*3.3. Modulation Transfer Function*

Six different MTFs for each filter scheme measured in the reconstructed images of M1 are shown in Figure 9. The higher the resolution of the reconstructed images, the better the MTF is preserved up to the high frequencies. In Figure 9f, *f* <sup>50</sup>, which indicates the specific spatial frequency when the MTF is dropped to 0.5, was 1.39, 1.40, 2.52, and 2.57 lp/mm for the configurations (1, 1), (1, 2), (2, 1), and (2, 2), respectively. The effect of detector resolution on the reconstruction image resolution was minor when we compared the curves between configurations (1, 1) and (1, 2) (or (2, 1) and (2, 2)).

**Figure 8.** (**a**) Reconstructed image of M1 with different configurations according to the different windows and (**b**) the radial profile of configuration (2, 2).

**Figure 9.** Modulation transfer functions (MTFs) for different filter schemes from the (**a**) Butterworth A, (**b**) Hanning, (**c**) Hamming, (**d**), Parzen, (**e**) Butterworth B, and (**f**) Flat Top windows with different configurations. The *f* <sup>50</sup>s measured in the images implemented with the flat top window were 1.39, 1.40, 2.52, and 2.57 lp/mm for the configuration (1, 1), (1, 2), (2, 1), and (2, 2), respectively.

The MTF curves measured in the reconstructed images of each material using configurations (1, 1) and (2, 2) are shown in Figures 10 and 11. As shown in Figure 11f, the *f* <sup>50</sup>s were 0.94, 1.86, 2.05, and 2.52 and 1.86 lp/mm from M1 to M5, respectively, which demonstrates that MTFs were preserved up to high frequencies of the order of M3, M4, M1, M5, and M2; that is, in the order of the relative contrast in Figure 6f. In contrast, the imaging configuration (1, 1) not only did not follow the order of contrast, but also presented different orders of *f* <sup>50</sup>s for the different filter schemes.

**Figure 10.** MTFs for different materials with the configurations (1, 1) using different filter schemes from the (**a**) Butterworth A, (**b**) Hanning, (**c**) Hamming, (**d**), Parzen, (**e**) Butterworth B, and (**f**) Flat Top windows. The orders of *f* <sup>50</sup>s as a function of different materials were different for each filter scheme.

**Figure 11.** MTFs for different materials with the configurations (2, 2) using different filter schemes from the (**a**) Butterworth A, (**b**) Hanning, (**c**) Hamming, (**d**), Parzen, (**e**) Butterworth B, and (**f**) Flat Top windows. The order of *f* <sup>50</sup>s as a function of different materials was M3, M4, M1, M5, and M2.

## *3.4. Normalized Noise Power Spectrum*

Figure 12 shows the radially averaged 1D NPS for each configuration with different filter schemes. The standard reconstructed image resolution (configuration (1, 1) and (2, 1)) gave higher noise properties compared to the high-resolution images (configuration (1, 2) and (2, 2)). We also observed that the peak of the 1D NNPSs from the higher detector resolution was at larger spatial frequencies, which demonstrates that the noise was distributed up to a higher frequency when the smaller pixels were used in the detector. The NPSs decreased as the intensity of high-frequency smoothing increased.

**Figure 12.** Radially averaged 1D NPS for each configuration from (**a**) (1, 1), (**b**) (1, 2), (**c**) (2, 1), and (**d**) (2, 2) with different filter schemes.

#### **4. Discussion**

We herein designed band-limited filters for all schemes. These can effectively retrieve the sampled projections because the projections are discretized into each detector pixel so that it is band limited in the Fourier domain [24]. As a result, band-limited filters lead to the removal of unnecessary noise signals at high frequencies.

There is no universal filter in CT imaging; therefore, the user should select an optimal smoothing window to observe the detailed internal structure with a purpose. Selecting an optimal window function is often based on experience rather than theory because we do not have a high level of knowledge about whether the imaging object is lying under a low-, mid-, or high-frequency range [25]. Thus, comparing the initial imaging performance of different filters and choosing the best solution for one's purpose is a good approach [25]. The most important factor when selecting the filter scheme is the manner in which the filter removes as many of the unnecessary components as possible in the frequency domain. In this experimental study, the signals near the edge of each material that we aimed to observe mostly lie in the low-frequency range, and show severe MTF distortion in the images applied with a high-pass filter, such as Butterworth A in Figure 9a. In contrast, the results in Figure 11f indicate that the flat-top window preserved the MTF up to a high frequency without an aliasing among the six filter schemes in our experiment. This is because the reconstructed images applied with the flat-top window not only resulted in uniform pixel values but also showed small oscillations (less noise) in both the target and background, as shown in the radial profiles in Figure 8b.

The flat-top window is used for cases in which a frequency component is required to be measured with great accuracy, e.g., a fixed-sine source [26]. Measuring the MTFs in the frequency domain could be interpreted as a discrimination of the signals spreading near the circular edge region. If a much larger signal difference exists between the target and the background, such as the tungsten edge, filter selection would not have been significant. However, we measured the MTFs for materials having no significant signal difference against the background material (low-contrast imaging); therefore, the amplitude accuracy was a key factor because the principal components in the Fourier domain were largely positioned in the low-frequency area [27].

The MTFs were preserved well at higher frequencies from the images reconstructed with a higher resolution. We observed that there was an MTF preservation loss up to 1.77 times by comparing the *f* <sup>50</sup> between configurations (1, 1) and (2, 2) in Figures 10f and 11f when using the same target material and detector resolution. Therefore, using a UHR imaging protocol rather than a standard imaging configuration is recommended to understand the fine sharpness of low-contrast material if the detector is available to be operated at a higher resolution.

However, the high-level smoothing window is not recommended for standard resolution imaging configuration, as shown by the disagreement in the order of relative contrast in Figure 10. As shown in Figure 10, the flat-top window provided little difference in *f* <sup>50</sup>s for different materials even though there was a clear discrimination in UHR imaging protocol. This was because the flat-top window overly smoothed the low-contrast object in the standard imaging, whereas the smoothing was still effective in UHR mode.

The trend of 1D NNPS in the configuration (2, 1) showed that the noise was distributed over all of the spatial frequencies. This demonstrates the back-projection from the high-resolution to small-image array would largely reduce the quantum noise and result in uniformly distributed noise.

The main drawback of this study is that all materials used to measure the MTFs had low contrast against the background PMMA intensity. This limits the study of higher-object-contrast materials such as bone and contrast-enhanced imaging. Our future study will be directed toward the effect of various filter setups on higher-object-contrast materials.
