## 3 Methodology
The DWT coefficients are computed using the Haar wavelet, and the formula for DWT is given by:  
\[C(i,j)=\frac{1}{2}*(A(i,2j)+A(i,2j+1)),\text{ for the LL sub-band}\]  
\[C(i,j)=\frac{1}{2}*(A(i,2j+1)-A(i,2j)),\text{ for the LH sub-band}\]  
\[C(i,j)=\frac{1}{2}*(A(2i,j)-A(2i+1,j)),\text{ for the HL sub-band}\]  
\[C(i,j)=\frac{1}{2}*(A(2i,2j)+A(2i+1,2j)-A(2i,2j+1)-A(2i+1,2j+1)),\text{ for the HH sub-band} \tag{5}\]  
where \(A(i,j)\) are the pixels of the denoised image and \(C(i,j)\) are the DWT coefficients. In inverse discrete wavelet transform (IDWT), the 2D IDWT is applied to combine the denoised detail sub-bands with the original approximation sub-band to reconstruct the final denoised CT scan images. Setting coefficients below a specific threshold value to zero helps remove noise by identifying regions affected by blurring while retaining image details. Image deblurring was applied on Haar transform coefficients for restoring lost high-frequency details, effectively reducing the blurring effect. The architecture of the proposed approach is shown in Figure 1.  
Step 1 (input original image): read authentic CT scan images. Step 2 (perform initial noise detection): Use the anisotropic Gaussian filter (AGF) to gauge the level of Gaussian noise in the initial test when checking for the type of noise in the images. It smoothens images while preserving the edges and details, effectively reducing noise levels. Figure 1: The architecture of the proposed ensemble denoising approach. Step 3 (add Gaussian blur noise): read noisy corrupted CT scan images. Step 4 (perform DnCNN): the general CNN process is given below:  
1. Design a denoising CNN with skip connections to preserve low-level image details during denoising. 2. Implement batch normalization and ReLU activation after each convolutional layer to improve training stability. 3. Use residual blocks to capture and learn essential image features. 4. Implement skip connections to pass relevant information across different layers. The CNN architecture uses convolutional layers with batch normalization and ReLU activation functions. The skip connections are implemented using residual blocks. The output of each residual block is obtained as follows:  
\[F(g)=\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{(g-y)^{2}}{2\sigma^{2}}} \tag{2}\]0  
where \(R(x)\) is the residual block output, \(F(x)\) is the output of the convolutional layers, and x is the input to the residual block. Gradient-based algorithms, such as stochastic gradient descent (SGD), optimize the DnCNN model in finding the best values for the filters used to train the weights. The following loss function is used to minimize the error:  
\[F(g)=\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{(g-y)^{2}}{2\sigma^{2}}} \tag{2}\]1  
The DnCNN architecture above has three layers, each corresponding to a different function. Layer 1: Convolutional layers with ReLU, 64 filters each of dimensions 3 \(\times\) 3 \(\times\) (No. of channels), weaving their intricate patterns to produce 64 feature maps, each imbued with the essence of rectified linear units. The gray image summons one channel, while the color image demands three (RGB) channels. Layer 2: Convolutional layers, with ReLU batch normalization (BN), added between each convolutional and ReLU layer. The 64 additional filters intertwine, taking the form of 3 \(\times\) 3 \(\times\) 64 dimensions. Layer 3: The final convolutional layer stands as the key to unlocking the realm of image reconstruction. The 64 elusive filters, each with a size of 3 \(\times\) 3 \(\times\) 64, collaborate to peer into the depths and bring forth reimagined images. Step 5 (perform denoising):  
1. Apply the designed CNN to each detail sub-band obtained from the wavelet decomposition (LH, HL, HH). 2. Set the denoising threshold for each sub-band based on the noise level (\(\sigma\)) obtained in the preprocessing step. For example, set the point as 0.1. The threshold value is a determinant of the image used for the experiments, and in this study, a value of 0.05 was used.