# Mueller Matrix Decomposition and Image for Non-Destructive Testing of UAVs Skin
## 2 Principle  
In the Cloude decomposition, the coherency matrix corresponding to the Mueller matrix is given by [8]:  
\[T=\frac{1}{2}\sum_{ij,j=1}^{4}M_{ij}(\sigma_{i}\otimes\sigma_{j}^{*}), \tag{1}\]where \(\sigma_{i,j}(i,j=1,2,3,4)\) denote Pauli matrices, and \(M_{ij}(i,j=1,2,3,4)\) denote Mueller matrix elements. Any Mueller matrix could be decomposed into up to four non-depolarizing Mueller matrices. The coherency matrix could be decomposed as [8,9]:  
\[T=U_{4} \begin{bmatrix}\lambda_{1}&0&0&0\\ 0&\lambda_{2}&0&0\\ 0&0&\lambda_{3}&0\\ 0&0&0&\lambda_{4}\end{bmatrix} U_{4}^{\dagger}=\lambda_{1}e_{1}e_{1}^{\dagger}+ \lambda_{2}e_{2}e_{2}^{\dagger}+\lambda_{3}e_{3}e_{3}^{\ddagger}+\lambda_{4}e _{4}e_{4}^{\dagger}, \tag{2}\]  
where \(\lambda_{i}(i=1,2,3,4)\) denote eigenvalues with \(\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\geq\lambda_{4}\geq 0\), the symbol \(\dagger\) denotes the transpose conjugate operation of a matrix, \(U_{4}\) is an eigenvector matrix, and \(e_{i}(i=1,2,3,4)\) represent the eigenvectors. The relationship between the Mueller matrix and the coherency matrix is given by:  
\[M_{ij}=\frac{1}{2}tr(T\mu_{4i-j+4}), \tag{3}\]  
where \(\mu_{4i-j+4}(i,j=1,2,3,4)\) are calculated with Pauli matrices. The first step in constructing the depolarized space is to generate a 4 \(\times\) 4 unitary matrix as follows [22]:  
\[ \begin{split} V_{4}&=\left[\begin{array}{cccc}\cos \alpha&\sin\alpha&0&0\\ -\sin a&\cos\alpha&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array}\right]\cdot\left[\begin{array}{cccc}1&0&0&0\\ 0&\cos\beta&\sin\beta&0\\ 0&-\sin\beta&\cos\beta&0\\ 0&0&0&1\end{array}\right]\cdot\left[\begin{array}{cccc}1&0&0&0\\ 0&1&0&0\\ 0&0&\cos\gamma&\sin\gamma\\ 0&0&-\sin\gamma&\cos\gamma\end{array}\right],\\ \cdot\left[\begin{array}{cccc}e^{-i\delta_{1}}&0&0&0\\ 0&e^{-i\delta_{2}}&0&0\\ 0&0&e^{-i\delta_{3}}&0\\ 0&0&0&e^{-i\delta_{4}}\end{array}\right]\end{split}  \tag{4}\]  
where \(\alpha\) represents an internal degree of freedom, \(\beta\) represents the physical rotation of the sensor coordinates,\(\gamma\) represents the asymmetry of the Jones matrix, and \(\delta_{i}(i=1,2,3,4)\) represent the phase angles. After matrix multiplication calculation is transformed as:  
\[ \begin{split} V_{4}&=\begin{bmatrix}e_{1}^{\dagger}\\ e_{2}^{\dagger}\\ e_{3}^{\dagger}\\ e_{4}^{\dagger}\end{bmatrix}\Rightarrow V_{4}^{\dagger}=[e_{1}&e_{2}&e_{3}&e_{4}], \\ e_{1}&=\begin{bmatrix}\cos\alpha e^{i\delta_{1}}\\ \cos\beta\sin\alpha e^{i\delta_{2}}\\ \cos\gamma\sin\beta\sin\alpha e^{i\delta_{3}}\\ \sin\gamma\sin\beta\cos\alpha e^{i\delta_{4}}\end{bmatrix},\;e_{2}=\begin{bmatrix} -\sin\alpha e^{i\delta_{1}}\\ \cos\beta\sin\alpha e^{i\delta_{2}}\\ \cos\gamma\sin\beta\cos\alpha e^{i\delta_{3}}\\ \sin\gamma\sin\beta\cos\alpha e^{i\delta_{4}}\end{bmatrix},\\ e_{3}&=\begin{bmatrix}0\\ -\sin\beta e^{i\delta_{2}}\\ \cos\gamma\cos\beta e^{i\delta_{3}}\\ \sin\gamma\cos\beta e^{i\delta_{4}}\end{bmatrix},\;e_{4}=\begin{bmatrix}0\\ 0\\ -\sin\gamma e^{i\delta_{3}}\\ \cos\gamma e^{i\delta_{4}}\end{bmatrix}\end{split}  \tag{5}\]  
Through matrix transformation [21], \(V_{4}\) can be transformed into a lower-order matrix:  
\[V_{4R}U_{4}= \begin{bmatrix}1&0^{t}\\ 0&U_{3}\end{bmatrix} \Rightarrow U_{4}=V_{4R}^{\dagger} \begin{bmatrix}1&0^{t}\\ 0&U_{3}\end{bmatrix} , \tag{6}\]where \(V_{4R}\) represents the unitary matrix of the system, and \(U_{3}\) represents the depolarization transformation. Similar to Equation (2), the coherency matrix is given by:  
\[T_{3}=U_{3}(\phi,\sigma)diag(\lambda_{1},\lambda_{2},\lambda_{3})U_{3}(\phi, \sigma)^{\dagger}, \tag{7}\] \[U_{3}(\phi,\sigma)= \begin{bmatrix}e_{1}&e_{2}&e_{3},\\ \cos\beta\sin\alpha e^{i\delta_{1}}\\ \sin\beta\sin\alpha e^{i\delta_{3}}\end{bmatrix} ,\ e_{2}= \begin{bmatrix}-\cos \phi\sin\alpha e^{i\delta_{1}}\\ \cos\phi\cos\beta\cos\alpha e^{i\delta_{2}}-\sin\phi\sin\beta e^{i(\delta_{2}+ \sigma)}\\ \cos\phi\sin\beta\cos\alpha e^{i\delta_{3}}+\sin\phi\sin\beta e^{i(\delta_{3}+ \sigma)}\end{bmatrix} \] \[e_{3}=