## 3 Experimental Results and Analysis
### Verification of Proposed Model
#### 3.2.1 Experimental Results  
Parts of the robustness results are given in Tables 2-4.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Experiments** & **Watemark Length (\(L\))** & **Watemark Repeat Time (\(N\))** & **Attacking Strength Interval** & **Comparison Algorithm** \\ \hline Experiment 1 & 200 & 4 & \(1/800\) & Algorithm A [11] \\ Experiment 2 & 400 & 5 & \(1/2000\) & Algorithm B [8] \\ Experiment 3 & 300 & 6 & \(1/1800\) & Algorithm C [10] \\ \hline \hline \end{tabular}
\end{table}
Table 1: Experimental parameters (\(L=200\) and \(N=4\)).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**S (Attack Strength)** & \multicolumn{2}{c}{**\(R\) (Robustness Index)**} & \multicolumn{2}{c}{**\(S\) (Attack Strength)**} & \multicolumn{2}{c}{**\(R\) (Robustness Index)**} \\ \cline{2-5}
**5 (Attack Strength)** & **Proposed Model** & **Algorithm A** & **Strength** & **Proposed Model** & **Algorithm A** \\ \hline
0.2000 & 1 & 1 & 0.7300 & 0.3493 & 0.3630 \\
0.5000 & 1 & 1 & 0.7350 & 0.1895 & 0.1800 \\
0.6800 & 1 & 1 & 0.7400 & 0.0832 & 0.0670 \\
0.6900 & 0.9994 & 1 & 0.7450 & 0.0287 & 0.0310 \\
0.7000 & 0.9910 & 0.9860 & 0.7500 & 0.0076 &  
From Table 5, it is reasonable to conclude that errors between the proposed model and watermarking algorithms are small enough to be ignored. The maximum errors do not exceed 0.05, and the mean values of errors are no more than 0.0002, with standard deviations less than 0.005, indicating that the errors have kept a low and stable level.  
From the perspective of probability, the robustness result from a single procedure of watermark embedding and extraction can be regarded as a result of the Bernoulli Process. With the number of experiments increasing, the empirical robustness results will get closer to the theoretical ones, which are the expected probability of watermark has been successfully extracted. Thus, the experiments conducted above belong to the Monte Carlo method for estimating the theoretical robustness values. By using the Chernoff bound [43], if Pr represents the probability, \(m\) represents the number of experiments, \(R\) means the theoretical value, \(R\)' represents the empirical result, \(\varepsilon\) represents mean error ratio and \(\delta\) represents the probability of false positive, the approximation for \(R\) satisfies:  
\[\Pr(\left|R^{\prime}-R\right|\geq\varepsilon R)\leq\delta, \tag{11}\]  
when:  
\[m\geq\frac{3\ln\frac{2}{\delta}}{\varepsilon^{2}R}. \tag{12}\]  
Hence, the condition on \(m\) provides an \((\varepsilon,\delta)\)-approximation for \(R\). Noted that \(m\) varies by different robustness result \(R\). Taking \(S=0.6700\) and \(R=0.5360\) in Experiment 1 as an example, according to Table 2, if the approximation need to satisfy \(\varepsilon=(0.5530-0.5360)/0.5360=0.0317\) and \(\delta\) is set to 0.05 empirically, then the bound of \(m\) should be:  
\[m\geq 9279 \tag{13}\]  
by Equation (12). Apparently, the actual number of experiments in Experiment 1, which is 1000, is much smaller than the probabilistic bound of Equation (13). This situation demonstrates that the robustness results of the proposed method are close to empirical ones even of fewer experiments.  
According to the statistic results and the probabilistic bound analyzed above, it is easy to deduce that the proposed robustness computation model is able to reveal the watermarking algorithmic robustness against image cropping effectively and accurately.