# PSF Estimation of Space-Variant Ultra-Wide Field of View Imaging Systems
## 6 Results
### Experimental Laboratory Results  
The second part of the results describes the performance of our method when dealing with real data. The experimental images were obtained with the setup illustrated in Figure 10. A small white LED diode was used as a light source, together with a pinhole 200 \(\upmu\)m in diameter. The image data were acquired with a G2-8300 astronomical camera [44], with a 3358 \(\times\) 2536 pixel CCD sensor, corresponding to a size of 18.1 \(\times\) 13.7 mm, which gives a pixel size of 5.39 \(\upmu\)m. The camera was equipped with a Sigma 10 mm EX DC HSM fisheye lens [45], which is the diagonal type of fisheye lens. A camera was mounted on a 2D rotational stage. This configuration, together with an observation distance of 5 m, gives FWHM size of a PSF of around 6 pixels. Different positions of the light source, to cover the entire FOV, were achieved by rotating the camera in both axes. The advantage of this approach is that the same distance is kept between the light source and the camera, assuming that nodal mounting has been used. Table 3 summarizes parameters of the simulated system, which are the same as for the simulated system in Section 6.1. However, it is necessary to take into account that the optical aberrations are different. The graphs in Figures 11-13 and Table 4 show the results of the fit with a different order of \(W_{klm}\) coefficients, a fitted curve illustrating the trend of the results, and standard deviation error bars. Figure 14 shows a comparison of the results with other PSF modeling methods. It can be seen that the best result is obtained with coefficients up to the 4 th order. Then, with a higher order  
Figure 10: The experimental setup, consisting of a camera stage and a light source with a small aperture. Figure 9: The result of fitting. A comparison between the original PSF and the PSF model of the system. The graph on the right shows the intensity of the difference between the original PSF and the model; it indicates a very good result for the goodness of fit. of the coefficients, we obtain slightly worse results, even for the PSFs placed on the optical axis. With increasing image distance, the situation seems to deteriorate, and the difference increases. One reason for this is the optimization convergence, which is worse for the more involved \(W_{klm}\) coefficients in 6 th and 8 th orders. Another explanation is provided by Figure 15 and Table 5. It can be seen that the absolute difference is slightly worse for higher orders of \(W_{klm}\) coefficients, but that the shape of the model or the precision of the obtained shape of the PSF is better. The image of the differences in Figure 15 provides information proving that the maximum of the difference is located in a smaller number of pixels in the central part of the PSF. Thus the model seems to be well described in terms of the shape description. We can conclude that the higher order (6 th and 8 th) \(W_{klm}\) coefficients provide better parameters of the model in terms of the shape description and better localization of aberrations, but sometimes at the cost of a slightly worse absolute difference. Table 4 shows another interesting phenomenon. The results obtained by 8 th order fitting are slightly better than the results obtained by 6 th order fitting. We observed that the 6 th order model aberrations are overfitted; the model adds aberrations which are actually not present. However, use of the 8 th order model compensates these overfitting effects. Finally, we have to admit that it is not always necessary to use high order Zernike polynomials, and our results illustrate that a simple solution including only basic aberrations (up to the 4 th order) provides results of several percent. Figure 12: Experimental results for the MAXDIFF difference up to the 6 th order. The graph contains results from Table 4 and ten other points according to the normalized image distance \(H\).