## 3 Methodology
The value was chosen to demonstrate the concept of thresholding and its impact on image denoising. Factors such as noise characteristics and specific image content should be empirically determined in an experiment. 3. Perform soft thresholding on the CNN output for each sub-band to reduce noise and preserve critical features. 4. Denoising uses soft thresholding on the CNN output for each detail sub-band. The soft thresholding formula for denoising a sub-band is given by:
5. C_denoised \((i,j)\) = sign \((C(i,j))*\max(\mid C(i,j)\mid-\lambda,0)\)
6. Where C_denoised \((i,j)\) are the denoised DWT coefficients, \(C(i,j)\) are the original DWT coefficients, and \(\lambda\) is the wavelet threshold. The relationship between the denoising threshold and the noise level (\(\sigma\)) for each sub-band in the denoising process can be summarized as follows:1. Denoising threshold: the denoising threshold (\(\lambda\)) is a parameter that determines the level at which noisy coefficients in each sub-band will be attenuated or suppressed during the denoising process. 2. Noise level (\(\sigma\)): The noise level (\(\sigma\)) represents the standard deviation of the noise present in the image. It characterizes the amount of noise contamination in the image, such as Gaussian blur noise. 3. Relationship: The denoising threshold (\(\lambda\)) is typically set based on each sub-band's estimated noise level (\(\sigma\)). The choice of the denoising threshold is critical because it determines which coefficients are considered noise and should be reduced or eliminated. 4. If \(\lambda\) is set too high, it may remove essential image details, leading to over-smoothing and loss of image information. 5. If \(\lambda\) is too low, it may not effectively suppress the noise, resulting in noisy artifacts in the denoised image. Therefore, to strike a balance, the denoising threshold is often determined empirically or based on the statistical properties of the noise in each sub-band. It should be chosen to reduce noise while preserving essential image features and fine details. The relationship between the denoising threshold (\(\lambda\)) and the noise level (\(\sigma\)) is that \(\lambda\) is a parameter adjusted based on the estimated noise level in each sub-band to achieve effective noise reduction without excessive loss of image quality. The specific threshold value may vary depending on the noise characteristics and the desired level of denoising. Step 6 (Haar wavelet transform)  
A wavelet-based Haar transform is used for further noise reduction, where the additive noise is decomposed into one low-frequency sub-band image (LL2) and six high-frequency sub-band images (LH1, HL1, HH1 LH2, HL2, and HH2) while using a two-level Haar transform. In the wavelet domain, horizontal (HL1, HL2) and vertical (LH1, LH2) sub-band images have the same energy. Horizontal and vertical sub-band images were applied to soft thresholds to remove additive Gaussian blur noise. Removing noise and lower-frequency components while retaining high-frequency information related to the edge helps enhance edge details and reduce blurring. The soft thresholding in additive Gaussian blur noise is applied to suppress noise in an image by shrinking a small-magnitude coefficient, leaving more significant magnitude coefficients unchanged. A threshold value is applied and sets all coefficients below this threshold to zero, effectively reducing the noise level while preserving important signal information. Step 7 (inverse wavelet transform):  
1. Combine the denoised detail sub-bands with the original approximation sub-band. 2. Perform the 2D inverse discrete wavelet transform (IDWT) to reconstruct the final denoised CT image. 3. The 2D IDWT combines the denoised detail sub-bands with the original approximation sub-band to reconstruct the final denoised CT image. Further, the DnCNN is combined with AGF and wavelet Haar transform for post-processing.