An advanced numerical technique for subdivision depth of non-stationary quaternary refinement scheme for curves and surfaces

Volume 36, Issue 4, pp 408--431 https://dx.doi.org/10.22436/jmcs.036.04.05

Publication Date: August 24, 2024 Submission Date: March 24, 2024 Revision Date: June 01, 2024 Accteptance Date: July 17, 2024

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Authors

A. Mahmood - Department of Mathematics, ‎The Islamia University of Bahawalpur, ‎Pakistan. G. Mustafa - Department of Mathematics, ‎The Islamia University of Bahawalpur, ‎Pakistan. F. Khan - Department of Mathematics, ‎The Islamia University of Bahawalpur, ‎Pakistan.

Abstract

‎Refinement schemes are fundamental in computer graphics for generating smooth curves and surfaces‎. ‎Quaternary non-stationary subdivision schemes‎, ‎in particular‎, ‎have gained prominence due to their ability to handle complex geometric structures‎. ‎However‎, ‎determining the subdivision depth for these schemes remains challenging and often requires extensive computational resources‎. ‎Our paper presents a complete methodology with a step-by-step explanation to explore the depth of these schemes‎. ‎Since our method relies on convolution techniques‎, ‎we explain these both theoretically and mathematically‎. ‎Additionally‎, ‎several algorithms have been designed to aid in understanding and implementing the method for finding error bounds and subdivision depth in quaternary non-stationary subdivision schemes‎. ‎These are numerical methods for efficiently computing the error bounds and subdivision depth‎. ‎The numerical applications of these methods are presented‎. ‎The proposed method significantly reduces the computational cost associated with determining subdivision depth‎. ‎These algorithms work when existing methods fail to compute bounds and depths‎.

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Keywords

• Convolution
• error bound
• subdivision depth
• quaternary non-stationary refinement schemes
• algorithm

MSC

•  65D17
•  65D05
•  65D10
•  65Y04

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