## 6 Quality Analysis of the Reconstructed Image
### Objective Quality Analysis  
Objective quality analysis is based on the difference of the original \(f_{i,j}\) and reconstructed image \(u_{i,j}\). Two metrics are already mentioned and they are \(AAE\), (13), and \(MSE\) (14). The signal-to-noise ratio (\(SNR\)) is a ratio of the power of the signal and power of the background noise:  
\[\text{SNR}(f,g)=10\cdot log_{10}\left(\frac{\sigma^{2}(g)}{\text{MSE}}\right). \tag{18}\]  
The amplitude of image elements has a range:  
\[\text{MAX}:=|0.2^{q}-1| \tag{19}\]  
where \(q\) is the number of bits needed to display the amplitudes of the original image. The MSE does not take MAX into consideration so the Peak Signal-to-Noise Ratio (PSNR) is introduced:  
\[\text{PSNR}\left(u,v\right):=10\log_{10}\frac{\text{MAX}^{2}}{\text{MSE}}=20 \log_{10}\frac{\text{MAX}}{\sqrt{\text{MSE}}}. \tag{20}\]  
_SSIM_ (Structure Similarity) is a novel method of calculating similarity of two images [27]. The main idea is that the human visual system is suited for structural information processing and this measurement strives to measure differences in this information between original and reconstructed images. Let \(x_{i}\) and \(y_{i}\) be two discrete signals where \(i=1,2,\ldots,N\). The average value of discrete signal \(x_{i}\) is  
\[\mu_{x}:=\frac{1}{N}\sum_{i}^{N}x_{i}. \tag{21}\]  
The average value of discrete signal \(y_{i}\) is  
\[\mu_{y}:=\frac{1}{N}\sum_{i}^{N}y_{i}. \tag{22}\]  
The variance of \(x_{i}\) is  
\[\sigma_{x}^{2}:=\frac{1}{N-1}\sum_{i}^{N}(x_{i}-\mu_{x})^{2}. \tag{23}\]  
The variance of \(y_{i}\) is  
\[\sigma_{y}^{2}:=\frac{1}{N-1}\sum_{i}^{N}(y_{i}-\mu_{y})^{2}. \tag{24}\]  
The covariance of \(x_{i}\) and \(y_{i}\) is  
\[\sigma_{xy}:=\frac{1}{N-1}\sum_{i}^{N}(x_{i}-\mu_{x})\big{(}y_{i}-\mu_{y} \big{)}. \tag{25}\]  
Figure 13: Test images: (**a**) horse, (**b**) beauty, (**c**) mask, and (**d**) pills. The _SSIM_ is locally calculated over a window of determined size (usually \(8\times 8\)). For every local window, three parameters are calculated: brightness, contrast, and structure. The value that depicts the change of brightness intensity is calculated by:  
\[l(x,y):=\frac{2\mu_{x}\mu_{y}}{\mu_{x}^{2}+\mu_{y}^{2}}. \tag{26}\]  
The contrast is calculated using the variance between the original and reconstructed image as  
\[\text{MAX}:=|0.2^{q}-1| \tag{19}\]0  
The structural similarity uses the covariance of two images and is calculated by  
\[\text{MAX}:=|0.2^{q}-1| \tag{19}\]1  
These three components are combined  
\[\text{MAX}:=|0.2^{q}-1| \tag{19}\]2  
where \(\alpha,\beta\), and \(\gamma\) are parameters that define the relative importance of each component. The special case is when \(\alpha=\beta=\gamma=1\); the resulting SSIM index is provided with the next expression. \[\text{MAX}:=|0.2^{q}-1| \tag{19}\]3  
_MS-SSIM_ (Multi-Scale Structure Similarity) is based on _SSIM_[53], but the contrast and structure are calculated at different frequency scales, so it tests the image quality at different observation distances. The brightness is calculated only at the last scale, and the MS-SSIM is calculated as a combination of all the previously calculated coefficients at all other scales. The low-pass filter is iteratively applied to the original and reconstructed images, and the filtered image is requantized by a factor of 2. The original image is referred to as scale 1 and the largest scale as scale \(M\). The change in brightness intensity (26) is calculated only at scale \(M\) and denoted as \(l_{M}(x,y)\). On a random scale \(j\), the contrast \(c_{j}(x,y)\) is computed with (27) and the structural similarity \(s_{j}(x,y)\) is computed with (28). \[\text{MAX}:=|0.2^{q}-1| \tag{19}\]4  
Similar to (29), the exponents \(\alpha_{M},\beta_{j}\), and \(\gamma_{j}\) are used to assign the relative importance to the different components. Just as with the SSIM, the result of this measurement can range from 0 (no similarity) to 1 (identical images). The disadvantage of this measurement is that the calculation requires longer than with SSIM.