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

Samsul Ariffin Abdul Abdul Karim,Azizan Saaban,Van Thien Nguyen

doi:10.3390/sym12071071

Samsul Ariffin Abdul Abdul Karim, Azizan Saaban + Show 1 more

Open Access

https://doi.org/10.3390/sym12071071

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Abstract

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 C 1 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 C 1 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 (R2) 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.

Highlights

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]
When the quartic Bézier triangular is used for scattered data interpolation, an optimization method is required to produce the interpolated surface, as discussed in Saaban et al [35], Piah et al [36] and Hussain et al [37,38]
After we derive the sufficient condition for the positivity of quartic triangular patch, the final C1 scattered data scheme for positivity preservation can be written as follows: X

Summary

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]. Symmetry 2020, 12, 1071 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).

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