## 3 Analysis of Edge Method Accuracy
### Edge Angle  
If the absolutely vertical upward edge is taken as the reference of 0 deg and the clockwise is positive, the angle range of all possible edges is [0, 360] deg, which can be divided into the vertical edge (\(\theta\)\(\in\) [\(-\)45, 45] \(\cup\) [135, 225] deg) and horizontal edge (\(\theta\)\(\in\) [45, 135] \(\cup\) [225, 315] deg). According to the axisymmetric and centrosymmetric properties of edge features, and referring to the angle transformation equation of reference [7], [0, 360] deg edges can be correspondingly transformed into [0, 45] deg vertical edges for analysis. The angle transformation diagram is shown in Figure 5. In this section, four  
Figure 4: Partial simulated BNDF images with different SNR when the edge angle is 25 deg: (**a**) the SNR is \(\infty\), (**b**) the SNR is 12.6, (**c**) the SNR is 10.0, (**d**) the SNR is 4.8, (**e**) the SNR is 4.2, (**f**) the SNR is 3.9. Figure 3: Partial simulated BDE@Con images with different contrasts when edge angle is 10 deg: (**a**) the contrast is 0.84, (**b**) the contrast is 0.68, (**c**) the contrast is 0.52, (**d**) the contrast is 0.36, (**e**) the contrast is 0.20, (**f**) the contrast is 0.15, (**g**) the contrast is 0.10, (**h**) the contrast is 0.04. Figure 2: Partial simulated BDE image with different angles (0–45 deg): (**a**) when edge angle is 0 deg, (**b**) when edge angle is 5 deg, (**c**) when edge angle is 15 deg, (**d**) when edge angle is 25 deg, (**e**) when edge angle is 35 deg, (**f**) when edge angle is 45 deg. representative edge methods are used to measure the MTF of BDE images with different edge angles of 0-45 deg. The simulation results are expressed in Cartesian coordinates and polar coordinates, as shown in Figure 6.  
Figure 6a-h show that the consistency of MTF curves obtained by [0.2, 14.5] deg edges is obviously higher than that of other angle interval edges in the whole frequency ranges. This conclusion is basically consistent with the view that accurate MTF measurement \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Noise Level** & **Noise Type** & **Gaussian Noise** & **Uniform Noise** & **Exponential** & **Speckle Noise** & **SNR (dB)** \\
**(Variance)** & **(Threshold)** & **Noise (Mean)** & **(Variance)** & **SNR (dB)** \\ \hline
00 (noise free) & 0.000 & 0 & 0 & 0.000 & \(\infty\) \\
01 & 0.001 & 1 & 1 & 0.001 & 12.6 \\
02 & 0.005 & 2 & 2 & 0.005 & 10.0 \\
03 & 0.010 & 3 & 3 & 0.010 & 4.8 \\
04 & 0.015 & 4 & 4 & 0.015 & 4.2 \\
05 (noise max) & 0.020 & 5 & 5 & 0.020 & 3.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Noise parameter and SNR of different noise levels for different noise types. Figure 5: Diagram of edge angle transformation. Figure 6: MTF for different tilt angle edge images vary from 0.20 deg to 44.5 deg (the angle interval is 0.50 deg) in Cartesian and polar coordinates using different modified edge methods: (**a–d**) ISO, ISO-cos, OMNI, OMNI-sin edge method in Cartesian coordinates, (**e–h**) ISO, ISO-cos, OMNI, OMNI-sin edge method in polar coordinates. results can be obtained by [2; 10] degree edge verified in literature [23]. It shows that MTF measurement with STA edges has high stability. When the edge angle is greater than 15 deg, the MTF curves appear a downward trend with the increase of edge angle in the middle and low frequency ranges (Roughly [0, 0.42] lp/pix). It can be clearly seen that the larger the angle is, the more obvious the MTF downward trend is. When the edge angle is greater than 30 deg, the MTF curves appear an upward trend with the increase of edge angle and tend to be aliased in the high frequency ranges (roughly [0.42, 0.50] lp/pix). The above simulation results further confirm the validity and rationality of ISO 12233's recommendation of determining STA edges to measure MTF in horizontal or vertical directions approximately. Figure 6e-h verify the effectiveness of measuring MTF in different directions with different angle edges.