## 4 Experimental Results and Analysis
### Quantitative Analysis  
The proposed approach was verified using a subjective analysis approach. The performance evaluators, such as peak signal-to-noise ratio (PSNR), structural similarity index measure (SSIM), signal-to-noise ratio (SNR), and mean squared error (MSE), were considered to quantify the result. The proposed wavelet-based image deblurring and restoration approach was evaluated together with other state-of-the-art denoising techniques using PSNR, SSIM, and MSE as representative quantitative measurements, as shown below.  
PSNR is the ratio between the maximum possible power of a signal and the power of corrupting noise. The PSNR measures the peak signal-to-noise ratio between two images, used as the quality measurement between two images. The higher the value of PSNR, the better the quality of the compact image. PSNR is usually expressed in terms of the logarithmic decibel scale.  
PSNR is calculated as:  
\[\text{PSNR}_{\text{dB}}=10\text{log}_{10}\bigg{(}\frac{MAX^{2}}{MSE}\bigg{)} \tag{8}\]  
\[=20\text{log}_{10}(MAX)-10\text{log}_{10}(MSE)\]  
where:  
\(MAX\) = maximum possible pixel value of the image,  
\(MSE\) = mean square error.  
Figure 3: CT scan images with added Gaussian blur noise at 5% to 30% noise intensities in figure (**a**–**f**).  
The mean square error (\(MSE\)) is the cumulative error between the compressed and original images. The lower the \(MSE\), the better the quality of the impermeable image. It is calculated as:  
\[MSE=\frac{1}{pq}\sum_{a=0}^{p-1}\sum_{b=0}^{q-1}[I(a,b)-K(a,b)]^{2} \tag{9}\]  
where \(p\)\(q\): dimensions of the image,  
\(I(a,b)\): intensity of pixels (\(a,b\)) in the original image,  
\(K(a,b)\): intensity of pixels (\(a,b\)) in denoised images.  
The SSIM yields an objective assessment of the quality of the images. The SSIM depends on luminance, contrast, and structural terms. The product of these parameters is the SSIM of the image. The parameter \(L(x_{1},x_{2})\) is the luminance assessment function that determines the quality of having only a tiny margin between two images in terms of mean luminance (\(\mu x_{1}\) and \(\mu x_{2}\)).  
The parameter \(C(x_{1},x_{2})\) is the contrast assessment function that computes the quality of having only a tiny margin between two images in terms of standard deviations (\(\sigma_{x1}\) and \(\sigma_{x2}\)). The parameter \(S(x_{1},x_{2})\) is the structure assessment function determining the correlation coefficient between two images regarding covariance (\(\sigma_{x1x2}\)).  
\[\text{SSIM}(x_{1},x_{2})=\lfloor L(x_{1},x_{2})\rfloor a_{1}\times\lfloor C(x _{1},x_{2})\rfloor b_{1}\times\lfloor S(x_{1},x_{2})\rfloor c_{1} \tag{10}\]  
where:  
\[L(x_{1},x_{2})=\frac{2\mu_{x_{1}}\mu_{x_{2}}+U_{1}}{\mu_{x_{1}}^{2}+\mu_{x_{2 }}^{2}+U_{1}}\]  
\[C(x_{1},x_{2})=\frac{2\sigma_{x_{1}}\sigma_{x_{2}}+U_{2}}{\sigma_{x_{1}}^{2}+ \sigma_{x_{2}}^{2}+U_{2}}\]  
\[S(x_{1},x_{2})=\frac{\sigma_{x_{1}x_{2}}+U_{3}}{\sigma_{x_{1}}\sigma_{x_{2}}+ U_{3}}\]  
where \(\mu x_{1}\) and \(\mu x_{2}\) are the local means, \(\sigma x_{1}\) and \(\sigma x_{2}\) are the standard deviations, and (\(\sigma_{x1x2}\)) is the image's cross-covariance of \(x_{1}\), \(x_{2}\). Assume \(a_{1}=b_{1}=c_{1}=1\), \(U_{1}=\left(R\times S_{1}\right)^{2}\) and \(U_{2}=\left(R\times S_{2}\right)^{2}\), where \(R\) is the size of the image (256 for 8-bit grayscale images), \(U_{1}\) is a small constant value at \(S_{1}<<1\), \(U_{2}\) is the positive constant value at \(S_{2}<<1\), and \(U_{3}=U_{2}/^{2}\).  
The simplified version of SSIM is as follows:  
\[\text{SSIM}(x_{1},x_{2})=\frac{(2\mu_{x_{1}}\mu_{x_{2}}+U_{1})(2\sigma_{x_{1} x_{2}}+U_{2})}{\left(\mu_{x_{1}}^{2}+\mu_{x_{2}}^{2}+U_{1}\right)\left(\sigma_{x_{1} }^{2}+\sigma_{x_{2}}^{2}+U_{2}\right)} \tag{11}\]