## 3 Extracting the Frequency Domain Features of Printed Images
### Image Spatial Frequency Characteristics Based on Discrete Cosine Transform (DCT)
Similarly, the energy calculation for the mid-frequency and high-frequency regions is computed as follows:  
\[\varphi_{M}=\sum\limits_{(u,\;v)\in R_{M}}p^{2}(u,\;v), \tag{30}\]  
\[\varphi_{H}=\sum\limits_{(u,\;v)\in R_{H}}p^{2}(u,\;v), \tag{31}\]  
Figure 9: The spatial CSF in LF, MF and HF regions for a 4 \(\times\) 4 DCT block (adapted from [29]): (**a**) the classification of frequency regions depending on the CSF obtained in [29], (**b**) the three frequency regions in a 4 \(\times\) 4 DCT block. Similar to the SSIM, the similarity index takes the form of \(\left(2\cdot ab+c\right)/\left(a^{2}+b^{2}+c\right)\). Therefore, the similarity index between \(I_{o}\) and \(I_{s}\) in the low-frequency domain is defined as follows:  
\[S_{L}=\frac{1}{n}\sum_{\left(u,\,v\right)\in R_{L}}\frac{2\varphi_{oL}\cdot \varphi_{sL}+C_{6}}{\varphi_{oL}{}^{2}+\varphi_{sL}{}^{2}+C_{6}}, \tag{32}\]  
where \(\varphi_{oL}\) and \(\varphi_{sL}\) represent the contrast energy maps of \(I_{o}\) and \(I_{s}\) in the low-frequency domain, respectively. \(n\) denotes the number of \(4\times 4\) pixel blocks into which the image is divided, and \(C_{6}\) is a constant that is used to maintain stability. Similarly, the similarity in the mid-frequency and high-frequency domains is given as follows:  
\[S_{M}=\frac{1}{n}\sum_{\left(u,\,v\right)\in R_{M}}\frac{2\varphi_{oM}\cdot \varphi_{sM}+C_{7}}{\varphi_{oM}{}^{2}+\varphi_{sM}{}^{2}+C_{7}}, \tag{33}\]  
\[S_{H}=\frac{1}{n}\sum_{\left(u,\,v\right)\in R_{H}}\frac{2\varphi_{oH}\cdot \varphi_{sH}+C_{8}}{\varphi_{oH}{}^{2}+\varphi_{sH}{}^{2}+C_{8}}, \tag{34}\]  
where \(\varphi_{oM}\) and \(\varphi_{sM}\), respectively, represent the contrast energy maps of \(I_{o}\) and \(I_{s}\) in the MF region. \(\varphi_{oH}\) and \(\varphi_{sH}\), respectively, represent the contrast energy maps of \(I_{o}\) and \(I_{s}\) in the HF region. \(n\) denotes the number of \(4\times 4\) pixel blocks into which the image is divided. \(C_{7}\) and \(C_{8}\) are two positive invariants controlling numerical stability. To determine the appropriate invariant constants, a series of tests were conducted. Various values of \(C_{6}\), \(C_{7}\), and \(C_{8}\) were examined, with the SROCC used as the metric, to evaluate the relationship between the spatial frequency similarity and the MOS. As shown in Figure 10, the blue curve represents the SROCC under different invariant constant values, while the red curve indicates the rate of change of the SROCC. Figure 10a,d represent the low-frequency region, Figure 10b,e represent the mid-frequency region, and Figure 10c,f represent the high-frequency region. The results indicate that with the increase in the invariant constant, the SROCC shows an upward trend in all three regions, but the growth rate gradually decreases. This suggests that excessively high values of the invariant constant may not significantly improve performance. Considering all factors, it was concluded that the optimal values are \(C_{6}=0.03\), \(C_{7}=0.04\), and \(C_{8}=0.03\). Figure 10: The impact of different invariant constants on spatial frequency similarity: (**a**) the SROCC in the low-frequency region, (**b**) the SROCC in the mid-frequency region, (**c**) the SROCC in the high-frequency region, (**d**) the growth rate of the SROCC in the low-frequency region, (**e**) the growth rate of the SROCC in the mid-frequency region, and (**f**) the growth rate of the SROCC in the high-frequency region.