Implementation of \(q\)-Wavelet Transform on Image Compressions
Dzaky Muhammad, Kistosil Fahim (*), Mahmud Yunus, Sunarsini

Department of Mathematics, Institut Teknologi Sepuluh Nopember, Kampus ITS Sukolilo,
Surabaya 60111, Indonesia

Abstract

Quantum calculus, or \(q\)-calculus, is a new branch in mathematics field. The \(q\)-calculus is more general than the ordinary calculus, since it uses a \(q\) number to define a mathematical object on it. One kind of property that \(q\)-calculus provided is if \(q\) number is very close to one, then the mathematical objects behave like its ordinary version. Such phenomenom is called as \(q\)-analogue. Now days \(q\)-calculus still be considered as an active research. Some of researchers have work on theoretically which is they concern about deriving some properties or establishing new \(q\)-analogues. Strictly speaking, now we focus on how to implement \(q\)-analogue of wavelet transform in a real world case. Even though \(q\)-wavelet transform has been discussed by Fitouhi and Bettaibi in 2006, actually we find out that it cannot be implemented flatly. Because of they only concern on \(q\)-analogue of the continuous form along with its properties and we consider that is still hardly to actualized them computationally. Fortunately, in 2024 Fahim et al already have made \(q\)-analogue of discrete wavelet transform, which they specifically adopt derivatization of the Haar wavelet. In this paper we engage it on to image compression scheme and compare the result against using the classic version. We suppose that it gives a good result as well like the classic version gives. Also the result may get more vary, since existence of interchangeably \(q\) number.

Keywords: \(q\)-Calculus- \(q\)-Wavelet- Image compression

Topic: Functional Analysis