# PSF Estimation of Space-Variant Ultra-Wide Field of View Imaging Systems
## 6 Results
### Numerical Stability Verification  
This section deals with verifying the stability of the optimization convergence. The test pattern, used for verifying the functionality of the algorithm was an image obtained by generating random \(W_{klm}\) coefficients (i.e., random optical aberrations) and placing PSFs in locations covering the entire field of view (see Figure 5), assuming a rotational symmetric imaging system. To verify the algorithm, we test PSFs in the locations marked with red circles. Table 1 summarizes the parameters of the simulated system. Table 2, Figures 6 and 7 show successive verification of the proposed algorithm by values of MAXDIFF and the RMSE operator, a fitted curve illustrating the trend of the results, and standard deviation error bars. We can see that the difference between the original and the model is within the order of thousandths. The difference is given only by quantization noise and the sampling of the test pattern. The sampling of the original PSF appears as a serious issue, and low resolution causes problems within the convergence in the optimization of the \(W_{klm}\) parameters. The resolution of the image may affect the accurate positioning of the PSF. For an explanation, we can find the maximum of the PSF by calculating the center of the mass precisely, if the sampling of the pattern is finer. Therefore, if we can find the maximum precisely, we can use precise positioning \(\hat{u},\hat{v}\) for calculating the PSF of the system. Figure 5: Field of simulated PSFs with randomly generated \(W_{klm}\) coefficients. We consider the simulated imaging system as rotationally symmetric. The points in the red circle were used for verifying the algorithm. Figure 6: The verification results contain all points used in the verification. The positions of all points are illustrated in Figure 5. MAXDIFF according to the normalized image distance. \begin{table}
\begin{tabular}{c c} \hline Resolution & 3358 \(\times\) 2536 px \\ Sensor size & 18.1 \(\times\) 13.7 mm \\ Pixel size & 5.39 \(\upmu\)m \\ Lens focus distance & 10 mm \\ FOV & \(110^{\circ}\) \\ \hline \end{tabular}
\end{table} 
Table 1: The sensor size, the resolution and the optical parameters of the simulated artificial imaging system. Figure 7: The verification results contain all points used in the verification. The positions of all points are illustrated in Figure 5. RMSE according to the normalized image distance. The graphs mentioned above provide information about the accuracy of the model against the positioning of the PSF in the image. However, it can also be useful to mention the accuracy of the model when the original image includes noise. For this reason, we performed a test where one PSF, in the center of the image, was affected by additional Gaussian noise. A test was performed in which the SNR in the image under test was from 28 dB to 9 dB (see Figure 8). We can see that the algorithm works quite well from 28 dB to 20 dB, and MAXDIFF is less than 3%. However, when SNR is below 20 dB, the error decreases rapidly. This is due to the fact that our model focuses on optical aberrations, not on modeling the noise. Figure 9 illustrates a direct comparison between the original PSF, the obtained model and the difference between them in relation to RMSE and the MAXDIFF operator. This verification process was performed to verify whether the algorithm is able to find the model \(W_{klm}\) parameters that are closest to the \(W_{klm}\) coefficients used in generating the pattern illustrated in Figure 5. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Metrics**} & \multicolumn{6}{c}{**Normalized Image Distance H (-)**} \\ \cline{2-7}  & **0** & **0.17** & **0.33** & **0.50** & **0.67** & **0.83** \\ \hline RMSE (\(10^{-5}\)) & 8.2 & 7.3 & 8.8 & 16 & 57 & 270 \\ MAXDIFF (\%\(\%\)) & 0.14 & 0.13 & 0.16 & 0.36 & 2.1 & 6.4 \\ \hline \hline \end{tabular}
\end{table}