## 2 Background  
Let \(N\) denote the dimension of a SAR stack. Given a pixel \(p\), its observation sample vector can be expressed as:  
\[\mathbf{y}(p)=[y_{1}(p),y_{2}(p),...,y_{N}(p)]^{T} \tag{1}\]  
where \({}^{T}\) stays for matrix transposition. \(y_{n}(p)\) represents the complex reflectivity value associated with the \(n\)-th SAR image. Let \(M\) be the number of samples associated with \(p\). Based on the central limit theorem, the observation samples, \(\mathbf{y}=[\mathbf{y}(p_{1}),\mathbf{y}(p_{2})\ldots\mathbf{y}(p_{M})]\), follow a zero-mean \(N\)-variate complex circular Gaussian distribution. Therefore, the probability density function (PDF) of the data conditioned on interferometric phase can be expressed as [36]:  
\[p(\mathbf{y}|\mathbf{\theta})\propto\exp\Bigl{(}-\mathbf{y}^{H}\mathbf{C}^{-1} \mathbf{y}\Bigr{)} \tag{2}\]  
where \({}^{H}\) denotes the Hermitian conjugation, \(\mathbf{C}\) is the coherence matrix, and \(\mathbf{\theta}=[\theta_{1},\theta_{2},\)\(\ldots,\theta_{N}]^{T}\), where \(\theta_{n}\) is \(p\)'s \"true\" phase value with respect to the \(n\)-th SAR image [19]. In real applications, by assuming that the sample vectors are statistically independent, the coherence matrix can be estimated as follows:  
\[\mathbf{\hat{C}}=\frac{1}{M}\sum_{m=1}^{M}\mathbf{y}(p_{m})\mathbf{y}(p_{m})^{H} \tag{3}\]  
Allowing the decomposition of \(\mathbf{C}\) to its modulus and argument, \(\mathbf{C}\) can be further expressed as:  
\[\mathbf{C}=\left[\begin{array}{cccc}1&\gamma_{1,2}e^{i\hat{\mathbf{\theta}}_{1,2}}&...&\gamma_{1,N}e^{i\hat{\mathbf{\phi}}_{1,N}}\\ \gamma_{2,1}e^{i\hat{\mathbf{\phi}}_{2,1}}&1&...&\gamma_{2,N}e^{i\hat{\mathbf{\phi}}_ {2,N}}\\...&...&...&...\\ \gamma_{N,1}e^{i\hat{\mathbf{\phi}}_{N,1}}&\gamma_{N,2}e^{i\hat{\mathbf{\phi}}_{N,2} }&...&1\end{array}\right] \tag{4}\]  
where \(\gamma_{m,n}\) indicates an estimate of the coherence value on the interferometric pair with respect to the \(m\)-th and the \(n\)-th SAR observations, and \(\phi_{m,n}=\theta_{m}-\theta_{n}\) stands for the interferometric phase.  
MLE based approaches factorize the coherence matrix as:  
\[\mathbf{C}=\Theta|\mathbf{C}|\Theta^{H} \tag{5}\]  
where \(\Theta=\text{diag}\{\exp(j\mathbf{\theta})\}\). The maximum likelihood estimation of \(\Theta\) is obtained by minimizing the absolute value of its logarithm:  
\[\hat{\mathbf{\theta}}=\arg\min_{\mathbf{\theta}}\Bigl{\{}\Theta^{H}\Bigl{(}\bigl{|} \mathbf{\hat{C}}\bigr{|}^{-1}\circ\mathbf{\hat{C}}\Bigr{)}\Theta\Bigr{\}} \tag{6}\]  
where \(\circ\) represents the Hadamard product. [PERSON] et al. [19] proposed a Broyden-Fletcher-Goldfarb-Shanno (BFGS) based optimizer to resolve the above problem, which is the most widely used method in the MLE based phase reconstruction.  
The EVD approach proposes a factorization of \(\mathbf{C}\) to two full-rank complex matrices \(\mathbf{U}=\{\mathbf{u}_{1},\mathbf{u}_{2},...,\mathbf{u}_{N}\}\) and one diagonal real matrix \(\Lambda=\{\lambda_{1},\lambda_{2},...,\lambda_{N}\}\):  
\[\mathbf{\hat{C}}=\mathbf{U}\Lambda\mathbf{U}^{H} \tag{7}\]where \(\mathbf{U}\) is constructed with the orthogonal signal subspace \(\{\mathbf{u}_{1}\}\) and noise subspaces \(\{\mathbf{u}_{2},\mathbf{u}_{3},...,\mathbf{u}_{N}\}\) for a set of SAR sample vectors. According to the principle of beamforming method [37] in the SAR tomography, the eigen-decomposition based phase estimator can be expressed by an objective function [25]:  
\[\mathbf{u}_{1}=\arg\max_{\mathbf{u}_{1}}\Bigl{\{}\mathbf{u}_{1}^{H}\hat{ \mathbf{C}}\mathbf{u}_{1}\Bigr{\}} \tag{8}\]  
which is constrained by \(\mathbf{u}_{1}\mathbf{u}_{1}^{H}=1\). Comparing Equation (8) to Equation (6), the first eigenvalue-eigenvector pair \((\lambda_{1},\mathbf{u}_{1})\) provides the best approximation of \(\mathbf{C}\) under a single elementary scattering mechanism. The dominant interferometric signals on a given pixel are hence derived, which can be noted as:  
\[\mathbf{v}=\lambda_{1}^{1/2}\mathbf{u}_{1} \tag{9}\]