# Objective Prediction of Human Visual Acuity Using Image Quality Metrics
## 3 Results and Discussion  
The 49 mathematically degraded images that resulted from the convolution of the _PSF_ of each eye with the original images were obtained. Subsequently, the values of all the metrics for each degraded image, their mean, and their standard deviation (SD) were computed. Due to the major difference between the values of the metrics, and to compare the coefficients that resulted from the distinct fittings, the metrics results were normalized within the range \([0,1]\). We called them _nMSE_, _nPSNR_, _nGMSD_, and _nPSNR-HVS_. Therefore, linear combinations of nMSE with the other metrics were chosen, i.e., \(Q_{L}=\beta_{1}nMSE+\beta_{2}Q\), with \(Q\) being any of the other six herein used metrics. First, we evaluated collinearity with the _VIF_ index (Table 1). As shown, the _VIF_ values were high, except for _nMSE_, which was related to the other metrics. The generalized reduced gradient resolution algorithm was used to find the minimum value of the sum of the squares of the deviations between the value of subjective VA and \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline  & _nMSE_ & _nPSNR_ & _SSIM_ & _nGMSD_ & _MSSSIM_ & _FSIM_ \\ \hline _nPSNR_ & 7.64 & & & & & \\ _SSIM_ & 4.15 & 21.7 & & & & \\ _nGMSD_ & 3.05 & 12.3 & 27.8 & & & \\ _MSSIM_ & 4.23 & 24.8 & 150 & 30.8 & & \\ _FSIM_ & 5.93 & 35.8 & 76.7 & 15.2 & 57.4 & \\ _nPSNR-HVS_ & 7.78 & 63.8 & 27.9 & 14.6 & 31.7 & 57.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: VIF values among metrics. the target value provided by the fitting to the logistic function (5). This fitting provided an optimal local solution. Figures 3A-G and 4A-F show the subjective VA for distance vision versus the value obtained for all the metrics and the linear combination of metrics. The fitted functions are plotted in red, and the resulting parameters appear in Table 2. In all cases, the average amount of residuals was practically zero. Figure 3: Subjective VA vs. the values from calculating the normalized single metrics. (**A**) _nMSE_ (Mean Square Error Normalized), (**B**) _nPSNR_ (Peak Signal-to-Noise Ratio Normalized), (**C**) _SSIM_ (Structural Similarity Index), (**D**) _GMSD_ (Gradient Magnitude Similarity Deviation), (**E**) _MSSIM_ (Multiscale Structural Similarity Index), (**F**) FSIM (Feature Similarity Index), (**G**) _nPSNR-HVS_ (Peak Signal-to-Noise Ratio based on the Human Visual System). Red lines represent the fitting to the logistic function (5). Figure 4: Subjective VA vs. the values from calculating the linear combinations of normalized single metrics. Q_L is the linear combination of _nMSE_ with the other metrics. (**A**) linear combination _nMSE_ (Mean Square Error Normalized) with _nPSNR_ (Peak Signal-to-Noise Ratio Normalized), (**B**) linear combination _nMSE_ with _SSIM_ (Structural Similarity Index), (**C**) linear combination _nMSE_ with _GMSD_ (Gradient Magnitude Similarity Deviation), (**D**) linear combination _nMSE_ with _MSSIM_ (Multiscale Structural Similarity Index), (**E**) linear combination _nMSE_ with _FSIM_ (Feature Similarity Index), (**F**) linear combination _nMSE_ with _nPSNR-HVS_ (Peak Signal-to-Noise Ratio based on the Human Visual System). Red lines represent the fitting to the logistic function (5). To establish the best fitting, we performed an analysis of variance (ANOVA) with the subjective VA and the values provided by the above fittings of metrics. The ANOVA results are found in Table 3. In all cases, regressions provided a very low probability (\(p<0.001\)) of accepting the null hypothesis. The F-number was used to determine whether the high coefficient of determination values occurred by chance. The critical value for a 95% confidence level and a number of points (eyes; n = 52) was 4.03. As the F-number is much higher than the critical value, a significant relationship between the variables in the model must be accepted. Therefore, all the metrics are useful for predicting the subjective VA value. Regarding the t statistic, the metrics provided t values above 20.