## 2 Extraction of Spatial Domain Features from Digital Print Images
### Gradient Features of Digital Print Images Based on the Sobel Filter  
In print quality assessment, gradient refers to the rate of change in color or brightness within an image. Gradient features reveal the sharpness of edges and details, as well as variations in texture. This information is essential for evaluating print quality. Image edges play a vital role in conveying visual information, with gradient features widely utilized in image quality assessment due to their ability to capture both edge structure and contrast variations effectively [12]. Various forms of gradient features have been integrated into image quality assessment metrics. For example, the Feature Similarity Index  
Figure 6: The impact of different values of \(C_{1}\) on the similarity of data chromaticity: **(a)** the SROCC varies with the parameter \(C_{1}\), **(b)** the rate of change of SROCC with respect to the parameter \(C_{1}\).  
(FSIM) incorporates gradient and phase consistency features to evaluate the local quality of distorted images [13], while the Directional Similarity Measure (DASM) combines gradient magnitude, anisotropy, and local orientation features [14; 15; 16; 17]. Commonly employed edge detection filters include those created by [PERSON], [PERSON], and [PERSON] [18]. The advantage of the Sobel operator lies in its robustness to noise and its relatively accurate computation of edge direction. In this study, the Sobel filter was chosen to compute the gradient of the luminance channel in both \(I_{o}\) and \(I_{s}\). The gradient magnitude calculation formulas for these images using the Sobel filter on the L channel are as follows:  
\[G_{o}(i)=\sqrt{\left(h_{x}*L_{o}(i)\right)^{2}+\left(h_{y}*L_{o}(i)\right)^{2}}, \tag{8}\]  
\[G_{s}(i)=\sqrt{\left(h_{x}*L_{s}(i)\right)^{2}+\left(h_{y}*L_{s}(i)\right)^{2}}, \tag{9}\]  
where \(G_{o}(i)\) and \(G_{s}(i)\) denote the gradient magnitude values at position index \(i\) of the pre-press digital image and the printed and scanned image, respectively. The symbol \(*\) indicates the convolution operation. \(h_{x}\) and \(h_{y}\) represent the horizontal and vertical Sobel filter templates, defined as follows:  
\[h_{x}=\frac{1}{4}\begin{bmatrix}1&0&-1\\ 2&0&-2\\ 1&0&-1\end{bmatrix},\quad\,h_{y}=\frac{1}{4}\begin{bmatrix}1&2&1\\ 0&0&0\\ -1&-2&-1\end{bmatrix}, \tag{10}\]  
The similarity between the gradient magnitudes of \(I_{o}\) and \(I_{s}\) is quantified using the standard deviation of the similarity:  
\[G(i)=\frac{2G_{o}(i)\times G_{s}(i)+C_{2}}{G_{o}^{2}(i)+G_{s}^{2}(i)+C_{2}}, \tag{11}\]  
\[S_{G}=\frac{1}{N}\sum_{i=1}^{N}G(i), \tag{12}\]  
In this context, \(C_{2}\) serves as a stabilizing constant within the fraction to prevent division by zero. The relationship strength between gradient similarity and the Mean Opinion Score (MOS) was assessed using the SROCC metric across various \(C_{2}\) values, as illustrated in Figure 7. A higher SROCC value approaching one signifies a stronger correlation between the two datasets. Notably, the maximum SROCC value is achieved when \(C_{2}\) is set to 66. Thus, for this study, \(C_{2}\) is determined to be 66. The red number and dot line indicate the positions of the inflection points of the curve, which are the turning points where the curve transitions from an upward trend to a downward trend.