# Prospects of Structural Similarity Index for Medical Image Analysis
## 2 Historical Review and Basic Principles of SSIM
Once these three factors have been obtained, the combination of the comparisons, as indicated in Equation (7), shows a quality score within the range of \(-1\) to \(1\) because of the structural influence. However, in various cases, the score is between \(0\) and \(1\). Therefore, the SSIM satisfies the following conditions:  
1. Symmetry: \(SSIM(\mathbf{x},\mathbf{y})=SSIM(\mathbf{y},\mathbf{x})\);
2. Boundedness: \(SSIM(\mathbf{x},\mathbf{y})\leq 1\);
3. Unique maximum: \(SSIM(\mathbf{x},\mathbf{y})=1\) if and only if \(\mathbf{x}=\mathbf{y}\). As in previous studies regarding the universal quality index (UQI) [14], SSIM is also effective in inspecting the image locally by implementing a sliding window. Hence, according to the original article on SSIM [10], the sliding window was \(11\times 11\). In addition, there was an improvement in the sliding window by applying a Gaussian weighting function \(\mathbf{w}=\{w_{i}|i=1,\ 2,\ 3,\ \dots,\ N\}\) with a standard deviation of \(1.5\). The value of \(\mathbf{w}\) should fulfill a unit sum of \(\sum_{i=1}^{N}w_{i}=1\). Because of this Gaussian weighting function with a \(11\times 11\) local window, the local statistics, such as \(\mu_{x}\), \(\mu_{y}\), \(\sigma_{x}\), \(\sigma_{y}\), and \(\sigma_{xy}\), have the following adjustments:  
\[\mu_{x}=\frac{1}{N}\sum_{i=1}^{N}w_{i}x_{i}, \tag{9}\]  
\[\mu_{y}=\frac{1}{N}\sum_{i=1}^{N}w_{i}y_{i}, \tag{10}\]  
\[Q=\frac{\sigma_{xy}}{\sigma_{x}\sigma_{y}}\cdot\frac{2\overline{x}\overline{y}}{ \overline{x}^{2}+\overline{y}^{2}}\cdot\frac{2\sigma_{x}\sigma_{y}}{\sigma_{x}^ {2}+\sigma_{y}^{2}}\, \tag{2}\]0  
\[Q=\frac{\sigma_{xy}}{\sigma_{x}\sigma_{y}}\cdot\frac{2\overline{x}\overline{y}}{ \overline{x}^{2}+\overline{y}^{2}}\cdot\frac{2\sigma_{x}\sigma_{y}}{\sigma_{x}^ {2}+\sigma_{y}^{2}}\, \tag{2}\]1  
\[Q=\frac{\sigma_{xy}}{\sigma_{x}\sigma_{y}}\cdot\frac{2\overline{x}\overline{y}}{ \overline{x}^{2}+\overline{y}^{2}}\cdot\frac{2\sigma_{x}\sigma_{y}}{\sigma_{x}^ {2}+\sigma_{y}^{2}}\, \tag{2}\]2  
\[Q=\frac{\sigma_{xy}}{\sigma_{x}\sigma_{y}}\cdot\frac{2\overline{x}\overline{y}}{ \overline{x}^{2}+\overline{y}^{2}}\cdot\frac{2\sigma_{x}\sigma_{y}}{\sigma_{x}^ {2}+\sigma_{y}^{2}}\, \tag{2}\]3 SSIM procedures to quantify the image quality (adapted from [10]). where \(K_{1}\) and \(K_{2}\) are 0.01 and 0.03, respectively. When completing the SSIM computation over the entire image using a local window, the mean SSIM (MSSIM) can be obtained as  
\[Q=\frac{\sigma_{xy}}{\sigma_{x}\sigma_{y}}\cdot\frac{2\overline{x}\overline{y}}{ \overline{x}^{2}+\overline{y}^{2}}\cdot\frac{2\sigma_{x}\sigma_{y}}{\sigma_{x}^ {2}+\sigma_{y}^{2}}\, \tag{2}\]4  
where \(\mathbf{X}\) and \(\mathbf{Y}\) denote the reference image and the image under testing, respectively, whereas \(\mathbf{x}_{j}\) and \(\mathbf{y}_{j}\) are the images at the \(j\)-th window when the local window slides over the original and distorted images, and \(M\) is the number of local windows in the image. MSSIM showed consistency when compared with human observers. In addition, these results were confirmed by implementing the PSNR as an evaluation tool to measure other IQAs, which, in this case, is MSSIM. Further, the previous UQI with \(K_{1}\) and \(K_{2}\) is 0, which presents the smallest correlation with the human observers. These results indicate that MSSIM can improve the UQI ability in avoiding a zero in the denominator by setting \(K_{1}\) and \(K_{2}\) as \(\ll 1\). Hence, SSIM has become popular as an objective investigative tool for other fields included in medical image analysis [58; 59; 60; 61; 62; 63; 64; 65; 66; 67]. To overcome the weakness of a single-scale SSIM associated with a limitation of view, a multi-scale SSIM (MS-SSIM) was established [22]. The viewing conditions are incorporated with the display resolution, distance when reading the image, luminance background, and other set environments that can affect the image investigation results. Figure 2 shows the procedure used by MS-SSIM in evaluating an image.