## 3 Materials and Methods
### Lightweight U-Net Model Design with Embedded Gabor Kernel
#### 3.2.2 Embedding of Gabor Prior Knowledge  
The two-dimensional Gabor filter shares some similarities with the biological visual system, such as its response to visual signals of varying directions and frequencies. Therefore, Gabor filters exhibit a favorable performance in simulating the characteristics of the biological visual system. The image processed by its kernel function is highly similar to what is observed by higher animals [27]. Using the Gabor filter as a bandpass filter can eliminate noise while preserving the true structure information of ridges and valleys. Additionally, some trained shallow convolutional kernels in the neural network are similar to the Gabor filter [28; 29]. In order to extract richer image features, we embedded a Gabor layer in the first layer of the network. The Gabor kernel function is given by the following formula:  
\[\mathrm{Gabor}=\mathrm{exp}\Bigg{(}-\frac{\mathrm{x^{\prime 2}+\gamma^{2}2 \mathrm{y^{\prime 2}}}}{2\sigma^{2}}\Bigg{)}\mathrm{exp}\bigg{(}\mathrm{i}*\bigg{(} 2\pi\frac{\mathrm{x^{\prime}}}{\lambda}+\psi\bigg{)}\Bigg{)} \tag{1}\]  
\[\mathrm{x^{\prime}}=\mathrm{xcos}\theta+\mathrm{ysin}\theta \tag{2}\]  
\[\mathrm{y^{\prime}}=-\mathrm{xsin}\theta+\mathrm{ycos}\theta \tag{3}\]  
In this context, \"1\" denotes the imaginary unit, \"\(\psi\)\" signifies the phase parameter of the cosine function in the Gabor kernel, \"\(\theta\)\" represents the orientation of the parallel stripes in the Gabor filter kernel, \"\(\lambda\)\" stands for the frequency of the grayscale variation in the image, and \"\(\sigma\)\" denotes the standard deviation of the Gaussian factor in the Gabor function, which is related to \"\(\lambda\)\", as follows:  
\[\frac{\sigma}{\lambda}=\frac{1}{\pi}\sqrt{\frac{ln^{2}}{2}\bigg{(}\frac{2^{b}+ 1}{2^{b}-1}\bigg{)}} \tag{4}\]  
The DC component represents the mean brightness value of the image and does not contain any frequency information. Removing the DC component in a Gabor filter can effectively reduce the influence of illumination or image background brightness. In the near-infrared light spectrum, there is little difference in the absorption rate of near-infrared light between the vein region and the non-vein regions, resulting in severe contrast degradation and blurred vein edges. In the image, the non-vein area acting as the background is relatively large. Subtracting the DC component is equivalent to performing a differential operation on the background, which enhances the image contrast and makes the vein texture more prominent. As for filters, it is best to keep the mean at 0 to maintain the original image intensity. We embedded the Gabor filter without the DC component as a convolution kernel in the shallow network, enhancing the features of the image and helping the deeper network to extract semantic information [30]. The formula for the Gabor convolution kernel without a DC component is as follows:  
\[\mathrm{Gabor}=\mathrm{exp}\Bigg{(}-\pi\frac{\mathrm{x^{\prime 2}+\gamma^{2}2 \mathrm{y^{\prime 2}}}}{2\sigma^{2}}\Bigg{)}\bigg{[}\mathrm{exp}\bigg{(} \mathrm{i}*\bigg{(}2\pi\frac{\mathrm{x^{\prime}}}{\lambda}+\psi\bigg{)}\bigg{)} -\mathrm{exp}\bigg{(}-\pi\frac{\sigma^{2}}{\lambda^{2}}+\mathrm{i}*\psi\bigg{)} \bigg{]} \tag{5}\]