# PSF Estimation of Space-Variant Ultra-Wide Field of View Imaging Systems
## 5 Proposed Method  
The method proposed in this paper is based on modeling the PSF of the system, and comparing it with real image data without requiring measurements of the wavefront aberrations. The fitting of the model takes place in the image plane. There are three conditions for acquiring the set of calibration images. The first criterion is that FWHM of the image of the light source should be sufficient for PSF estimation. The experimental results show that FWHM size greater than 5 px is enough for the algorithm. FWHM of the diffractive image of the source should be smaller than 1 px. Otherwise the diffractive function has to be added to the model of the system. The image of the 200 \(\upmu\)m pinhole has FWHM size 0.6 \(\upmu\)m which is less than the size of the pixel of our camera. The influence of the source shape has to be taken into account. The second criterion is the Signal-to-Noise Ratio (SNR), which should be greater than 20 dB. Section 6 contains a comparison of results obtained with different SNR. The third criterion is that the number of test images depends on the space variance of the system, i.e., a heavily distorted optical system will require more test images to satisfy this criterion. We have to choose the distance of two different PSFs in such a way that  
\[RMSE_{IN}(f_{i},f_{i+1})=\sqrt{\frac{1}{M\times N}\sum\limits_{u=0}^{N-1}\sum \limits_{v=0}^{M-1}(f_{i}-f_{i+1})^{2}}<5\%\,. \tag{15}\]where \(f_{i}\) and \(f_{i+1}\) are the images of the point light source in the image plane. \(M\) and \(N\) are the sizes of the \(f_{i}\) and \(f_{i+1}\) images. In our example, a grid of 24 positions of the PSFs in one quadrant of the optical system is sufficient to estimate the model. The wavefront is modeled using Zernike polynomials and known optical parameters. In addition to the input image data, we need to know camera sensor parameters such as resolution, the size of the sensor and optical parameters such as the focal length (crop factor, if included), the F-number and the diameter of the exit pupil. The obtained model of the PSF of the optical system is based on the assessment of differential metrics. We can describe the modeling of real UWFOV systems as a procedure with three main parts: the optical part, the image sensor, and the influence of the sensor noise. The space-variant impulse response \(h(u,v,\xi,\eta)\) can include the influence of the image sensor (e.g., pixel shape and sensitivity profile, noise or quantization error). The sensor has square pixels with uniform sensitivity. Then, the PSF of the imaging system can be described as  
\[h(u,v,\xi,\eta)=h_{opt}(u,v,\xi,\eta)*h_{sen}(u,v), \tag{16}\]  
where \(h_{sen}(u,v)\) is the PSF of the sensor and \(h_{opt}(u,v,\xi,\eta)\) is the PSF of the optical part. Symbol \({}^{\star}\) describes the convolution. \(h_{opt}(u,v,\xi,\eta)\) can be calculated from the system parameters and wavefront deformations using the Fourier transform as described in Equation (13). The wavefront deformation is modeled using Zernike polynomials for the target position in the image plane (see Equation (7)). Ultra-wide field images typically have angular dependent PSF. High orders of Zernike polynomials are therefore used in the approximation. The wavefront approximation is used up to the 8 th order plus the 9 th spherical aberration of the expansion function. The field dependence of the coefficients is formulated in [22]. In our work, the set of field-dependent coefficients was expanded up to the 8 th order plus the 9 th spherical aberration. The actual table of the used \(W_{klm}\) coefficient is attached in Appendix A as Table A1. Let us assume an imaging system with an unknown aberration model. We then obtain with this system a grid of \(K\) test images of a point light sources covering the entire FOV as  
\[\{f_{i}(u,v)\}_{i=1}^{K}. \tag{17}\]  
Then, let  
\[\{f_{d}(\hat{u},\hat{v})\}_{d=1}^{L}, \tag{18}\]  
be the \(d\)-th realization of the model in the corresponding position in the image as the original object.