# Quality Control in Dental Cone-Beam Computed Tomography
## 2 Parameters and Methods
### Spatial Resolution  
The spatial resolution was defined by the 10% point of the modulation transfer function (MTF) [24; 25]. An advantage of MTF is that the detailed information at all spatial frequencies can be acquired, thus avoiding the influence of subjective factors, such as naked-eye observation. Additionally, more information can usually be obtained by this method than otherwise [26].  
The DVT_AP phantom was placed at the center of the FOV, and scanning was performed under routine conditions for adults. The scan region was made free of any substances interfering with the beam. The region of interest (ROI) was delineated on the reconstructed axial slices for the phantom, as shown in Figure 2.  
The four sides of this ROI should be parallel with the pixel rows (columns), and only the polyvinyl chloride (PVC) and the air were included. The edge of transition between the PVC and the air should be parallel with the upper and lower sides of the ROI. Then the phantom was repositioned, and the images were acquired repeatedly to optimize the parallelism of the edge. The width of the upper and lower sides of the ROI corresponded to the width of 5 mm in the phantom (the tolerance was one pixel). Each segment of the vertical side in PVC and the air should be at least 3 mm.  
The spatial resolution was defined by the 10% point of MTF, as calculated using the following steps:  
1. Data acquisition The mean pixel values of all rows parallel with the edge and included in ROI were calculated (\(M_{1}\), \(M_{2}\), \(M_{3}\), \(\ldots\), \(M_{n}\)).
2. Finding the differential  
Figure 2: An example of locating and delineating the ROI for determining the modulation transfer behavior.  
The first-order differential was calculated by subtracting the mean pixel values in adjacent rows (\(D_{1}\), \(D_{2}\), \(D_{3}\), \(\ldots\), \(D_{n-1}\)), as shown in Formula (4).  
\[D_{m}=M_{m+1}-M_{m} \tag{4}\]  
where \(D_{m}\) is the first-order differential of the mean pixel value of the rows, and \(M_{m}\) and \(M_{m+1}\) are the mean pixel values of the \(m\)th and the \((m+1)\)th rows, respectively.  
(3) Setting the limits  
The maximum value \(D_{k}\) was chosen from \(D_{1}\), \(D_{2}\), \(D_{3}\), \(\ldots\), and \(D_{n-1}\). Starting from \(D_{k}\), the symmetric adjacent data band was redefined as \(D_{k-l}\), \(D_{k-l+1}\), \(\ldots\), \(D_{k}\), \(\ldots\), \(D_{k+l-1}\), and \(D_{k+l}\).  
(4) Fourier transform  
If the number of data \(D_{k-l}\), \(D_{k-l+1}\), \(\ldots\), \(D_{k}\), \(\ldots\), \(D_{k+l-1}\), and \(D_{k+l}\) obtained by step 3 was not the power of 2, then zero padding was performed. Next, discrete Fourier transform was applied to the sequence of values, with normalization of the Fourier coefficient to its maximum to obtain \(\overline{F_{0}}\), \(\overline{F_{1}}\), \(\ldots\), \(\overline{F_{m}}\).  
Similarly, discrete Fourier transform was applied to the arithmetically symmetric sequence \(|D_{k-l}+D_{k+l}|/2\), \(|D_{k-l+1}+D_{k+l-1}|/2\), \(\ldots\), \(|D_{k}|\), \(\ldots\), \(|D_{k-l+1}+D_{k+l-1}|/2\), and \(|D_{k-l}+D_{k+l}|/2\), with normalization to its maximum to obtain \(\overline{F_{0}}\), \(\overline{F_{1}}\), \(\ldots\), \(\overline{F_{m}}\).  
(5) Averaging  
The average \(F_{p}=(\overline{F}+\widetilde{F}_{p})/2\) was allocated to its corresponding spatial frequency \(v_{p}=p\times v_{n}/m\), where \(p\in[0,1,\ldots,m]\), and \(v_{n}\) is the Nyquist frequency.  
(6) Plotting  
(\(F_{p};v_{p}\)) was plotted with \(v_{p}\) on the \(x\)-axis and \(F_{p}\) on the \(y\)-axis, and connected with a straight line. The 10% MTF point was found to define the spatial resolution.