# Collusion Attack Resilient 3D Mesh Watermarking Based on Anti-Collusion Fingerprint Code
## 3 Proposed Method
### Detector Response Modeling and Error Minimization  
To reflect the statistical characteristics of the actual noise and interference, the threshold \(\tau_{b}\) has to be controlled to minimize the bit error ratio. In this section, we model a distribution of the responses from the extracted bit, and determine \(\tau_{b}\) by analyzing the proposed model. The averaging attack of the anti-collusion code has the same effect as a bitwise AND operation of every bit position. Therefore, bit information '1' can survive during the averaging when '0' was not exist in the same bit position of the attacker's code. The probability of '1' from the code set \(\mathbf{N}\) is \(\frac{n-1}{n}\)[26], where \(n\) denotes the number of attackers covered by \(\mathbf{N}\). After the collusion with \(n\) attackers, the expected value of bit information '1' can be modeled as a binomial distribution as  
\[E(n)=\sum_{k=0}^{n}\binom{n}{k}\frac{1}{n}^{k}(1-\frac{1}{n})^{n-k}, \tag{13}\]  
where \(k\) is the number of the '1' bit in the same position of the attacker's code. To reflect the statistical characteristic of the actual noise and interference, the detector response \(\psi\) was modeled as a mixture of the Gaussian distribution. Similar to Equation (13), the embedded signal from Equation (4) is attenuated by the number of attackers (\(n\)). Therefore, the distribution of the attenuated signal of bit '1' can be modeled as  
\[\Psi(x|n,\sigma)=\sum_{k=0}^{n}\left(\binom{n}{k}\frac{1}{n}^{k}(1-\frac{1}{n })^{n-k}\cdot f(x|\frac{k}{n},\sigma^{2})\right), \tag{14}\]  
where \(f(x|\mu,\sigma^{2})\) denotes the probability density function of the Gaussian distribution, and \(\sigma\) is determined by the watermark-to-noise-ratio (WNR) \(\sigma=||\alpha||^{2}/\sigma_{d}^{2}\), where \(\sigma_{d}^{2}\) is the strength of the additive noise. Figure 2 shows simulated plots of the \(E(n)\), and \(\Psi(x|\mu,\sigma)\), where \(n=8\) and \(\sigma=0.1\). To verify the proposed model in Equation (14), we measured the detector response from 1000 sample models with the same parameters (\(n=8\), \(\sigma=0.1\)). As a result of the tests, Figure 2(a) shows a histogram of the measured responses that has structure very similar to that of the proposed model (Figure 2(b)). Using the model \(\Psi\), we determine the threshold \(\tau_{b}\) by minimizing the code extraction error as follows:  
\[\tau_{b}=\underset{t}{\text{minimize}}\left(e_{1}+e_{2}\right), \tag{15}\]  
where \(e_{1}\) and \(e_{2}\) are false positive error and false negative error, respectively. Each error probability was derived from \(\Psi\) as follows,  
\[e_{1}=\int_{0}^{t}\left(\frac{1}{n}\right)^{n}\cdot f(x|1,\sigma^{2})dx, \tag{16}\]  
\[e_{2}=\int_{t}^{\infty}\sum_{k=0}^{n-1}\left(\binom{n}{k}\frac{1}{n}\left(1- \frac{1}{n}\right)^{n-k}\cdot f(x|\frac{k}{n},\sigma^{2})\right)dx. \tag{17}\]  
From now on, we focus on mathematical derivation, in order to provide the threshold value \(\tau_{b}\) illustrated in Figure 3b. From Equation (15), \(\tau_{b}\) is value that makes the derivative of \(e_{1}+e_{2}\) by \(t\) to be zero. \[\frac{de_{1}}{dt}=\left(\frac{1}{n}\right)^{n}\cdot f(t|1,\sigma^{2}) \tag{18}\]  
Figure 3: (**a**) Histogram of the detector responses obtained from the 1000 sample models (\(n=8\), \(\sigma=0.1\)); and (**b**) obtained \(\tau_{b}\) using Equation (15) with various \(\sigma\), \(n\) values. \[\frac{de_{2}}{dt}=-\sum_{k=0}^{n-1}\left(\binom{n}{k}\frac{1}{n}^{k}(1-\frac{1}{n} )^{n-k}\cdot f(t|\frac{k}{n},\sigma^{2})\right) \tag{19}\]  
Substitute \(\binom{n}{k}\frac{1}{n}^{k}(1-\frac{1}{n})^{n-k}\) to \(\alpha_{k}\) for \(k=1\) to \(n\).