# Mueller Matrix Decomposition and Image for Non-Destructive Testing of UAVs Skin
## 2 Principle
\begin{bmatrix}\sin\phi\sin\alpha e^{i(\delta_{1}-\sigma)} \\ -\cos\phi\sin\beta e^{i\delta_{2}}-\sin\phi\cos\beta\cos\alpha e^{i(\delta_{2}- \sigma)}\\ \cos\phi\cos\beta e^{i\delta_{3}}-\sin\phi\sin\beta e^{i(\delta_{3}-\sigma)} \end{bmatrix} ,\]  
where the rank-3 depolarizer is built around a polarized system with two angles and \(\lambda_{i}(i=2,3)\). Similar to Equation (6), \(U_{3}\) is expressed as:  
\[V_{3}U_{3}= \begin{bmatrix}1&0^{\dagger}\\ 0&U_{2}\end{bmatrix} \Rightarrow U_{3}=V_{3R}^{\dagger} \begin{bmatrix}1&0^{ \dagger}\\ 0&U_{2}\end{bmatrix} ,\ U_{2}= \begin{bmatrix}\cos\phi&\sin\phi e^{i\epsilon}\\ \sin\phi e^{i\epsilon}&\cos\phi\end{bmatrix}  \tag{8}\]  
The unitary matrix \(V_{3}\) also has a diagonal phase shift matrix with the form \(D_{3}=diag(e^{i\theta_{1}},e^{i\theta_{2}},e^{i\theta_{3}})\), which adds three degrees of freedom (DOF). However, if we further consider that the determinant of the unitary matrix is 1, we are only interested in the depolarization part of the Hermitian matrix and its conjugate transpose operation, and \(D_{3}\) is always hidden in the construction of the depolarized space. According to the group theory and the Cloude method [22], the polarized parameters are reduced to three parameters \(\lambda_{1}\), \(\phi\), \(\epsilon\), and the depolarization parameters are reduced to three eigenvalues \(\lambda_{i}(i=2,3,4)\). For the Mueller matrices of multiple scattering materials, the depolarization spectral lines can be represented by eigenvalues without considering eigenvectors. Thus, the rank-4 depolarizer is built around a polarized system with only three eigenvalues. Then, the coherency matrix is decomposed as:  
\[T=(1-\frac{\lambda_{2}+\lambda_{3}+\lambda_{4}}{3})U_{4}(\theta, \epsilon)diag(\lambda_{1},0,0,0)U_{4}(\theta,\epsilon)^{\dagger}+diag(\frac{ \lambda_{2}+\lambda_{3}+\lambda_{4}}{3},\lambda_{2},\lambda_{3},\lambda_{4}), \tag{9}\] \[U_{4}(\theta,\epsilon)= \begin{bmatrix}\cos\theta&-\sin\theta e ^{\delta}&0&0\\ \sin\theta e^{\delta}&\cos\theta&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix} ,\]  
where \(U_{4}(\theta,\delta)\) represents the polarization eigenvectors without depolarization parameters, and \(\theta\) and \(\delta\) are the polarization parameters of scattering related to the material composition, particle size, phase angle, etc. [23]. Depolarization induced by multiple scattering is only related to \(\lambda_{i}(i=2,3,4)\). The Mueller matrix is decomposed as:  
\[M=(1-d) \begin{bmatrix}1&-\cos 2\varphi&0&0\\ -\cos 2\varphi&1&0&0\\ 0&0&\sin 2\varphi\cos\Delta&\sin 2\varphi\sin\Delta\\ 0&0&-\sin 2\varphi\sin\Delta&\sin 2\varphi\cos\Delta\end{bmatrix} +M_{d}, \tag{10}\] \[M_{d}=diag(d,\lambda_{2}-\frac{d}{2},\lambda_{3}-\frac{d}{2}, \lambda_{4}-\frac{d}{2})\]  
where \(\varphi=\cos^{-1}(\sin 2\theta\cos\delta)/2\), \(\Delta=cos^{-1}(\tan^{-1}2\theta)\), and \(d=2(\lambda_{2}+\lambda_{3}+\lambda_{4})/3\). \(d\) represents the total degree of depolarization [24]. The first term on the right side of the equation represents the Mueller-Jones matrix related to the ideal reflector, and the second one represents the depolarization matrix related to multiple scattering. From Equation (10), the depolarization is completely caused by the non-coherent superposition of multiple scattering. When the depolarization under different polarized states is uniform, \(\lambda_{i}(i=2,3,4)\)are equal, and then \(M_{d}\) is transformed into the complete depolarizer with the form of \(diag(1,0,0,0)\). In such case, the given method in the work is in agreement with the methods reported in the optical literatures [10,24]. In general, \(\lambda_{i}(i=2,3,4)\) are not equal, and then the relative error of \(\lambda_{i}(i=2,3,4)\) related to multiple scattering is expressed as:  
\[\eta_{i}=\frac{2\lambda_{i}-d}{d} \tag{11}\]  
When \(\lambda_{i}(i=2,3,4)\) are not equal, the multiple scattering has a non-uniform effect on the depolarization in the Mueller matrix. The trace of the depolarization matrix in this work is \(d\), and the sum of \(\eta_{i}(i=2,3,4)\) is equal to 0.