## 3 Methods
### Calculation of Bending Energy Based on Thin-Plate Spline (TPS) Method in the Spatial Domain
Because the main energy is concentrated in the low-frequency range, which could interfere with the main frequency component of \(f_{0}\), logarithm transformation is applied to \(PWS_{I}(u,v)\), and we obtain a new matrix \(P(x,y)\), known as the power spectral image (PSI), which is expressed as  
\[P(x,y)=\log PWS_{I}(u,v). \tag{9}\]  
Figure 3c indicates the structure of tags on the PSI with a typical grid pattern for the normal liver or abnormal liver on the first frame of the MR tagging image before transformation. In Figure 4, the points on the u axis are the principal frequency of the tag (Region 4) and its high harmonic components in the transverse direction, whereas points on the \(v\) axis are the principal frequency of the tag (Region 2) and its high harmonic components in the longitudinal direction. We should note that the intersectional tags in the diagonal directions also appear periodically; therefore, other points (Region 1 and Region 3) are conspicuously plotted on the PSI as well, which makes the power spectral value of the tag appear equidistantly in the frequency domain. Their corresponding power spectra are plotted using all cases (normal or abnormal) to determine the most sensitive frequency component among Regions 1-4. To avoid the unwanted components of the power spectrum in the PSI that pertain to other organs or tissues, the liver region (bottom of Figure 5a,b) is manually segmented from the original MR tagging images (top of Figure 5a,b) beforehand. Note that the PSI has better quality with segmented liver images (bottom of Figure 5c,d) than with the original images (top of Figure 5c,d). With the deformation of the liver, the displacement of the tag differs between the normal and abnormal livers in the spatial domain, which will also result in differences in the power spectrum in the frequency domain. It is obvious that the movement of the tags from Frame 1 to Frame 9 in the longitudinal direction (Region 4 in Figure 4) is greater than that in the transverse direction; thus, the principal frequency \(f_{0}\) in the transverse direction (the right square in Figure 6a,b) is selected as a region of interest (ROI) for calculating the magnitude of the power spectrum. A larger deformation of the liver results in a larger difference in the power spectrum of the two ROIs. Therefore, the difference in the power spectral (DPS) values of the tag region between the first frame and the ninth frame of the MR tagging image in the 2D frequency domain is high on the normal liver (Figure 6a) and low on the abnormal liver (Figure 6b). In a \(5\times 5\) ROI, the power spectral value in the coordinate \((x,y)\), which has the maximum power spectral value \(p_{1}(x,y)\), is located in the first frame of the MR tagging image, whereas the power spectral value at the same coordinate of \((x,y)\) is \(p_{9}(x,y)\) in the ninth frame of the MR tagging image. The DPS value between \(p_{1}(x,y)\) and \(p_{9}(x,y)\) is determined by \(p_{1-9}(x,y)=p_{1}(x,y)-p_{9}(x,y)\). Figure 3: Discrete Fourier transform (DFT) of a 2D MR tagging image with its (**a**) original power spectrum; (**b**) frequency range replaced by a power spectrum; (**c**) power spectral image (PSI) after logarithm transformation. Figure 4: Four principal frequencies in four directions are marked as Regions 1–4 (**a**), and their corresponding power spectra are plotted using all the cases to determine the most sensitive frequency component (**b**) for differentiating normal cases (*****) from abnormal cases (**a**). Figure 5: Segmentation of the liver region from an original MR tagging image in Frame 1 (**a**) and the corresponding PSIs (**b**); (**c**,**d**) spatial and frequency images derived from Frame 9. Note that the PSIs have better contrast in the segmented liver images (bottom) than in the original images (top).