In this paper, we present the \(q\) -analogue of the Haar wavelet transform, formulated by extending the classical Haar wavelet construction with two parameters, \(m\) and \(q\) . This approach establishes a novel framework for signal analysis, termed \(q\) -multiresolution analysis, which is based on a newly designed \(q\) -Haar scaling function. Both the \(q\) -Haar scaling and wavelet functions facilitate efficient signal decomposition and reconstruction. To demonstrate its practical utility, we implement the proposed \(q\) -Haar wavelet transform for image denoising. Experiments are conducted on grayscale images ( \(512 \times 512\) pixels) corrupted with white Gaussian noise at various levels ( \(\sigma ^2=10,20,30\) ). The performance of the \(q\) -Haar wavelet transform is evaluated in terms of peak signal-to-noise ratio (PSNR) and visual perception. For comparative analysis, the classical Haar wavelet is also applied to the same denoising tasks to highlight the distinctive properties and advantages of its \(q\) -analogue. Moreover, in this experiment, we use different parameter values: \( q = 0.2, 0.5, 0.7, 0.999 \) and \( m = 1, 3, 5, 7, 9 \) . Various combinations of these values are tested, and the resulting PSNR scores are compared. Results show that, there exist parameters m and q such that the proposed \( q \) -Haar wavelet transform provides better results than the classical Haar wavelet transform.
