# Fragile Watermarking of 3D Models in a Transformed Domain
## 6 Experimental Results  
This section summarizes the results of a number of experiments we executed to test the performance of the proposed algorithm. In order to measure the resulting quality, we considered four objective parameters, namely the sensitivity, the Mean Absolute Error (MAE), the Root Mean Squared Error (RMSE), and the Peak Signal-to-Noise Ratio (PSNR). Indeed, the sensitivity gives a measure of the capability of the algorithm to detect tampering attacks. Thus, the meanings of the parameters used to measure the performance of the proposed algorithm are:  
* Sensitivity (defined as in [24]): the sensitivity of level \(\pm\delta\) is defined as the fraction of vertices that are detected as tampered in the verification phase when one byte of each EU is modified by \(\pm\delta\); this is a very powerful fragility measure, in particular for small values of \(\delta\) as one or two, because it is actually the least tampering an attacker can perform: any other attack, such as noise addition, adding, removing, or shifting vertices or surfaces, etc., would modify the mantissa values of the vertex coordinates of more than just \(\pm\delta\);
* Mean Absolute Error (MAE): measures the average absolute difference between vertex attributes of the host and the watermarked model, \[\text{MAE}=\frac{\sum_{\mathbf{v}\in\mathbf{Y}}\sum_{\mathbf{c}\in\Omega}| \mathbf{v}_{c}-\mathbf{v}_{c^{\prime}}|}{\text{card}(\mathbf{Y})\text{ card}(\Omega)},\] (8) where \(\mathbf{v}\) scans the set Y of all vertices, \(c\) scans the set \(\Omega\) of the attributes that can be modified by the embedder (i.e., in the present embodiment \(x,y,z\)), the \({}^{\prime}\) symbol refers to the modified attribute in the watermarked model, and the function \(\text{card}()\) returns the cardinality of a set;
* Root Mean Squared Error (RMSE): measures the distortion between vertex coordinates of the host and the watermarked model, \[\text{RMSE}=\sqrt{\frac{\sum_{\mathbf{v}\in\mathbf{Y}}\|\mathbf{v}-\mathbf{v }^{\prime}\|^{2}}{\text{card}(\mathbf{Y})}},\] (9) where \(\mathbf{v}^{\prime}\) represent the modified vertices and the operator \(\|.\|\) computes the Euclidean distance;
* Peak Signal-to-Noise Ratio (PSNR): measures the distortion with respect to the maximum elongation of the vertices from the centroid of the model, \[\text{PSNR}=10\log_{10}\frac{\left(\max_{\mathbf{v}\in\mathbf{Y}}\|\mathbf{v}- \mathbf{c}\|\right)^{2}}{\frac{1}{\text{card}(\mathbf{Y})}\sum_{\mathbf{v}\in \mathbf{Y}}\|\mathbf{v}-\mathbf{v}^{\prime}\|^{2}},\] (10) where \(\mathbf{v}^{\prime}\) represent the modified vertices, \(\mathbf{c}\) is the centroid of the host model, and the operator \(\|.\|\) computes the Euclidean distance. We chose a setting for the genetic algorithm to be used in all experiments; in particular, we derived the setting values from many experiments devised to have fast convergence to a solution resulting in a high quality watermarked model. The settings were population size = 300, \(p_{c}=0.9\), \(p_{m}=0.25\), maximum generations = 2000. We ran the embedding algorithm on ten 3D models (shown in Figure 3), and the objective results are reported in Table 3. The payload in all experiments was set to 5 bpv. From Table 3, it can be seen that, given the very high values of sensitivity for small modifications (more than 99% of vertices detected when altering them by only \(\pm 2\) in one of the bytes of their coordinates), the proposed algorithm is highly reliable in tampering detection, i.e., it is very fragile. Moreover, the very small values of MAE and RMSE and the very high values of PSNR proved that the 3D models underwent a negligible modification, which may be generally accepted apart from specific applications, in which case we think that a reversible algorithm should be applied (but this is not the objective of the present proposal). As concerns the complexity of the proposed method, let us consider separately the preprocessing phase, the embedding phase, and the verification phase.