## 2 Preamble
### Extension of the Orthogonal Wavelet Representation to the Two-Dimensional Case  
In this section, we briefly develop the two-dimensional case for acoustic radiation images corresponding to squeal noise. More details can be found in [28; 29].  
We focus on the extension for a function in \(\mathbf{L}^{2}(\mathbb{R}^{2})\) for which the scalar product is given by  
\[\forall(f,g)\in\left(\mathbf{L}^{2}(\mathbb{R}^{2})\right)^{2},<f,g>=\iint_{ \mathbb{R}^{2}}f(x,y)g(x,y)dxdy \tag{16}\]  
For multiresolution approximations, each vector space \(\mathbf{V}_{2^{j}}\) can be decomposed as a tensor product of two identical sub-spaces of \(\mathbf{L}^{2}(\mathbb{R})\), such as \(\forall j\in\mathbb{Z}\), \(\mathbf{V}_{2^{j}}=V_{2^{j}}\otimes V_{2^{j}}\). Then the scaling function \(\Phi(x,y)\) whose dilatation and translation given an orthonormal basis of each space \(\mathbf{V}_{2^{j}}\), can be defined by \(\Phi(x,y)=\phi(x)\phi(y)\), where \(\phi(x)\) is the one-dimensional scaling function of the multiresolution approximation \(V_{2^{j}}\). The, the orthogonal basis of \(\mathbf{V}_{2^{j}}\) can be defined by  
\[\forall j\in\mathbb{Z},\ \left(\Phi_{j,kJ}(x,y)\right)_{(kJ)\in\mathbb{Z}^{2}}= \left(2^{-j}\phi_{2^{j}}(x-2^{-j}k)\phi_{2^{j}}(y-2^{-j}l)\right)_{(kJ)\in \mathbb{Z}^{2}}=\left(\varphi_{j,k}(x)\varphi_{j,l}(y)\right)_{(kJ)\in \mathbb{Z}^{2}} \tag{17}\]  
It should be noted that relation between the multiresolution approximation \(\mathbf{V}_{2^{j}}\) and \(\mathbf{V}_{2^{j+1}}\) can be easily defined by [28; 29]  
\[\mathbf{V}_{2^{j+1}}=\mathbf{V}_{2^{j}}\oplus\left(V_{2^{j}}\otimes O_{2^{j} }\right)\oplus\left(O_{2^{j}}\otimes V_{2^{j}}\right)\oplus\left(O_{2^{j}} \otimes O_{2^{j}}\right) \tag{18}\]  
where \(O_{2^{j}}\) defines the orthogonal complement of \(V_{2^{j}}\) in \(V_{2^{j+1}}\) as previously discussed in Section 3.2. As previously discussed in Section 3.2 for the one-dimensional case, the detail signal at the resolution \(2^{j}\) is still equal to the orthogonal projection of the signal on the orthogonal complement of \(\mathbf{V}_{2^{j}}\in\mathbf{V}_{2^{j+1}}\). [PERSON] [28; 29] demonstrated that this orthogonal complement \(\mathbf{O}_{2^{j}}\) can be defined by scaling and translating the three following wavelets functions  
\[\begin{array}{l}\Psi^{1}(x,y)=\phi(x)\psi(y)\\ \Psi^{2}(x,y)=\psi(x)\phi(y)\\ \Psi^{3}(x,y)=\psi(x)\psi(y)\end{array} \tag{19}\]  
where \(\phi(x)\) and \(\psi(y)\) are, respectively, the one-dimensional scaling function of the multiresolution approximation \(\mathbf{V}_{2^{j}}\) and \(\mathbf{O}_{2^{j}}\).  
Then, it is possible to define the operator \(\mathbf{A}_{2^{j}}f\), which approximates the function \(f(x,y)\) at a given resolution \(2^{j}\), such as  
\[\forall j\in\mathbb{Z},\ \mathbf{A}_{2^{j}}f=\left(<f(x,y),\Phi_{j,kJ}>\right)_{(kJ )\in\mathbb{Z}^{2}} \tag{20}\]The difference of information between \(\mathbf{A}_{2^{j+1}}f\) and \(\mathbf{A}_{2j}f\) can also by defined by the three following detailed images (in the three orthonormal directions)  
\[\forall j\in\mathbb{Z},\mathbf{D^{1}}_{2j}f=\left(<f(x,y),\mathbf{\Psi^{1}_{j,k }}>\right)_{(k,l)\in\mathbb{Z}^{2}} \tag{21}\]  
\[\forall j\in\mathbb{Z},\mathbf{D^{2}}_{2j}f=\left(<f(x,y),\mathbf{\Psi^{2}_{j,k }}>\right)_{(k,l)\in\mathbb{Z}^{2}} \tag{22}\]  
\[\forall j\in\mathbb{Z},\mathbf{D^{3}}_{2j}f=\left(<f(x,y),\mathbf{\Psi^{3}_{j,k }}>\right)_{(k,l)\in\mathbb{Z}^{2}} \tag{23}\]  
Finally, it can be concluded that an two-dimensional image is completely represented by the \(3J+1\) discrete images defined by the decomposition  
\[\mathcal{N}(f)_{J}=\left(\mathbf{A}_{2^{-j}}f,(\mathbf{D^{1}}_{2^{k}}f, \mathbf{D^{2}}_{2^{k}}f,\mathbf{D^{3}}_{2^{k}}f)_{-J\leq k\leq-1}\right) \tag{24}\]  
where \(J\) corresponds to the number of iterations for the decomposition of the original image. Figure 6 illustrates the decomposition process and the disposition of \(\mathbf{A}_{2^{-j}}f\), \(\mathbf{D^{1}}_{2^{k}}f\), \(\mathbf{D^{2}}_{2^{k}}f\) and \(\mathbf{D^{3}}_{2^{k}}f\) for three iterations.