## 3 Methodology  
CT images often suffer from Gaussian blur noise and streak artifacts that reduce image quality, affect image analysis, obfuscate image information, and compromise diagnostic confidence. With that assumption, a wavelet-AGF-based image deblurring and restoration ensemble approach is proposed. Here, a denoising DnCNN that leverages deep learning architectures and image processing techniques that reduce the blurring effect and edge preservation is tailored for this study [54]. This framework is trained to denoise images corrupted with additive Gaussian blur noise (AGBN). Unlike the conventional denoising models, the combination of the anisotropic Gaussian filter (AGF) and Haar wavelet transform allows different standard deviations along different directions, making it capable of preserving edges while simultaneously reducing blurring. The Haar wavelet transform is preferred over other transform methods because it effectively preserves edges and fine details in images. This is particularly advantageous when dealing with CT images, which often contain essential anatomical and structural edges that must be preserved during denoising. Combining Haar wavelet, anisotropic Gaussian filters, and DnCNN leverages the strength of each component: Haar wavelet for preserving structural details, anisotropic Gaussian filters for noise reduction while maintaining edges, and DnCNN for capturing complex noise patterns. Here, both spatial and frequency domain information are addressed. The additive Gaussian blur noise is evenly distributed through the imagery plane with various density values following the normal distribution \"bell-shaped curve\". which has a shape reminiscent of a bell. Mathematically, the addition of Gaussian blur noise (GBN) can be shown as:  
\[m(x,y)=i(x,y)+n(x,y) \tag{1}\]  
where \(i(x,y)\) is the original signal, \(n(x,y)\) is the added noise, and \(m(x,y)\) is the final image, with \((x,y)\) determining the pixel location in the viewpoint plane. The bell-shaped curve follows a probability density function (PDF) representing an image's statistical distribution of pixel intensities. It is represented as:  
\[F(g)=\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{(g-y)^{2}}{2\sigma^{2}}} \tag{2}\]  
where \(g\) = gray level, \(F(g)\) represents the probability density function, \(\mu\) = mean, and \(\sigma\) = standard deviation of the noise. The noisy CT image \"image1.jpg\" is loaded, and the noise level (\(\sigma\)) is extracted from the image, where \(\sigma\) = 0.15. The anisotropic Gaussian filter is used to suppress Gaussian blur noise from the input image to enhance the quality of the CT scan medical images. The mathematical formula for the anisotropic Gaussian filter is given by:  
\[G(x,y)=\exp\left(-\frac{x^{2}+y^{2}}{2*\sigma^{2}}\right) \tag{3}\]  
where \(G(x,y)\) is the Gaussian filter kernel, \((x,y)\) are the spatial coordinates, and \(\sigma\) is the noise standard deviation from the noisy image. Here, the AGF is applied as a preprocessing step to reduce AGBN in the image by adaptively adjusting the filter parameters based on local image features. Noise standard deviation: \(\sigma\) = 0.15 (extracted from the noisy image)  
Anisotropic Gaussian filter kernel: \(G(x,y)=\exp(-2\sigma^{2}\times 2+y^{2})\)  
\[G(xy)=\exp\left(-\frac{x^{2}+y^{2}}{2\sigma^{2}}\right) \tag{4}\]  
These parameters describe the noise standard deviation (\(2\sigma\)) and the mathematical formula for the anisotropic Gaussian filter kernel (\(G(x,y)\)). The filter kernel suppresses Gaussian blur noise in the input image to enhance the quality of the CT scan medical images. After noise reduction with the AGF, the Haar transform is used as a preprocessing step for further noise reduction by performing a 2D discrete wavelet transform (DWT) on the denoised image using the Haar wavelet transform. The 2D DWT Haar wavelet transform decomposes the image into approximation (LL) and detail (LH, HL, HH) sub-bands.