## 3 Proposed Method
### The Reasoning Process and Spoof Scores
Therefore, in this subsection, the probability density function \(G_{d}\), of the bona fide sample is expressed as  
\[G_{d_{r}}=\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-(x_{i}-\mu)^{2}/2\sigma^{2}}. \tag{17}\]  
The closer the test sample is to the bona fide sample, the higher the similarity between the high-dimensional data distribution of the test sample and \(G_{d_{r}}\). In this subsection, using the similarity between the distribution mean \(\mu\) and variance \(\sigma\) of the test sample and those of the bona fide sample, two types of anomaly scores are defined to determine the authenticity of the test sample. Equation (18) represents the determination of the spoof score based on the probability density function of the bona fide sample,  
\[SpoofScore_{G}=1-G_{d_{r}}(\frac{1}{N}\sum R_{test}) \tag{18}\]  
where SpoofScore\({}_{G}\) represents the probability that the test sample is a bona fide sample and \(R_{test}\) represents the input of the test sample. The smaller the value of SpoofScore\({}_{G}\), the greater the probability that the test sample is a bona fide sample. Therefore, when the probability is higher than a predetermined threshold, the model is detected as a spoof  
Figure 12: Density graph of pixel reconstruction distance and density graph of fused-feature reconstruction distance after Gaussian fitting. sample; otherwise, it is judged as a bona fide sample. Regarding the distribution mean \(\mu\) and variance \(\sigma\), another type of spoof score detection is designed in this section:  
\[\mu Score =\frac{|\mu_{test}-\mu_{min}|}{|\mu_{max}-\mu_{min}|} \tag{19}\] \[\sigma Score =\frac{|\sigma_{test}-\sigma_{min}|}{|\sigma_{max}-\sigma_{min}|}\]  
where \(\mu_{test}\) and \(\sigma_{test}\) are the mean and variance of the data distribution of the test samples, and they are compared with the predetermined thresholds. Samples exceeding the thresholds are judged as spoof samples. To normalize the spoof scores, this section uses a normalization method to convert the scores into probabilities and combines the mean and variance to detect spoof samples,  
\[SpoofScore_{\mu-\sigma}=1-S(\mu Score<\phi_{1})\cdot S(\sigma Score<\phi_{2}) \tag{20}\]  
where \(S(\cdot)\) is the sign function. \(\phi_{1}\) and \(\phi_{2}\) are predefined thresholds. When both \(\mu Score\) and \(\sigma Score\) are less than the thresholds, the test sample is judged as a spoof sample. This paper proposes a new classification constraint method. As shown in Figure 13a, represents the existing unsupervised anti-spoofing boundary discrimination method, which usually distinguishes spoof samples by constraining the reconstruction distance of bona fide samples. However, this method may cause some spoof samples with similar structures and too small reconstruction distances to be wrongly aligned with bona fide samples, and it is unable to constrain spoof samples, thus interfering with the anti-spoofing method. Therefore, as shown in Figure 13b, this chapter combines the spoof scores of SpoofScore\({}_{\mu-\sigma}\) and SpoofScore\({}_{G}\). By imposing data distribution constraints on the high-dimensional features of input samples, it can make the scattered spoof samples and bona fide samples align with different data distributions, respectively, thereby generating a clearer discrimination boundary in the input samples. Figure 13: Spoofness score determination constraint. (**a**) Reconstruction distance constraint. (**b**) Data distribution constraint.