# A Fragile Watermarking Scheme for Authenticity Verification of 3D Models in GLB Format
## 3 Materials and Methods
### Interference Measures  
Standard vertex attributes, such as position, normal vector, or texture coordinates, are vector values whose coordinates are real numbers. Therefore, most metrics, such as Peak Signal-to-Noise Ratio (PSNR), Signal-to-Noise Ratio (SNR), Root Mean Square Error (RMSE) or Hausdorff distance (HD), can be applied to each of these attributes. However, modifying the normal vector or texture coordinates does not affect the model's geometry but rather its visual layer. In this study, PSNR, SNR, RMSE and HD were used to evaluate the level of distortion introduced during the watermarking of 3D models.  
The selected metrics are among the most popular and widely used in the 3D watermarking domain. The application of PSNR, SNR, RMSE, and HD is sufficient to conduct a thorough evaluation of the presented method. Each metric measures a different aspect in the context of the introduced changes to the model.  
PSNR measures the distortion with respect to the maximum elongation of the vertices from the centroid of the model [28]. It is expressed in decibels (dB).  
\[PSNR=10\cdot\log_{10}\frac{\left(\max\limits_{\mathbf{v}\in\mathbf{Y}}\|\mathbf{ v}-\mathbf{c}\|\right)^{2}}{\frac{1}{card(\mathbf{Y})}\sum\limits_{\mathbf{v}\in \mathbf{Y}}\|\mathbf{v}-\mathbf{v}^{\prime}\|^{2}}, \tag{2}\]  
where \(\mathbf{v}^{\prime}\) is a vertex of the modified model corresponding to \(\mathbf{v}\), \(\mathbf{Y}\) is the set of vertices forming the original model, \(card(\mathbf{Y})\) is the cardinality of set \(\mathbf{Y}\), and \(\mathbf{c}\) is the centroid of the original model.  
SNR is a general measure of the ratio between the power of a signal and the power of background noise present in the signal. It is also commonly expressed in decibels (dB). In contrast to PSNR, SNR focuses on the total power of the signal and the total power of the noise. Mathematically, SNR is defined as [36]:  
\[SNR=10\cdot\log_{10}\frac{\sum\limits_{\mathbf{v}\in\mathbf{Y}}\|\mathbf{v}\|^ {2}}{\sum\limits_{\mathbf{v}\in\mathbf{Y}}\|\mathbf{v}-\mathbf{v}^{\prime}\| ^{2}}. \tag{3}\]  
In both PSNR and SNR, higher values indicate better performance, as lower noise levels (i.e., \(\sum\limits_{\mathbf{v}\in\mathbf{Y}}\|\mathbf{v}-\mathbf{v}^{\prime}\|\to 0\)) result in these metrics tending toward infinity.  
RMSE measures absolute error between the original and watermarked model.  
Figure 5: Scheme of the extraction algorithm.  
\[RMSE=\sqrt{\frac{\sum\limits_{\mathbf{v}\in\mathbf{Y}}\|\mathbf{v}-\mathbf{v}^{ \prime}\|^{2}}{card(\mathbf{Y})}}. \tag{4}\]  
Hausdorff distance (HD or \(d_{H}\)) measures how far two subsets of a metric space are from each other. In computer graphics, the Hausdorff distance is used to measure the difference between two different representations of the same 3D object [12; 37].  
Let \(S_{1}\), \(S_{2}\) denote the set of vertices (planes) of the modified and original models, respectively; then,  
\[d_{H}=\max{(E(S_{1},S_{2}),E(S_{2},S_{1}))}, \tag{5}\]  
where \(E(S_{x},S_{y})=\max\limits_{p\in S_{x}}e(p,S_{y})\) is the unilateral distance between two planes \(S_{x}\), \(S_{y}\), where \(e(p,S)=\min\limits_{p^{\prime}\in S}d(p,p^{\prime})\) is the distance between point \(p\) and plane \(S\), where \(d(\cdot,\cdot)\) is the Euclidean distance.