*2.3. Ramp Filter Design in Spatial Domain and Six Di*ff*erent Window Functions*

Linear filtering can be categorized into two methods: applying the convolution kernel in the spatial domain and linear multiplication of a transfer function in the Fourier domain. A band-limited ramp filter constructed in the Fourier domain is defined as follows:

$$\text{RAMP} \left( \omega \right)\_{A} = \begin{cases} \left| \omega \right|\_{\nu} & \text{if } \left| \omega \right| \le 0.5 \text{ lp/mm} \\ 0, & \text{otherwise} \end{cases} \tag{1}$$

where ω is the discretized spatial frequency by considering the Nyquist frequency. However, the ramp filter in Equation (1) has a zero at ω = 0 lp/mm such that the signals at the DC offset (zero frequency component) after linear multiplication go to zero. The Fourier transform of the ramp convolution kernel constructed in the spatial domain can be defined as follows [17]:

$$\text{RAMP}(\omega)\_B = \text{FT}\{\text{ramp}(n)\} = \int\_{-\infty}^{\infty} \text{ramp}(n)e^{-i2\pi\omega}d\omega \tag{2}$$

$$nump(n) = \begin{cases} \begin{array}{ll} 1/4, & \text{if } n = 0 \\ 0, & \text{if n is an even number} \\ -1/\left(n\pi\right)^{2}, & \text{if n is an odd number} \end{array} \tag{3}$$

The ramp filter in Equation (2) does not include a zero, as shown in the comparison of the two shapes in Figure 2. Filtering with non-zero conditions avoids the zero signals that might have occurred if the filters were used with zero conditions.

**Figure 2.** (**a**) Comparison of the ramp filters designed in different domains and (**b**) its magnified plot near the DC (zero frequency) component.

Many window functions have been introduced depending on the strength of noise suppression at different cutoff frequencies for each purpose [18]. However, the reduction of the critical signal is inevitable during noise suppression; therefore, the optimal window function is often heuristically chosen after multiple reconstruction trials. Six different smoothing windows were implemented herein in the ramp filter. Each window function was followed by the equation summarized in Table 4, where a term *L* in (b), (c), (d), and (f) indicates the length of the window.

## *2.4. Modulation Transfer Function (MTF)*

1 √1 + ( ) 2 ⁄ 0.5 (1 + cos ( 2 )) 0.54 − 0.46 cos ( 2 ) Spatial resolution for each imaging configuration and each filter scheme was evaluated by the MTF measurement of the cylindrical materials as conducted by Richard et al. [19]. After subtracting the two-dimensional planar fit from the original region of interest (ROI) of each targeted cylinder, the radial pixel values around the edge of the circular shape were rearranged to yield a one-dimensional edge spread function (ESF). When converting the image grid from a Cartesian to polar map, the center

) + 0.27 cos (

3

2 (1 −

2 − 1 || 2⁄ ) 3

1 √1 + (

4 − 1

⁄

{ 1 − 6 ( || 2⁄ ) 2

0.21 − 0.41 cos (

) − 0.08 cos (

 ) 2

if 0 ≤ || ≤ ( − 1)⁄4 if ( − 1)⁄4 ≤ || ≤ ( − 1)⁄2

> 6 − 1

) + 0.006 cos (

8 − 1 ) of each disk was measured on a binary image through a gray-level threshold. The ESF, which is equivalent to the radial profile of the circle, was resampled with one-tenth of the reconstructed pixel size to reduce the non-uniformly distributed pixel noise [20]. The final ESF was derived by averaging the ESFs measured from consecutive axial slices. The MTF was the Fourier amplitude of the derivative of the ensemble-averaged ESF. In addition, the high-frequency noise of the ESF derivative was relieved through a Hanning window having the same length as the ESF size. The overall process of radial MTF measurement is depicted in Figure 3.


**Table 4.** Description of each window that was implemented with the ramp filter.

**Figure 3.** A depicted workflow for 1D edge spread function (ESF) measurement of the targeted low-contrast material.
