## 2 Measurement Principles and Methods
### ESF Processing Methods  
Extracting the ESF curve is a crucial step in ensuring the computational accuracy of the slanted-edge method. The optical imaging system's conversion of optical signals into electrical signals on the detector results in a down-sampling phenomenon [39]. This varies based on pixel size and leads to the edge image data manifesting as a discrete sequence of points rather than a continuous curve. Consequently, part of the ESF curve information is missing, which directly impacts the accuracy of the MTF. To address this issue, a multi-line edge alignment method is employed to create a more densely sampled ESF curve. Firstly, the pixels within the edge image's ROI are projected. This projection can be executed in two ways: horizontally or perpendicular to the slanted edge, as illustrated in Figure 6. While these projection methods yield similar results when the edge angle is small, they diverge significantly as the edge angle increases. In this paper, the ROI image is projected orthogonally to its edge. The subsequent one-dimensional (1D) pixel array requires sub-pixel oversampling, where the bin width is a critical parameter. In the methods outlined in ISO 12233, ISO-cos, and OMNI, a 4x OSR is utilized. The primary distinction among these methods lies in the variation in bin width relative to the edge's inclination angle. Specifically, ISO 12233 and OMNI employ a constant bin width of 1/4 pixel, independent of the angle, whereas ISO-cos  
Figure 5: RMSE of three methods for detecting edges under different noise conditions. Figure 6: Comparison of different projection methods: (**a**) horizontal direction; (**b**) vertical slanted-edge direction. on the neighboring pixel dimensions at the projection level, resulting in a bin width of \((\cos\ \theta)/4\) pixel. At any edge angle, when \(\tan\ \theta=1/4\), the ISO-cos method aligns the edge image's pixel projection intervals with the oversampling bin width. However, for angles exceeding \(14.04^{\circ}\), where the projection interval of each pixel column fails to align with the bin width, resulting in a gap larger than the bin width, the assumptions about the random phases of the imaging system's sampling points are violated. This misalignment leads to inaccuracies in the MTF estimation. Since \(\cos\ \theta\leq 1\), the bin width in the OMNI method is always greater than or equal to that in ISO-cos at a fixed OSR. This reduces the likelihood of empty bins in edge image projections at larger edge angles. Nonetheless, in scenarios involving smaller angles, the OMNI method's accuracy for sub-pixel oversampling binning does not match that of the ISO-cos method. [PERSON] [25] proposed the OMNI-sine method, an enhancement of the OMNI approach, which adapts the bin width in response to the edge angle. In this method, the sampling interval for each row of the ROI aligns with the bin width at intervals of \(\cos\ \theta\) pixel, and for each column, it aligns at \(\tan\ \theta\) pixel intervals. However, challenges arise when the slanted-edge angle \(\theta\) approaches \(0^{\circ}\), causing the bin width to diminish towards zero and potentially leading to an infinitely large OSR. When the angle is less than \(4^{\circ}\), the OSR can exceed \(14^{\circ}\). While, theoretically, a higher OSR correlates with greater computational accuracy, practically, an excessively high rate increases computational cost with marginal gains in accuracy. Conversely, a very low rate leads to data underutilization and compromised accuracy, particularly in accurately determining the system MTF at the Nyquist frequency. To mitigate these issues, the bin width is confined within \(1/n\) bin to minimize the occurrence of empty bins. The OSR, denoted as \(\
u_{\text{bin}}\), is defined in Equation (14) and ensures computational efficiency comparable to the ISO method across all angles. \[