## 3 Analysis of Edge Method Accuracy  
The principle of the edge method is not complicated, but its MTF measurement results often show low robustness and non-uniqueness. For example, it is difficult to keep the MTF results of the same edge image measured many times by the same algorithm completely consistent. The accuracy of the edge method is limited by the randomness of edge images used in different imaging fields and the strong coupling between algorithm steps. This section analyzes the main factors that affect the accuracy of the edge method: edge angle (Ang), oversampling rate (OSR), height and width of ROI (ROI-H and ROI-W), edge contrast (Con), random noise (noise level can be characterized by SNR) and gives suggestions on parameter determination with universal reference significance. In this paper, four representative edge methods are used for simulation analysis: ISO 12233: 2017 international standard edge method [4], ISO-cos modified edge method [22], OMNI modified edge method [11], and OMNI-sin modified edge method with variable OSR [7], which are abbreviated as ISO, ISO-cos, OMNI, and OMNI-sin, respectively. In order to compare the accuracy of different modified edge methods, the non-critical analysis steps are unified: ROIs with the same shape, size, and position are determined, and the widely used row difference linear fitting method is used to detect the edge angle [4], the oversampled discrete ESF is differentiated to obtain LSF, and the same MTF compensation model is used [2]. The similarities and differences adopted in the main steps of the above methods are summarized in the following Table 1:  
**Ideal discrete edge (IDE) image:** The IDE image with specified size, angle, bright/dark gray values, and resolution can be simulated and generated by the heaviside function. It should be noted that the simulated IDE image contains the discrete sampling degradation only (i.e., \(MTF_{IDE}(f)=MTF_{samp}(f)\)), so \(MTF_{IDE}(f)\) is slightly less than or equal to 1 in the whole SF ranges. The size of the simulated IDE image is 400 \(\times\) 400 pixels, the resolution is 300 dpi, and the gray values of the bright and dark edges are 180 and 60, respectively. It should be noted that, in this paper, the reference value of \(MTF_{samp}(f)\) for IDE images with different edge angles is approximately obtained by the mean value of MTF measurements of the above four edge methods. **Blurry degraded edge (BDE) image:** The types and parameters of the blurry kernel do not affect the analysis of the edge method, so the Gaussian blurry kernel, which is the most widely used, with a size of 5 \(\times\) 5 pixels, the mean value of 0 and the variance of 1 is determined. Based on the IDE image, the corresponding BDE image is obtained by convolution of 5\(\times\)5 Gaussian blurry kernel. It should be noted that the simulated BDE image contains the discrete sampling degradation and blurring degradation (i.e., \(MTF_{BDE}(f)=MTF_{samp}(f)\cdot MTF_{blur}(f)\)). The theoretical value of \(MTF_{blur}(f)\) can be calculated by the modulus of FFT of the Gaussian fuzzy kernel. The simulated partial BDE images with different edge angles of 0-45 deg are shown in Figure 2 (The simulated angle interval is 0.1 deg. ). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Process** & **Method** & **ISO 12233** & **ISO-cos** & **OMNI** & **OMNI-sin** \\ \hline Project to & \(\parallel X\)-axis & \(\perp\) Edge & \(\perp\) Edge & \(\perp\) Edge \\ OSR & 4 & 4 & 4 & Variable OSR [7] \\ Bin Width (pixel) & 1/4 & cos(\(\theta\))/4 & 1/4 & 1/(Variable OSR) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Projection and binning process comparison among four edge methods. **Blurry degraded edge with different contrasts (BDE@Con):** Based on the BDE image, eight kinds of BDE@Con edge images (0.04, 0.10, 0.15, 0.20, 0.36, 0.52, 0.68, 0.84) were obtained by adjusting the gray values of the bright (B) and dark (D) edges (B130/D120, B138/D112, B144/D106, B150/D100, B170/D80, B190/D60, B210/D40, B230/D20).