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.
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.
A. Mahmood, G. Mustafa, F. Khan, An advanced numerical technique for subdivision depth of non-stationary quaternary refinement scheme for curves and surfaces, Journal of Mathematics and Computer Science, 36 (2025), no. 4, 408--431
Mahmood A., Mustafa G., Khan F., An advanced numerical technique for subdivision depth of non-stationary quaternary refinement scheme for curves and surfaces. J Math Comput SCI-JM. (2025); 36(4):408--431
Mahmood, A., Mustafa, G., Khan, F.. "An advanced numerical technique for subdivision depth of non-stationary quaternary refinement scheme for curves and surfaces." Journal of Mathematics and Computer Science, 36, no. 4 (2025): 408--431
