## 3 Analysis of Edge Method Accuracy
### Oversampling Ratio  
Binning, OSR, and phase shift (PS) play a decisive role in the accuracy of ESF construction. Adopting 4\(\times\) OSR has the advantages of moderate calculation and obtaining valid data before the cut-off frequency, but it may not be suitable for edges with a too-small or too-large angle. Related research [7] proves that the accuracy of the edge method can be  
Figure 7: Diagrams of degradation due to sampling for different tilt angle edges: (**a**) when edge angle is 0 deg, (**b**) when edge angle is 8 deg, (**c**) when edge angle is 22.5 deg, (**d**) when edge angle is 30 deg, (**e**) when edge angle is 38 deg, (**f**) when edge angle is 45 deg. further improved by adopting different OSRs for different edge angles, so it is necessary to propose a more adaptable OSR model. The row and column projection path distance formed by projecting the row and column pixels of ROI along the edge to the direction perpendicular to the edge plays an important role in the reasonable determination of OSR. For [0, 45] deg vertical edge, the projection path distance of adjacent pixels in each row (or adjacent columns) is \(\cos(\theta)\)(pix), and the projection path distance of adjacent pixels in each column (or adjacent rows) is \(\sin(\theta)\)(pix). Based on the ISO-cos binning strategy, it is necessary to bin multi-row projection data with a projection pitch of \(\sin(\theta)\)(pix) that fall within the width of \(\cos(\theta)\)(pix). The optimal adaptive binning of the projection pixels can be realized when the OSR is determined as \(1/\tan(\theta)=(\cos(\theta)/\sin(\theta))\). Figure 8 shows a comparative schematic diagram of the ISO-cos binning process when different OSRs are determined for different edge angles. It can be seen that \(1/\tan(\theta)\) determines the minimum number of effective rows to be determined and the optimal OSR for adaptive binning. When the actual OSR is higher than the optimal OSR, there will be empty bins, while when it is lower than the optimal OSR, the data utilization rate will be insufficient. The relationship between the ideal OSR and edge angle is shown in Figure 9. When the edge angle is less than 3 deg, the minimum effective rows and ideal OSR are greater than 20 and increase sharply with the decrease of the edge angle. When the edge angle is greater than 25.56 deg, the minimum effective rows and ideal OSR are less than 2 and gradually decrease to 1 with the increase of edge angle. The practical application of the edge method requires that the OSR should not be too high or too low. OSR higher than 10 will obviously increase the calculation cost, but the accuracy will not be improved qualitatively. Meanwhile, an OSR lower than 2 will decrease the accuracy and make it difficult to obtain MTF accurately before Nyquist frequency. In this section, the ISO-cos edge method is used to measure the MTF of BDE images with 3, 5, 10, 14, 25, and 40 deg, and three kinds of OSR, ideal, low, and high, are used to simulate and verify the influence of different OSR on the accuracy of edge method. \(1/\tan(\theta)\) is used to calculate the ideal optimal OSR (OptOSR/OptOSR2) corresponding to different edge angles. When the ideal OSR is greater than 8, the ideal suboptimal OSR (OptOSR1) will be calculated by \(1/[2\cdot\tan(\theta)]\), and when the suboptimal OSR is still greater than 8, 8 will be determined as the advised OSR (AdvOSR3). When the ideal OSR (OptOSR1) is less than 2, 2 will be determined as the advised oversampling rate (AdvOSR), and the problem of an empty bin will be solved by interpolation. It should be noted that the MTF reference value MTF(Ref) of the BDE image is calculated by \(MTF_{samp}(f)\cdot MTF_{blur}(f)\) (for the methods of obtaining \(MTF_{samp}(f)\) and \(MTF_{blur}(f)\), see the previous description in Section 3). The simulation results are shown in Figure 10.  
Figure 8: Diagrams of OSR determination when edge angle differs: (**a**) when edge angle is 14.04 deg, (**b**) when edge angle is 20 deg.