Constructing new control points for Bézier interpolating polynomials using new geometrical approach

Interpolation is a mathematical technique employed for estimating the value of missing data between data points. This technique assures that the resulting polynomial passes through all data points. One of the most useful interpolating polynomials is the parametric interpolating polynomial. Bézier in...

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Main Author:	Fadhel, Mustafa Abbas Fadhel
Format:	Thesis
Language:	eng eng
Published:	2022
Subjects:	QA Mathematics
Online Access:	https://etd.uum.edu.my/10310/1/s902524_01.pdf https://etd.uum.edu.my/10310/2/s902524_02.pdf
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id	my-uum-etd.10310
record_format	uketd_dc
spelling	my-uum-etd.103102023-02-12T07:54:08Z Constructing new control points for Bézier interpolating polynomials using new geometrical approach 2022 Fadhel, Mustafa Abbas Fadhel Omar, Zurni Awang Had Salleh Graduate School of Arts & Sciences Awang Had Salleh Graduate School of Arts & Sciences QA Mathematics Interpolation is a mathematical technique employed for estimating the value of missing data between data points. This technique assures that the resulting polynomial passes through all data points. One of the most useful interpolating polynomials is the parametric interpolating polynomial. Bézier interpolating curves and surfaces are parametric interpolating polynomials for two-dimensional (2D) and three-dimensional (3D) datasets, respectively, that produce smooth, flexible, and accurate functions. According to the previous studies, the most crucial component in deriving Bézier interpolating polynomials is the construction of control points. However, most of the existing strategies constructed control points that produce partial smooth functions. As a result, the approximate values of the missing data are not accurate. In this study, nine new strategies of geometrical approach for constructing new 2D and 3D Bézier control points are proposed. The obtained control points from each new strategies are substituted in the relevant Bézier curve and surface equations to derive Bézier piecewise and non-piecewise interpolating polynomials which leads to the development of nine new methods. The proposed methods are proven to preserve the stability and smoothness of the generated Bézier interpolating curves and surfaces. In addition, the numerical results show that most of the resulting polynomials are able to approximate the missing values more accurately compared to those derived by the existing methods. The Bézier interpolating surfaces derived by the proposed method with highest accuracy for 3D datasets are then applied to upscale grey and colour images by the factors of two and three. Not only does the proposed method produces higher quality upscaled images, the numerical results also show that it outperforms the existing methods in terms of accuracy. Therefore, this study has successfully proposed new strategies for constructing new 2D and 3D control points for deriving Bézier interpolating polynomials that are capable of approximating the missing values accurately. In terms of application, the derived Bézier interpolating surfaces have a great potential to be employed in image upscaling. 2022 Thesis https://etd.uum.edu.my/10310/ https://etd.uum.edu.my/10310/1/s902524_01.pdf text eng 2025-06-21 staffonly https://etd.uum.edu.my/10310/2/s902524_02.pdf text eng public other doctoral Universiti Utara Malaysia
institution	Universiti Utara Malaysia
collection	UUM ETD
language	eng eng
advisor	Omar, Zurni
topic	QA Mathematics
spellingShingle	QA Mathematics Fadhel, Mustafa Abbas Fadhel Constructing new control points for Bézier interpolating polynomials using new geometrical approach
description	Interpolation is a mathematical technique employed for estimating the value of missing data between data points. This technique assures that the resulting polynomial passes through all data points. One of the most useful interpolating polynomials is the parametric interpolating polynomial. Bézier interpolating curves and surfaces are parametric interpolating polynomials for two-dimensional (2D) and three-dimensional (3D) datasets, respectively, that produce smooth, flexible, and accurate functions. According to the previous studies, the most crucial component in deriving Bézier interpolating polynomials is the construction of control points. However, most of the existing strategies constructed control points that produce partial smooth functions. As a result, the approximate values of the missing data are not accurate. In this study, nine new strategies of geometrical approach for constructing new 2D and 3D Bézier control points are proposed. The obtained control points from each new strategies are substituted in the relevant Bézier curve and surface equations to derive Bézier piecewise and non-piecewise interpolating polynomials which leads to the development of nine new methods. The proposed methods are proven to preserve the stability and smoothness of the generated Bézier interpolating curves and surfaces. In addition, the numerical results show that most of the resulting polynomials are able to approximate the missing values more accurately compared to those derived by the existing methods. The Bézier interpolating surfaces derived by the proposed method with highest accuracy for 3D datasets are then applied to upscale grey and colour images by the factors of two and three. Not only does the proposed method produces higher quality upscaled images, the numerical results also show that it outperforms the existing methods in terms of accuracy. Therefore, this study has successfully proposed new strategies for constructing new 2D and 3D control points for deriving Bézier interpolating polynomials that are capable of approximating the missing values accurately. In terms of application, the derived Bézier interpolating surfaces have a great potential to be employed in image upscaling.
format	Thesis
qualification_name	other
qualification_level	Doctorate
author	Fadhel, Mustafa Abbas Fadhel
author_facet	Fadhel, Mustafa Abbas Fadhel
author_sort	Fadhel, Mustafa Abbas Fadhel
title	Constructing new control points for Bézier interpolating polynomials using new geometrical approach
title_short	Constructing new control points for Bézier interpolating polynomials using new geometrical approach
title_full	Constructing new control points for Bézier interpolating polynomials using new geometrical approach
title_fullStr	Constructing new control points for Bézier interpolating polynomials using new geometrical approach
title_full_unstemmed	Constructing new control points for Bézier interpolating polynomials using new geometrical approach
title_sort	constructing new control points for bézier interpolating polynomials using new geometrical approach
granting_institution	Universiti Utara Malaysia
granting_department	Awang Had Salleh Graduate School of Arts & Sciences
publishDate	2022
url	https://etd.uum.edu.my/10310/1/s902524_01.pdf https://etd.uum.edu.my/10310/2/s902524_02.pdf
_version_	1776103787853774848

Constructing new control points for Bézier interpolating polynomials using new geometrical approach

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