Scattered Data Interpolation Using Quartic Triangular Patch for Shape-Preserving Interpolation and Comparison with Mesh-Free Methods
Scattered data interpolation is important in sciences, engineering, and medical-based problems. Quartic Bézier triangular patches with 15 control points (ordinates) can also be used for scattered data interpolation. However, this method has a weakness; that is, in order to achieve <inline-formula...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-06-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/12/7/1071 |
| _version_ | 1797563747402776576 |
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| author | Samsul Ariffin Abdul Karim Azizan Saaban Van Thien Nguyen |
| author_facet | Samsul Ariffin Abdul Karim Azizan Saaban Van Thien Nguyen |
| author_sort | Samsul Ariffin Abdul Karim |
| collection | DOAJ |
| description | Scattered data interpolation is important in sciences, engineering, and medical-based problems. Quartic Bézier triangular patches with 15 control points (ordinates) can also be used for scattered data interpolation. However, this method has a weakness; that is, in order to achieve <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>C</mi> <mn>1</mn> </msup> </mrow> </semantics> </math> </inline-formula> continuity, the three inner points can only be determined using an optimization method. Thus, we cannot obtain the exact Bézier ordinates, and the quartic scheme is global and not local. Therefore, the quartic Bézier triangular has received less attention. In this work, we use Zhu and Han’s quartic spline with ten control points (ordinates). Since there are only ten control points (as for cubic Bézier triangular cases), all control points can be determined exactly, and the optimization problem can be avoided. This will improve the presentation of the surface, and the process to construct the scattered surface is local. We also apply the proposed scheme for the purpose of positivity-preserving scattered data interpolation. The sufficient conditions for the positivity of the quartic triangular patches are derived on seven ordinates. We obtain nonlinear equations that can be solved using the regula-falsi method. To produce the interpolated surface for scattered data, we employ four stages of an algorithm: (a) triangulate the scattered data using Delaunay triangulation; (b) assign the first derivative at the respective data; (c) form a triangular surface via convex combination from three local schemes with <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>C</mi> <mn>1</mn> </msup> </mrow> </semantics> </math> </inline-formula> continuity along all adjacent triangles; and (d) construct the scattered data surface using the proposed quartic spline. Numerical results, including some comparisons with some existing mesh-free schemes, are presented in detail. Overall, the proposed quartic triangular spline scheme gives good results in terms of a higher coefficient of determination (R<sup>2</sup>) 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. |
| first_indexed | 2024-03-10T18:47:46Z |
| format | Article |
| id | doaj.art-bcf2a2c13fce46fc8706b54133f9a54d |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T18:47:46Z |
| publishDate | 2020-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-bcf2a2c13fce46fc8706b54133f9a54d2023-11-20T05:24:49ZengMDPI AGSymmetry2073-89942020-06-01127107110.3390/sym12071071Scattered Data Interpolation Using Quartic Triangular Patch for Shape-Preserving Interpolation and Comparison with Mesh-Free MethodsSamsul Ariffin Abdul Karim0Azizan Saaban1Van Thien Nguyen2Fundamental and Applied Sciences Department and Centre for Smart Grid Energy Research (CSMER), Institute of Autonomous System, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, Seri Iskandar 32610, Perak Darul Ridzuan, MalaysiaSchool of Quantitative Sciences, UUMCAS, Universiti Utara Malaysia, Kedah 06010, MalaysiaFPT University, Education Zone, Hoa Lac High Tech Park, Km29 Thang Long Highway, Thach That Ward, Hanoi 10000, VietnamScattered data interpolation is important in sciences, engineering, and medical-based problems. Quartic Bézier triangular patches with 15 control points (ordinates) can also be used for scattered data interpolation. However, this method has a weakness; that is, in order to achieve <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>C</mi> <mn>1</mn> </msup> </mrow> </semantics> </math> </inline-formula> continuity, the three inner points can only be determined using an optimization method. Thus, we cannot obtain the exact Bézier ordinates, and the quartic scheme is global and not local. Therefore, the quartic Bézier triangular has received less attention. In this work, we use Zhu and Han’s quartic spline with ten control points (ordinates). Since there are only ten control points (as for cubic Bézier triangular cases), all control points can be determined exactly, and the optimization problem can be avoided. This will improve the presentation of the surface, and the process to construct the scattered surface is local. We also apply the proposed scheme for the purpose of positivity-preserving scattered data interpolation. The sufficient conditions for the positivity of the quartic triangular patches are derived on seven ordinates. We obtain nonlinear equations that can be solved using the regula-falsi method. To produce the interpolated surface for scattered data, we employ four stages of an algorithm: (a) triangulate the scattered data using Delaunay triangulation; (b) assign the first derivative at the respective data; (c) form a triangular surface via convex combination from three local schemes with <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>C</mi> <mn>1</mn> </msup> </mrow> </semantics> </math> </inline-formula> continuity along all adjacent triangles; and (d) construct the scattered data surface using the proposed quartic spline. Numerical results, including some comparisons with some existing mesh-free schemes, are presented in detail. Overall, the proposed quartic triangular spline scheme gives good results in terms of a higher coefficient of determination (R<sup>2</sup>) 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.https://www.mdpi.com/2073-8994/12/7/1071quartic splinetriangulationscattered datacontinuitysurface reconstructionpositivity-preserving |
| spellingShingle | Samsul Ariffin Abdul Karim Azizan Saaban Van Thien Nguyen Scattered Data Interpolation Using Quartic Triangular Patch for Shape-Preserving Interpolation and Comparison with Mesh-Free Methods Symmetry quartic spline triangulation scattered data continuity surface reconstruction positivity-preserving |
| title | Scattered Data Interpolation Using Quartic Triangular Patch for Shape-Preserving Interpolation and Comparison with Mesh-Free Methods |
| title_full | Scattered Data Interpolation Using Quartic Triangular Patch for Shape-Preserving Interpolation and Comparison with Mesh-Free Methods |
| title_fullStr | Scattered Data Interpolation Using Quartic Triangular Patch for Shape-Preserving Interpolation and Comparison with Mesh-Free Methods |
| title_full_unstemmed | Scattered Data Interpolation Using Quartic Triangular Patch for Shape-Preserving Interpolation and Comparison with Mesh-Free Methods |
| title_short | Scattered Data Interpolation Using Quartic Triangular Patch for Shape-Preserving Interpolation and Comparison with Mesh-Free Methods |
| title_sort | scattered data interpolation using quartic triangular patch for shape preserving interpolation and comparison with mesh free methods |
| topic | quartic spline triangulation scattered data continuity surface reconstruction positivity-preserving |
| url | https://www.mdpi.com/2073-8994/12/7/1071 |
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