# Screen-Cam Robust Image Watermarking with Feature-Based Synchronization
## 4 Parameter Settings
### The Selection of the Threshold for Watermark Detection  
The selection of the threshold \(T\) determines the false-positive rate and the true-positive rate. \(T\) needs to be set low enough to ensure that the watermark can be detected from watermarked LSFRs and high enough to ensure that the watermark cannot be detected from unwatermarked LSFRs.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \(k_{1}\) & **4.5** & **5** & **5.5** & **6** & **6.5** & **7** & **7.5** & **8** & **8.5** & **9** \\ \hline Side length between 240 and 300 & 3.78 & 3.62 & 3.53 & 0.88 & 0.88 & 1.32 & 1.77 & 1.77 & 0.88 & 0.88 \\ Side length over 300 & 0.00 & 0.00 & 2.65 & 2.65 & 2.20 & 1.77 & 1.77 & 1.77 & 1.77 \\ Total & 3.78 & 3.62 & 3.53 & 3.53 & 3.53 & 3.52 & 3.53 & 3.53 & 2.65 & 2.65 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Average number of constructed LSFRs with a different \(k_{1}\).  
Figure 12: Erroneous bits corresponding to different message extraction thresholds.  
Messages extracted from an unwatermarked image can be considered as independent random variables [43]. Therefore, the probability that a single bit match is 0.5. The relationship between the false-positive rate of single detection \(P_{f}\) and the threshold \(T\) is:  
\[P_{f}=\sum_{i=1-T}^{l}\left(0.5\right)^{l}\cdot\left(\frac{l!}{i!(l-i)!}\right) \tag{19}\]  
As mentioned in Section 3.3.3, each LSFR will be iteratively detected at different radii and angles. The maximum number of iterations is 231 times. Suppose we complete all iterative detection, the false-positive rate of the detection of one LSFR \({P^{\prime}}_{f}\) is:  
\[{P^{\prime}}_{f}=1-\left(1-P_{f}\right)^{231} \tag{20}\]  
The false-positive rate curve with different thresholds is shown in Figure 12(a). In order to choose an appropriate threshold, we further analyzed the influence of different secret keys and different host images on the positive detection rate. The eight host images were all embedded with other three different keys: \(K_{2}\), \(K_{3}\), \(K_{4}\). Each watermarked image was captured 24 times with different shooting conditions. The experimental setting is the same as in Section 4.3.  
Based on the 768 detection results of watermarked images with four different keys, eight different host images, and 24 different shooting conditions, we can calculate the true-positive rate with different thresholds. The true-positive rate curves between different keys and different host images are shown in Figure 12(b),c. The true-positive rate can be seen to be stable for different embedding messages. However, different images have different variations during screen-cam, so the true-positive rate is also different when \(T\) is below 10.  
According to the result, we set threshold \(T\) to 8, which means when the number of erroneous bits is below 8, the detection is successful. According to Formula 20, the false-positive rate of the detection of one LSFR is \(8.86\times 10^{-8}\). The true-positive rate with \(K_{1}\), \(K_{2}\), \(K_{3}\) and \(K_{4}\) is \(98.44\%\), \(96.88\%\), \(97.40\%\), and \(96.35\%\), respectively. Furthermore, the true-positive rate of the eight host images is \(100\%\), \(92.71\%\), \(97.92\%\), \(100\%\), \(94.79\%\), \(100\%\), \(93.75\%\), and \(98.96\%\), respectively.