## 4 Practical Multidirectional MTF Measurement Method
### Modified Adaptive OSR Model  
Based on the analysis conclusion in Section 3.2, this paper proposes an OSR determination model which is more in line with the principle of multidirectional projection and the actual calculation demand, as shown in Equation (13).  
\[OSR=\begin{cases}8&,\theta\in[0,3.18)\ or\big{[}0,\arctan\Big{(}\frac{1}{18} \Big{)}\Big{)}\text{deg}\\ \frac{1}{2\cdot\tan(\theta)}&,\theta\in[3.18,6.34)\ or\big{[}\arctan\Big{(} \frac{1}{18}\Big{)},\arctan\Big{(}\frac{1}{18}\Big{)}\Big{)}\text{deg}\\ \frac{1}{\tan(\theta)}&,\theta\in[6.34,14.04]\ or\Big{[}\arctan\Big{(}\frac{1}{ 9}\Big{)},\arctan\Big{(}\frac{1}{4}\Big{)}\Big{]}\text{deg}\\ \max\Big{[}\frac{1}{\tan(\theta)},2\Big{]}&,\theta\in(14.04,45]\ or\Big{(} \arctan\Big{(}\frac{1}{4}\Big{)},\arctan(1)\Big{]}\text{deg}\end{cases} \tag{13}\]  
When \(\theta\in[0,3.18)\) deg, the edge angle is \"too small,\" and the projection path is too dense. The ideal optimal OSR in this angle interval is all greater than 18. The minimum effective number of ROI rows (\(R_{min}=1/\tan(\theta)\)) corresponding to the small angle edge is relatively large. If the ideal high OSR is still adopted, the calculation cost will be increased, and the accuracy will not be improved obviously. Therefore, considering the accuracy and calculation cost of MTF measurement for small angle edges, the OSR is determined as 8, which can sufficiently ensure the accuracy of the edge method.  
When \(\theta\in[3.18,6.34)\) deg, the edge angle is \"relatively small,\" and the projection path is still relatively dense. The ideal optimal OSR in this angle interval lies between (9, 18). If the ideal OSR in this interval is all reduced by 0.5 times, the OSR will be between (4.5, 9), which is moderate and conforms to the ideal suboptimal OSR model \(1/[k\cdot\tan(\theta)]\) (\(k\in\)\(Z_{+}\)). Therefore, the actual OSR will be determined according to the ideal suboptimal OSR model \(1/[2\cdot\tan(\theta)]\).  
When \(\theta\in[6.34,14.04]\) deg, the edge angle is \"moderate,\" and the projection path distance is moderate. The ideal OSR of this angle interval is moderate, which lies between [4,9] and meets the actual needs of engineering. Therefore, the actual OSR will be determined according to the ideal optimal OSR model \(1/\tan(\theta)\).  
When \(\theta\in(14.04,45]\) deg, the edge angle is \"too big,\" and the projection path is sparse. The ideal optimal OSR corresponding to this angle interval is small, lies between [1, 4), which shows the projection data used for ESF construction is insufficient. In order to obtain the MTF before Nyquist frequency (0.5 lp/pix) accurately, the actual OSR should be above 2. When the ideal optimal OSR is less than 2, the actual OSR is determined as 2, and the empty bin in the construction of oversampled ESF can be filled with the vacant value by linear interpolation. When the ideal optimal OSR is greater than or equal to 2, the actual OSR will be determined according to the ideal optimal OSR model \(1/\tan(\theta)\). Therefore, the actual OSR will be determined according to the \(\max[1/\tan(\theta),2]\) model to ensure that the OSR is greater than or equal to 2.