The q-Analogue of the Haar Wavelet Transform: A Novel Approach to Image Denoising

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 (512×512 pixels) corrupted with white Gaussian noise at various levels (σ2=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.