## 3 Results  
In comparison with CSAM, the required phase-encoding lines and image acquisition times (TA) in the CSAK method for the undersampling ratios of 25%, 35%, and 45% were reduced by approximately 10%, 15%, and 22%, and 16%, 23%, and 32%, respectively, as shown in Table 2.  
The results of the simulation performed using the full k-space sampling dataset obtained from the human participant showed that the CSA images became noisy as the undersampling ratio increased, and the SNR also decreased as the number of sampling lines increased. The SNRs in the white and gray matter regions were 10.41, 9.80, 8.86, and 5.57, and 9.69, 9.35, 8.46, and 5.59, at 25%, 35%, 45%, and 100% (i.e., full k-space sampling), respectively (Table 3). Note that the SNRs in the white and gray matter were 5.57 and 5.59, respectively, for the full k-space. In addition, it was visually confirmed that a circular ringing artifact occurred at the top of the CSAM image, which was most severe at the \begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**CS Technique**} & \multicolumn{3}{c}{**Undersampling Ratio**} \\ \cline{2-4}  & **25\%** & **35\%** & **45\%** \\ \hline \multirow{2}{*}{
\begin{tabular}{c} CSAM \\ CSAK \\ \end{tabular} } & 06 m 04 s (579) & 08 m 24 s (804) & 10 m 48 s (1038) \\  & 05 m 05 s (523) & 06 m 26 s (680) & 07 m 18 s (811) \\ \hline \end{tabular}
\end{table} 
Table 2: Acquisition times for CSA techniques used in brain phantom imaging together with the numbers of phase-encoding lines in parentheses. Figure 2: Simulation for k-space undersampling effects. (**a**) A full sampling image; (**b**) image reconstructed from the 25% undersampling mask applied to the full sampling dataset; (**c**) image reconstructed from the 35% undersampling mask applied to the full sampling image; (**d**) image reconstructed from the 45% undersampling mask applied to the full sampling image. 25% undersampling ratio (Figure 2). Ringing and blurring artifacts were also found in the CSAM and CSAKS images, although there were some differences depending on the undersampling ratio. However, they were not found in the CSAKC image (Figure 3). Figure 4 shows the CSAM, CSAKS, and CSAKC images, line profiles, and MTFs at the 25% undersampling ratio. Ringing and blurring artifacts were clearly seen in the CSAM images (Figure 4a). The slope of the line profile was steepest in the CSAK image, and the edge rise distances for a pixel level between 10% and 90% were 6, 7, and 3 pixels for CSAM, CSAKS, and CSAKC, respectively (Figure 4b). The 50% MTFs were 0.38 line pairs per millimeter (lp/mm), 0.43 lp/mm, and 0.54 lp/mm for CSAM, CSAKS, and CSAKC, respectively, which is 42.86% higher for CSAKC than for CSAM. The 10% MTFs were 0.59 lp/mm, 0.70 lp/mm mm, and 0.97 lp/mm, respectively, showing that CSAKC had 63.64% better inter-tissue contrast than CSAM (Figure 4c). Furthermore, despite using the same CSAK method, no artifacts were found when using CSAKC. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**ROI** & **Undersampling Ratio** & **Mean (Signal)** & **STD (Noise)** & **SNR** \\ \hline \multirow{4}{*}{WM} & 25 & 78.44 & 7.54 & 10.41 \\  & 35 & 78.75 & 8.04 & 9.80 \\  & 45 & 80.46 & 9.08 & 8.86 \\  & Full (100) & 66.03 & 11.85 & 5.57 \\ \hline \multirow{4}{*}{GM} & 25 & 73.05 & 7.54 & 9.69 \\  & 35 & 75.17 & 8.04 & 9.35 \\  & 45 & 76.88 & 9.08 & 8.46 \\ \cline{1-1}  & Full (100) & 66.16 & 11.85 & 5.59 \\ \hline \hline \end{tabular} Abbreviations: WM, white matter; GM, gray matter; SNR, signal-to-noise ratio; STD, standard deviation.  
\end{table} 
Table 3: SNR depending on the undersampling ratio in gray and white matter regions. Figure 3: CSAM, CSAKS, and CSAKC images at different undersampling ratios in the brain phantom. Unlike CSAKC, linear and blurring artifacts can be seen with CSAM and CSAKS. Artifacts are most severe in the CSAM image.