## 2 Measurement Principles and Methods
### Simulation Experiment Analysis  
To precisely assess the accuracy of the algorithm, the RMSE between the test and theoretical MTF curves across various frequencies is utilized as the evaluation metric. This method of accuracy calculation, in contrast to the commonly employed absolute error measurement of MTF values at the Nyquist frequency, encompasses a broader range of  
Figure 8: Processing steps of the algorithm in this paper.  
Figure 7: Hamming window and Tukey window frequency and time\(-\)domain response curves: (**a**) the time-domain response curve; (**b**) the frequency-domain response curve.  
spatial frequencies. It more effectively captures the congruence between the computed and theoretical MTF curves, thereby providing a more comprehensive representation of the algorithm's accuracy.  
In line with imaging system theory, the response of an imaging system to spatial frequency can be simulated using a mathematical function, provided the system's parameters are known. These parameters include the aperture diameter \(D\), focal length \(f\), and wavelength \(\lambda\). Furthermore, Equation (16) defines the cutoff frequency \(f_{\text{cut}}\) in the context of the diffraction limit.  
\[f_{\text{cut}}=\frac{1}{\lambda}\frac{D}{f} \tag{16}\]  
The OTF in the diffraction limit can be expressed in Equation (17) as:  
\[\text{OTF}_{\text{diff}}(f)=\frac{2}{\pi}\left(\arccos\left(\frac{f}{f_{\text{ cut}}}\right)-\frac{f}{f_{\text{cut}}}\sqrt{1-\left(\frac{f}{f_{\text{cut}}} \right)^{2}}\right) \tag{17}\]  
The imaging system's image sensor size is designated as 8 \(\upmu\)m, with an aperture diameter of 30 mm, a focal length of 25 mm, and a diffraction wavelength of 10 \(\upmu\)m. Utilizing these parameters, the system's OTF is derived in accordance with Equation (17). Subsequently, the PSF is computed through an inverse Fourier transform of the OTF, simulating the system's response to a point light source. The cumulative sum of these PSFs yields the ESF, which represents the system's response to an idealized edge. This ESF is then projected onto a two-dimensional grid, determined by setting the edge angle and the image dimensions (\(H\) and \(V\)). The ESF effectively models the imaging system's response to the slanted edge, culminating in the generation of slanted-edge images at angles of 5\({}^{\circ}\), 10\({}^{\circ}\), 14\({}^{\circ}\), and 26\({}^{\circ}\), as shown in Figure 9.  
In this section, edge images are generated using software, after which Gaussian noise with a variance of 0.005 is introduced to the images. Subsequently, the ISO-cos, OMNI-sine, and the adaptive OSR method proposed in this paper are employed to calculate the MTF of the edge images at various angles. Simulations are conducted to validate the accuracy of the algorithm presented in this paper, as illustrated in Figure 10.  
Figure 10 clearly demonstrates that the MTF curves calculated using the algorithm proposed in this paper align more closely with the theoretical values compared to the other two methods. Specifically, Figure 10a indicates that the OMNI-sine method is prone to MTF aliasing at high frequencies with slanted-edge angles. Similarly, the ISO-cos method exhibits aliasing in MTF curves at high frequencies when the slanted-edge angle is substantial. The system MTF value at the Nyquist frequency, as depicted in Figure 10d, is 0.1586. Additionally, a comparison of the RMSE between the MTFs and the system values for each method across different frequencies is presented in Table 2.  
Figure 9: Images of slanted edge with different angles: (**a**) slanted-edge angle of 5\({}^{\circ}\); (**b**) slanted-edge angle of 10\({}^{\circ}\); (**c**) slanted-edge angle of 14\({}^{\circ}\); (**d**) slanted-edge angle of 26\({}^{\circ}\).  
It can be seen that the MTF value measured by the algorithm in this paper has the smallest mean square error with the theoretical value, which is closer to the real result.