# PSF Estimation of Space-Variant Ultra-Wide Field of View Imaging Systems
## 5 Proposed Method
* The next calibration point will be placed on the \(\hat{u}\)-axis and next to the first point. * All \(W_{klm}\) coefficients from the first point fit will be used as start conditions in the next step of the fit. * In the next step, we will fit all the points along the \(\hat{u}\)-axis by increasing distance \(H\). * The previous result is used as the start condition for the next point. \[R_{f_{i},f_{d}}(\hat{u},0)\Rightarrow W_{klm}(d),\Delta\hat{u},\Delta\hat{v}.\] (23)
* Then, we can continue along the \(\hat{v}\)-axis by increasing distance \(H\). This procedure gives the first view of the model. \[R_{f_{i},f_{d}}(0,\hat{v})\Rightarrow W_{klm}(d),\Delta\hat{u},\Delta\hat{v}.\] (24)
* where \(W_{klm}(d)\) is the d-th realization of the fit. * After fitting all the on-axis points, we will start to fit all the off-axis points. * The example in this paper uses 24 points. \[h(u,v,\xi,\eta)=h_{opt}(u,v,\xi,\eta)*h_{sen}(u,v), \tag{16}\]0 (25)
* After fitting all the points, we need to evaluate the output \(W_{klm}\) coefficients which can describe the field dependency of our model. We verified experimentally that the median applied to the set of estimated \(W_{klm}\) coefficients provides better results of the output model than other statistical methods. Thus, we need to find the median of every \(W_{klm}\) coefficient over all fit realizations (the number of realizations is L) of the used points. This step will eliminate extreme values of \(W_{klm}\) coefficients which can occur at some positions of the PSF due to convergence issues caused by sampling of the image or overfitting effects caused by high orders polynomials. Extreme values indicate that the algorithm found some local minimum of the cost function and not the global minimum. The values of the \(W_{klm}\) coefficients are then significantly different from the coefficients obtained in the previous position. These variations are given by the goodness of fit. The output set of \(W_{klm}\) coefficients then consists of values verified over the field. As is illustrated in Figure 4, the described procedure repeats in every order (as defined in Equation (10)) of \(W_{klm}\) coefficients. The estimates of higher order coefficients (6 th and 8 th) come from lower order coefficients that have already been estimated from lower orders. If we want to describe our optical system with coefficients up to the 8 th order, we will first need to obtain coefficients of the 4 th order. Then, it is necessary to repeat the procedure from Equation (22) to Equation (25) of the previously described procedure assuming 4 th order \(W_{klm}\) coefficients as a starting condition. Following the whole procedure, we will estimate the 6 th order coefficients. Then, repeating the procedure again with 6 th order coefficients as a starting condition, we can finally calculate 35 coefficients of the 8 th order. The result is a set of coefficients (the number of coefficients depends on the order that is used) related to the optical system with the field dependency described in Section 3. \[h(u,v,\xi,\eta)=h_{opt}(u,v,\xi,\eta)*h_{sen}(u,v), \tag{16}\]1 Diagram of the proposed algorithm. Note that the \(W_{klm}\) coefficients from 4 th order estimate are used as a start condition for the 6 th order estimate, and the 6 th order \(W_{klm}\) coefficients are used as a start condition for an estimate of the 8 th order \(W_{klm}\) coefficients.