 0.3259 | 0.166279 |

0.1886 | 0.3189 | 0.154486 | 0.5219 | 0.5516 | 2.273778 | 0.8792 | 0.8319 | 0.003798 |

0.2017 | 0.9668 | 0.011703 | 0.5349 | 0.4039 | 1.858252 | 0.8838 | 0.0509 | 0.000664 |

0.21 | 0.7572 | 0.042784 | 0.5483 | 0.1654 | 0.895154 | 0.89 | 0.9708 | 0.000509 |

0.2147 | 0.2017 | 0.025997 | 0.557 | 0.2965 | 1.02504 | 0.897 | 0.5121 | 0.745189 |

0.2204 | 0.3232 | 0.180361 | 0.5639 | 0.366 | 1.370095 | 0.9045 | 0.286 | 0.076266 |

0.2344 | 0.4369 | 0.662063 | 0.5785 | 0.0367 | 0.734861 | 0.9084 | 0.9582 | 0.00026 |

0.241 | 0.8908 | 0.035317 | 0.5864 | 0.9502 | 0.688545 | 0.9204 | 0.6183 | 0.372734 |

0.2528 | 0.0647 | 0.047165 | 0.5929 | 0.2638 | 0.725544 | 0.9348 | 0.378 | 0.356442 |

0.2571 | 0.5693 | 0.673111 | 0.5988 | 0.9277 | 0.613938 | 0.9435 | 0.401 | 0.459525 |

0.2733 | 0.2947 | 0.174707 | 0.6118 | 0.5378 | 1.60735 | 0.949 | 0.9479 | 7.49E-05 |

0.2854 | 0.4332 | 0.760009 | 0.6252 | 0.7375 | 0.521789 | 0.957 | 0.7425 | 0.039667 |

0.2902 | 0.3347 | 0.323199 | 0.6331 | 0.4675 | 1.417194 | 0.9772 | 0.8883 | 0.00041 |

0.2965 | 0.7436 | 0.169566 | 0.6399 | 0.9186 | 0.375998 | 0.9983 | 0.5497 | 0.66287 |

0.302 | 0.1066 | 0.141203 | 0.6489 | 0.0417 | 0.330061 | |||

0.3126 | 0.8845 | 0.173287 | 0.6559 | 0.1291 | 0.29764 |

Size Data | Interpolation Points | Function | CPU Time (in Seconds) | |
---|---|---|---|---|

The Proposed Scheme | Quartic Bézier [35] | |||

36 | 1296 | F_{1} | 0.6587 | 1.5653 |

33 | 1296 | F_{2} | 1.0159 | 2.6322 |

25 | 377 | F_{3} | 0.0935 | 1.1535 |

100 | 1697 | F_{4} | 0.5168 | 18.5996 |

Number of Evaluation Points | Function | Max Error | R^{2} | ||
---|---|---|---|---|---|

The proposed scheme | Quartic Bézier [35] | The proposed scheme | Quartic Bézier [35] | ||

1296 | F_{1} | 0.2729197868113 | 0.282452475456 | 0.9800905 | 0.9760174674571 |

1296 | F_{2} | 0.6346772647732 | 0.633667822156 | 0.819140703 | 0.8244631293045 |

377 | F_{3} | 0.2716512377297 | 0.311611498819 | 0.93773671136 | 0.935885979498 |

1697 | F_{4} | 0.64080639264362 | 0.4808917225171 | 0.98734351099 | 0.988298717425 |

**Table 12.**Coronavirus disease 2019 (COVID-19) cases at Selangor and Klang Valley in Malaysia until 15 April 2020.

Label | District | Longitude | Latitude | COVID-19 Cases |
---|---|---|---|---|

A | Hulu Langat | 101.7620249 | 3.0727692 | 440 |

B | Petaling | 101.664208 | 3.086134 | 363 |

C | Klang | 101.449611 | 3.043125 | 171 |

D | Gombak | 101.714574 | 3.233044 | 142 |

E | Sepang | 101.709401 | 2.800862 | 68 |

F | Hulu Selangor | 101.641482 | 3.52361 | 49 |

G | Lembah Pantai | 101.672189 | 3.104444 | 577 |

H | Kuala Selangor | 101.34555 | 3.362102 | 35 |

I | Kuala Langat | 101.496182 | 2.836562 | 25 |

J | Sabak Bernam | 101.058059 | 3.687115 | 23 |

K | Kepong | 101.623581 | 3.2059 | 142 |

L | Titi Wangsa | 101.695278 | 3.173573 | 129 |

M | Cheras | 101.71649 | 3.107178 | 78 |

N | Putrajaya | 101.684046 | 2.918 | 54 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Abdul Karim, S.A.; Saaban, A.; Nguyen, V.T.
Scattered Data Interpolation Using Quartic Triangular Patch for Shape-Preserving Interpolation and Comparison with Mesh-Free Methods. *Symmetry* **2020**, *12*, 1071.
https://doi.org/10.3390/sym12071071

**AMA Style**

Abdul Karim SA, Saaban A, Nguyen VT.
Scattered Data Interpolation Using Quartic Triangular Patch for Shape-Preserving Interpolation and Comparison with Mesh-Free Methods. *Symmetry*. 2020; 12(7):1071.
https://doi.org/10.3390/sym12071071

**Chicago/Turabian Style**

Abdul Karim, Samsul Ariffin, Azizan Saaban, and Van Thien Nguyen.
2020. "Scattered Data Interpolation Using Quartic Triangular Patch for Shape-Preserving Interpolation and Comparison with Mesh-Free Methods" *Symmetry* 12, no. 7: 1071.
https://doi.org/10.3390/sym12071071