## 3 Extracting the Frequency Domain Features of Printed Images  
### Image Multiscale and Multiorientation Texture Features Based on Gabor Filters  
Texture features can capture the surface structure and details of printed samples, which are crucial for evaluating the fidelity and consistency of print quality. Texture reflects certain patterns of color and grayscale variations on the surface of an object, serving as effective information to distinguish objects with similar spectral properties but different spatial distribution characteristics. It is widely used in the extraction of image information. Multiscale texture analysis methods include wavelet transform, Gabor transform, pyramid decomposition, and contourlet transform, among others. In multiscale image analysis, the support region of the Gabor filter's main function has a rectangular structure, with its length and width varying with the scale. Therefore, it can optimally and sparsely describe the edges of an image and the approximate singular curves with the minimum number of coefficients [19]. Gabor filters can precisely adjust their response to specific orientations, making them highly effective in capturing directional information in images. Each Gabor filter is highly sensitive to local frequency components along a specific orientation. In contrast, wavelet transforms (such as Haar wavelets, Daubechies wavelets, etc.) typically have limited directional selectivity, mainly capturing horizontal, vertical, and diagonal orientations. Although extensions like 2D directional wavelets exist to provide more directional sensitivity, they are generally more complex than Gabor filters in terms of orientation control. Gabor filters are often used to simulate biological visual perception due to their similarity to the response characteristics of simple cells in the human visual system, especially in processing visual textures and edges [19; 20; 21; 22; 23]. While wavelet transforms excel in image compression and denoising, their biological relevance is typically not as strong as that of Gabor filters. Gabor transformation can effectively analyze grayscale variations in images at various scales and orientations. A two-dimensional Gabor filter consists of a series of bandpass filters with given orientation angles and spatial frequencies. For a given frequency and orientation angle, it is defined as follows:  
\[g(x,\,y,\,\lambda,\,\theta)=\frac{1}{2\pi\sigma_{x}\sigma_{y}}e^{\left[-\frac {1}{2}\left(\frac{x^{2}}{\sigma_{x}^{2}}+\frac{y^{2}}{\sigma_{y}^{2}}\right) \right]\times\left[i\left(\frac{2\pi\gamma}{\lambda}\right)\right]}, \tag{21}\]  
In the formula  
\[x=x_{0}\text{cos}\,\theta+y_{0}\text{sin}\,\theta, \tag{22}\]  
\[y=-x_{0}\text{sin}\,\theta+y_{0}\text{cos}\,\theta, \tag{23}\]  
\((x_{0},\,y_{0})\) is the spatial position coordinate of the image pixel. \(\lambda\) is the wavelength, which is the reciprocal of spatial frequency. \(\theta\) is the angle between the filter orientation and the \(y_{0}\) axis. \(\sigma_{x}\) and \(\sigma_{y}\) are the standard deviations of the Gaussian envelope in the \(x\) and \(y\) directions, calculated as follows:  
\[\sigma_{x}=\frac{\lambda}{\pi}\times\sqrt{\frac{\text{ln}\,2}{2}}\times\frac{ 2^{B_{w}}+1}{2^{B_{w}}-1}, \tag{24}\]  
\[\sigma_{y}=\frac{\sigma_{x}}{\gamma}, \tag{25}\]  
where \(B_{w}\) and \(\gamma\) are the passband width and the aspect ratio of the filter frequency, respectively. Here, \(g(x,\,y,\,\lambda,\,\theta)\) represents a complex matrix, wherein the real and imaginary components are convolved separately with the information from \(I_{o}\) and \(I_{s}\) on the L channel. Subsequently, the real and imaginary parts are combined, and the modulus is calculated. In Figure 8, we present an example of an original image extracted from the TID2013 database alongside the image's response after being filtered by a Gabor filter bank with five scales and four orientations.