# Bayesian Filtering Multi-Baseline Phase Unwrapping Method Based on a Two-Stage Programming Approach
## 3 Bayesian Filtering MBPU Method
### Multi-Baseline UIFPU Algorithm  
UIF is a newly proposed nonlinear filtering method based on the UKF. UIF improves the robustness of the UKF model by adding the information matrix and its corresponding information state vector to the UKF model. UIF performs deterministic sampling based on UT transformation, and uses a series of specified samples to approximate the posterior probability density of the system state. Therefore, TSPAUIF is more suitable to obtain the phase of complex noise regions. The sigma points can be obtained by the following formula:  
\[\left\{ \begin{array}{l}\xi_{0}(k-1)=\overset{\rightarrow}{\psi_{i}}(k-1) \\ \xi_{j}(k-1)=\overset{\rightarrow}{\psi_{i}}(k-1)+\left\{\sqrt{(n_{\psi}+ \lambda)P_{i_{\psi\psi}}(k-1)}\right\}_{j}\ \ \ \ \ j=1,2,\ldots,n_{\psi}\\ \xi_{j}(k-1)=\overset{\rightarrow}{\psi_{i}}(k-1)-\left\{\sqrt{(n_{\psi}+ \lambda)P_{i_{\psi\psi}}(k-1)}\right\}_{j}\ \ \ \ \ j=n_{\psi}+1,n_{\psi}+2,\ldots,2n_{\psi}\end{array} \right. \tag{17}\]  
where \(n_{\psi}\) is the dimension of the state variable (i.e., \(n_{\psi}=1\)); \(\lambda=\omega^{2}\big{(}n_{\psi}+\kappa\big{)}-n_{\psi}\), \(\omega\), \(l\), and \(\kappa\) are parameters used to adjust the sigma points. In this paper, \(\omega=0.01\), \(l=2\), and \(\kappa=0\). Therefore, we can obtain the one-step prediction equation as:\[\left\{ \begin{array}{l}\chi_{j}^{-}(k)=f_{i}\big{[}\xi_{j}(k-1)\big{]}\\ \vvet{\psi_{i}^{-}}(k)=\sum\limits_{i=0}^{2n_{\psi}}b_{j}^{m}\chi_{j}^{-}(k)\\ P_{i_{i\psi\psi}^{-}}^{-}(k)=\sum\limits_{i=0}^{2n_{\psi}}b_{j}^{m}\left[\chi_{ j}^{-}(k)-\vvet{\psi_{i}^{-}}(k)\right]\left[\chi_{j}^{-}(k)-\vvet{\psi_{i}^{-}}(k) \right]^{T}+Q_{i}(k-1)\end{array} \right. \tag{18}\]  
We let \(\varphi(k)\) and \(\hat{\varphi}(k)\) be the information matrix and its corresponding information state vector, respectively, at pixel \(k\). We can obtain \(\varphi(k)\) and \(\hat{\varphi}(k)\) by the following formula:  
\[\left\{ \begin{array}{l}\varphi(k){=}\left[P_{i_{\psi\psi}^{-}}^{-}(k)\right] ^{-1}\\ \hat{\varphi}(k)=\varphi(k)\vvet{\psi_{i}^{-}}(k)\end{array} \right. \tag{19}\]  
We can obtain \(\phi_{i}^{-}(k)\) and \(P_{i_{\phi\phi}}^{-}(k)\) using the following formula:  
\[\left\{ \begin{array}{l}\xi_{j}^{-}(k)=h_{i}\Big{[}\chi_{j}^{-}(k)\Big{]}\\ \phi_{i}^{-}(k)=\sum\limits_{j=0}^{2n_{\psi}}b_{j}^{m}\xi_{j}^{-}(k)\\ P_{i_{\phi\phi}}^{-}(k)=\sum\limits_{j=0}^{2n_{\psi}}b_{j}^{c}[\xi_{j}^{-}(k)- \phi_{i}^{-}(k)]\cdot[\xi_{j}^{-}(k)-\phi_{i}^{-}(k)]^{T}+R_{i}(k)\end{array} \right. \tag{20}\]  
where \(b_{j}^{m}\) and \(b_{j}^{c}\) are the weights. \(b_{0}^{m}=\lambda/(n_{\psi}+\lambda)\), \(b_{0}^{c}=\lambda/(n_{\psi}+\lambda)+(1-\omega^{2}+l)\) and \(b_{j}^{m}=b_{j}^{c}=1/2(n_{\psi}+\lambda)\) can obtain the weights. In this paper, \(j=1,2\). We assume that \(\varsigma(k)\) represents the information state contribution and \(\mathbf{N}(k)\) represents and the corresponding associated information matrix. Thus, we can obtain the following:  
\[\left\{ \begin{array}{l}\varsigma(k)=\varphi(k)P_{i_{\psi\phi}}^{-}(k)[R(k) ]^{-1}\cdot\left\{\phi_{i}(k)-\phi_{i}^{-}(k)+[P_{i_{\psi\phi}}^{-}(k)]^{T}[ \varphi(k)]^{T}\psi_{i}^{-}(k)\right\}\\ \mathbf{N}(k)=\varphi(k)P_{i_{\psi\phi}}^{-}(k)[R(k)]^{-1}\cdot[P_{i_{\phi\phi }}^{-}(k)]^{T}[\varphi(k)]^{T}\end{array} \right. \tag{21}\]  
Through the above analysis, we can obtain \(\psi_{i}(k)\) and \(P_{i_{\psi\phi}}(k)\) as follows:  
\[\left\{ \begin{array}{l}\dfrac{\partial(k)=\hat{\varphi}(k)+\varsigma(k)}{ \overline{\varphi}(k)=\varphi(k)+\mathbf{N}(k)}\\ P_{i_{\psi\phi}}(k)=\left[\overline{\varphi}(k)\right]^{-1}\\ \psi_{i}(k)=\partial(k)/\overline{\varphi}(k)\end{array} \right. \tag{22}\]  
The above three models are MBPU methods, based on the Bayesian filtering method proposed in this paper. All three methods can simultaneously perform PU and filtering. For the convenience of expression, the above equations are all expressed in a one-dimensional shape. We can replace \(k\) in the formula with \((m,n)\) to represent the two-dimensional method.