## 3 Extracting the Frequency Domain Features of Printed Images
### Image Spatial Frequency Characteristics Based on Discrete Cosine Transform (DCT)  
Spatial frequency features reveal the distribution of various frequency components in printed samples, which helps evaluate image sharpness and detail fidelity. Contrast sensitivity is a critical visual attribute of the human visual system (HVS), where different sensitivities exist for various distortions in images. Hence, we utilize discrete cosine transform (DCT) on the luminance channel L of the image, leveraging the sub-band features derived from the image's DCT [24; 25; 26; 27] to emulate the contrast sensitivity function (CSF) within the HVS. Performing 4 \(\times\) 4 DCT and 8 \(\times\) 8 DCT transformations on the TID2013 dataset, we calculated the spatial frequency similarity and the SROCC indicator between MOSs. The results, as shown in Table 3, indicate that the 4 \(\times\) 4 transformation yields better results compared to the 8 \(\times\) 8 transformation. The computation complexity of 4 \(\times\) 4 DCT blocks is lower, and due to their smaller size, they can handle local changes and details in the image more accurately, especially in regions with rich, high-frequency details. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **4 \(\times\) 4 DCT** & **8 \(\times\) 8 DCT** \\ \hline Low frequency & 0.559 & 0.535 \\ Mid frequency & 0.595 & 0.510 \\ High frequency & 0.465 & 0.401 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: A comparison of the results for DCT variations at two scales. Figure 8: The texture information of a reference image after filtering using a Gabor filter bank in five scales. According to the research by [PERSON] [28], we partition the spatial frequency domain into regions with varying sensitivities to distortion. Three indicators of spatial frequency similarity are represented by comparing the contrast energy values of low-frequency (LF), mid-frequency (MF), and high-frequency (HF) regions within 4 \(\times\) 4 discrete cosine transform (DCT) blocks. Figure 9 illustrates the spatial contrast sensitivity function (CSF) within the LF, MF, and HF regions of the 4 \(\times\) 4 DCT blocks. The discrete cosine transforms of two-dimensional images are as follows:  
\[ \begin{split} F(u,\;v)=c(u)c(v)\sum\limits_{x=0}^{M-1}\sum \limits_{y=0}^{N-1}f(x,\;y)cos\frac{\pi(2x+1)u}{2M}cos\frac{\pi(2y+1)v}{2N},\\ u=0,\;1,2\ldots,\;\text{M}-1\;\;\;v=0,\;1,2\ldots,N-1,\end{split}  \tag{27}\]  
where  
\[c(u)=\left\{ \begin{array}{l}\frac{1}{\sqrt{N}},\;u=0\\ \sqrt{\frac{2}{N}},\;u=1,2\ldots,M-1\end{array} \right.\quad c(v)=\left\{ \begin{array}{l}\frac{1}{\sqrt{N}},v=0\\ \sqrt{\frac{2}{N}},v=1,2\ldots,N-1\end{array} \right. \tag{28}\]  
The key advantage of two-dimensional discrete cosine transform (DCT) lies in its ability to concentrate a significant portion of the image's energy within specific frequency ranges [25]. DCT transforms signals or images from the time domain (spatial domain) to the frequency domain [26]. This facilitates a better understanding of the frequency components present in the signal or image. In the frequency domain, one can observe the contributions of various frequency components in the signal or image, aiding in further analysis and processing. The contrast energy map extracted from the LF region of the image is defined as described in [25]:  
\[\varphi_{L}=\sum\limits_{(u,\;v)\in R_{L}}p^{2}(u,\;v), \tag{29}\]  
The energy calculation for the mid-frequency and high-frequency regions is computed as follows: \(p(u,\;v)\) represents the magnitude of the DCT coefficients at \((u,\;v)\) and \(R_{L}\) denotes the mid-frequency region.