## 2 Materials and Methods
### Gabor Filters-Based Feature Extraction Method  
Gabor filters (also known as Gabor wavelets) have properties similar to the human visual system, particularly for frequency and orientation representations. They are suitable for texture representation and discrimination. Gabor filters extract features directly from grayscale images using statistical information about character structures. However, in order to improve performance on low-quality images, the Gabor filter outputs are subjected to an adaptive sigmoid function [2,12-14]. A 2D Gabor filter is a complicated modulated sinusoidal function of a Gaussian kernel with a spatial response and frequency defined by Equations (1) and (2) (See Figure 5):  
\[h(x,y;\lambda,\phi,\sigma,\sigma)=\frac{1}{2\pi\sigma_{x}\sigma_{y}}exp\left\{ -\frac{1}{2}\left[\frac{R_{1}^{2}}{\sigma_{x}^{2}}+\frac{R_{2}^{2}}{\sigma_{y}^ {2}}\right]\right\}\times exp\left[i\cdot\frac{2\pi R_{1}}{\lambda}\right] \tag{1}\]  
where  
\[R_{1}=x\cos\phi+y\sin\phi\]  
\[R_{2}=-x\sin\phi+y\cos\phi\]  
\[H\left(u,v;\lambda,\phi,\sigma_{x},\sigma_{y}\right)=exp\left\{-2\pi^{2} \left(\sigma_{x}^{2}\left(F_{1}-\frac{1}{\lambda}\right)^{2}+\sigma_{y}^{2}(F_ {2})^{2}\right)\right\}\times C \tag{2}\]  
where  
\[F_{1}=u\cos\phi+v\sin\phi\]  
\[F_{2}=-u\sin\phi+v\cos\phi\]  
\[C=\text{constant}\]  
where \(\Delta x\) and \(\Delta y\) are denoted as the spatial localization of the Gabor filter that is computed with the help of spatial width, which is depicted in Equation (3). \[\left(\Delta x\right)^{2}=\frac{\int_{-\infty}^{+\infty}hh^{*}(R_{1})^{2}d(R_ {1})}{\int_{-\infty}^{+\infty}hh^{*}d(R_{1})} \tag{3}\]  
Figure 4: Structure of hybrid deep neural model. \[\Delta x=\sigma_{x}/\sqrt{2},\Delta y=\sigma_{y}/\sqrt{2} \tag{4}\]  
Distances between Gabor filters adjacent to an image are mentioned as spatial sampling intervals and are defined by \(\text{D}x\) and \(\text{D}y\), respectively. In order to avoid unintentional image data loss, the following relationships between effective spatial sampling interval and widths, as shown in Figures 5 and 6, must satisfy the condition shown in Equation (5):  
\[R_{1}=x\cos\phi+y\sin\phi\]0  
Spatial sampling intervals are critical parameters to consider when designing a Gabor filter. However, in previous studies [15, 16], it was not considered, resulting in poor performance and significant image detail loss. Gabor filter spatial-frequency localization can also be expressed using the efficient bandwidth measures \(\Delta v\) and \(\Delta u\). In order to accomplish this, Equation (3) is transformed into Equation (6):  
\[R_{1}=x\cos\phi+y\sin\phi\]1  
Depending on spatial-frequency bandwidth, another concept known as the orientation bandwidth can be obtained [15], as indicated in Figure 7b. \[ \begin{split}\Delta\theta&\approx 2\text{arcsin}(( \Delta v_{V}/2)/(1/\lambda))\\ &=2\text{arcsin}\Big{(}\lambda/\left(4\sqrt{2}\pi\sigma_{y} \right)\Big{)}\end{split}  \tag{7}\]  
In this study, we express spatial-frequency localization in 2D space in two different ways: line orientation selectivity and line-width selectivity, as depicted in Figure 7. During the analysis, \(h\left(x,y:\lambda,\phi,\sigma_{x},\sigma_{y}\right)\) is highly sensitive compared to the orientation \((\phi+\pi/2)\) with \(\lambda/2\). Figure 5: The proposed hybrid model steps. (**a**) spatial field viewpoint of a Gabor filter; (**b**) spatial frequency field. A feature extraction method based on Gabor filters is used to extract and locate initial features from the face region [17]. Gabor filters' main advantage is their resistance to translation, rotation, and scale. They also resist photometric disturbances such as lighting variations and image noise [2; 12; 18; 19; 20]. The Gabor filters' properties are extracted directly from grayscale photographs. As shown in Algorithm 1, a 2D Gabor filter is a Gaussian kernel controlled by a complex sinusoidal plane wave in the spatial domain.