## 2 Background
### Multiresolution Wavelet Decomposition  
Multiresolution analysis applied to 3D meshes has been widely used in the literature since it guarantees a good tradeoff between the mesh complexity and the processing of the available resource [17]. It produces different representations of the 3D mesh starting from the low frequencies (coarse mesh) to a set of medium and high frequencies representing detailed information at different resolution levels. Each level represents the same 3D mesh but with a different complexity. The advantage of using multiresolution analysis in 3D watermarking lies in the fact that it offers several embedding locations and provides high robustness and imperceptibility requirements. In addition, such analysis renders the watermarking method useful for several applications. Indeed, the coarsest level is used to insert a robust watermark for copyright protection, a fragile watermark can be embedded into a dense mesh to ensure authentication, etc. Wavelet transformation is a common  
Figure 1: Three-dimensional meshes and their corresponding mesh saliency: (**left**) original 3D meshes; (**right**) 3D mesh saliency.  
tool for performing mesh multiresolution analysis. The mathematical formula of synthesis and wavelet analysis of 3D models was introduced by [PERSON] et al. [PERSON] _et al._ (2017). The principle of lazy wavelet decomposition for semi-regular triangular meshes is sketched in Figure 2. One iteration of lazy wavelet transform consists of merging each of the four triangles in one triangle at low-resolution level \(j+1\), i.e., three of the six initial vertices are conserved in the lower resolution and so on. The prediction errors for the deleted vertices \((v_{4}^{j},v_{5}^{j},v_{6}^{j})\) represent the wavelet coefficients \((W_{1}^{j+1},W_{2}^{j+1},W_{3}^{j+1})\). Note that such analysis can only be applied on a dense mesh with semi-regular connectivity. Figure 3 illustrates the wavelet decomposition of a bunny mesh. The multiresolution representation of the six vertices \((v_{1}^{j},v_{2}^{j},v_{3}^{j},v_{4}^{j},v_{5}^{j},v_{6}^{j})\) can be expressed as follows:  
\[V^{j+1}=A^{j+1}V^{j} \tag{6}\]  
\[W^{j+1}=B^{j+1}v^{j} \tag{7}\]  
where \(V^{j}=\left[v_{1}^{j},v_{2}^{j},\ldots,v_{k}^{j}\right]^{T}\) refers to the vertices coordinates at resolution level \(j\), \(k^{j}\) represents the number of vertices at level \(j\), and \(W^{j+1}=\left[w_{1}^{j+1},w_{2}^{j+1},\ldots,w_{t}^{j+1}\right]^{T}\) is the wavelet coefficient vector at resolution level \(j+1\). \(t^{j+1}\) is the number of wavelet coefficient vectors at resolution level \(j+1\), where \(t^{j+1}=k^{j}-k^{j+1}\). \(A^{j+1}\) is a non-square matrix that represents the triangle reduction by joining four triangles into one. Non-square matrix \(B^{j+1}\) produces the wavelet vectors that start from the midpoint of the edge in the lower resolution \(j+1\) and ends at the vertices that are deleted at the same level \(j+1\).  
Figure 3: Wavelet decomposition of the semi-regular Bunny mesh: (**a**) dense mesh; (**b**) after one wavelet decomposition; (**c**) after two wavelet decompositions; (**d**) coarsest representation of Bunny.  
Figure 2: Illustration of the lazy wavelet mechanism applied to a triangular semiregular mesh [PERSON] _et al._ (2017).