C 1 positivity preserving scattered data interpolation using rational Bernstein-Bézier triangular patch

2009 ◽  
Vol 35 (1-2) ◽  
pp. 281-293 ◽  
Author(s):  
Malik Zawwar Hussain ◽  
Maria Hussain
Keyword(s):  
Scattered Data ◽  
Data Interpolation ◽  
Scattered Data Interpolation ◽  
Positivity Preserving ◽  
Triangular Patch

scholarly journals Range restricted positivity-preserving scattered data interpolation

2014 ◽  
Vol 2 (1-2) ◽  
Author(s):  
Abd. Rahni Mt. Piah ◽  
Azizan Saaban ◽  
Ahmad Abd. Majid
Keyword(s):  
Convex Combination ◽  
Sufficient Conditions ◽  
Original Data ◽  
Scattered Data ◽  
Piecewise Polynomial ◽  
Data Interpolation ◽  
Scattered Data Interpolation ◽  
Positivity Preserving ◽  
Triangular Patch ◽  
C1 Continuity

The construction of a range restricted bivariate C1 ( or G1 ) interpolant to scattered data is considered in which the interpolant is positive everywhere if the original data are positive. This study is motivated by earlier work in which sufficient conditions are derived on Bézier points in order to ensure that surfaces comprising cubic Bézier triangular patches are always positive and satisfy C1 continuity conditions. In the current work, simpler and more relaxed conditions are derived on the Bézier points. The gradients at the data sites are then calculated (and modified if necessary) to ensure that these conditions are satisfied. Each triangular patch of the interpolating surface is formed as a convex combination of three quartic Bézier triangular patches. Its construction is local and easily extended to include as upper and lower constraints to the interpolant surfaces of the form z = P(x,y) where P is a polynomial of degree less or equal to 4. Moreover, C1 ( or G1 ) piecewise polynomial surfaces consisting of polynomial pieces of the form z = P(x,y) on the triangulation of the data sites are also admissible constraints. A number of examples are presented.

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

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1071
Author(s):  
Samsul Ariffin Abdul Abdul Karim ◽  
Azizan Saaban ◽  
Van Thien Nguyen
Keyword(s):  
Convex Combination ◽  
Sufficient Conditions ◽  
Maximum Error ◽  
Scattered Data ◽  
Coefficient Of Determination ◽  
Control Points ◽  
Data Interpolation ◽  
Scattered Data Interpolation ◽  
Positivity Preserving ◽  
Mesh Free

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.

scholarly journals Scattered Data Interpolation Using Cubic Trigonometric B閦ier Triangular Patch

2021 ◽  
Vol 69 (1) ◽  
pp. 221-236
Author(s):  
Ishak Hashim ◽  
Nur Nabilah Che Draman ◽  
Samsul Ariffin Abdul Karim ◽  
Wee Ping Yeo ◽  
Dumitru Baleanu
Keyword(s):  
Scattered Data ◽  
Data Interpolation ◽  
Scattered Data Interpolation ◽  
Triangular Patch

Positivity-Preserving Scattered Data Interpolation

2005 ◽  
pp. 336-349 ◽  
Author(s):  
Abd. Rahni Mt. Piah ◽  
Tim N. T. Goodman ◽  
Keith Unsworth
Keyword(s):  
Scattered Data ◽  
Data Interpolation ◽  
Scattered Data Interpolation ◽  
Positivity Preserving

Positivity-preserving scattered data interpolation scheme using the side-vertex method

2012 ◽  
Vol 218 (15) ◽  
pp. 7898-7910 ◽  
Author(s):  
Muhammad Sarfraz ◽  
Malik Zawwar Hussain ◽  
Muhammad Arfan Ali
Keyword(s):  
Scattered Data ◽  
Data Interpolation ◽  
Interpolation Scheme ◽  
Scattered Data Interpolation ◽  
Positivity Preserving

scholarly journals Error estimates for scattered data interpolation on spheres

1999 ◽  
Vol 68 (226) ◽  
pp. 733-748 ◽  
Author(s):  
Kurt Jetter ◽  
Joachim Stöckler ◽  
Joseph D. Ward
Keyword(s):  
Error Estimates ◽  
Scattered Data ◽  
Data Interpolation ◽  
Scattered Data Interpolation
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