# Bayesian Filtering Multi-Baseline Phase Unwrapping Method Based on a Two-Stage Programming Approach
## 1 Introduction  
Interferometric synthetic aperture radar (InSAR) has become a valuable and important method for digital elevation models (DEMs) production [1; 2; 3; 4]. Phase unwrapping (PU), as a difficulty in the research of InSAR data processing, has received widespread attention in recent decades [5; 6; 7]. Traditional single-baseline PU (SBPU) obtains the unwrapped results based on a phase continuity assumption (PCA). As shown in Equation (1), the gradients of adjacent pixels are limited to \(\pi\). Therefore, due to the limitation of the PCA, SBPU cannot obtain better results in regions with largegradient changes [8; 9]. Compared to SBPU, the multi-baseline (MB) PU method will increase the gradient ambiguity coefficient from \([-1,1]\) to \([-n,n]\), so that the MBPU method can get rid of the PCA limitation [10; 11]. Therefore, the method can obtain ideal results, even in areas with terrain exhibiting phase layover and phase undersampling, which has caused MBPU to receive increasing attention in recent years [12; 13; 14]. \[\hat{\Delta}_{\psi}(k,k-1)= \left\{ \begin{array}{ll}\varphi(k)-\varphi(k-1)&\left|\varphi(k) -\varphi(k-1)\right|\leq\pi\\ \varphi(k)-\varphi(k-1)-2\pi&\varphi(k)-\varphi(k-1)>\pi\\ \varphi(k)-\varphi(k-1)+2\pi&\varphi(k)-\varphi(k-1)<-\pi\end{array} \right. \tag{1}\]  
where \(\hat{\Delta}_{\psi}(k,k-1)\) is the phase gradient between neighboring pixels. In recent years, more and more MBPU methods have been proposed. These can be divided into two categories. One of methods requires parameter phase estimation, which mainly includes maximum likelihood (ML) estimation [15] and maximum a posterior estimation (MAP) [16]. This type of method solves MB interferograms according to the probability density function, and can obtain the final DEM without PU. However, the noise robustness of the ML method is poor, and the method is extremely susceptible to noise. A small amount of noise will cause an obvious \"burr\" phenomenon in the unwrapped results, and it requires a sufficient number of high-quality interferograms to obtain satisfactory noise robustness. The MAP method is obtained by establishing a Markov random field, based on the ML method. The MAP method can obtain more accurate DEM reconstruction results, but its reconstruction time is longer, and the optimization steps are more complicated. These shortcomings have limited the application of the ML and MAP methods, and improved methods have been proposed [17; 18]. For example, [PERSON] [19] proposed an efficient ML estimation approach of MB SAR interferometry for refined topographic mapping in mountainous areas. The Chinese remainder theorem (CRT) is a representative of the second category of MBPU method [20]. The second category of method is different from the first type, in that it does not require parameter phase estimation. The CRT method can accurately obtain the MBPU results in the absence of noise, but noise is inevitable in real data. To solve this issue, a closed-form CRT-based algorithm was proposed [21]. The MBPU problem was transformed into a cluster-analysis problem, which reduced the shortcomings of the MB unwrapping algorithm with poor noise robustness [10; 22]. [PERSON] [13] proposed a recursive MBPU method that analyzes the relationship between the long-baseline and the short-baseline interferograms, and then uses the short-baseline to obtain the PU results of the long-baseline. [PERSON] [23] proposed an MB InSAR PU method based on a mixed integer optimization model. [PERSON] [12] proposed a groundbreaking MBPU framework that can introduce an SBPU model to an MBPU method, which has important implications for MBPU research. [PERSON] [24] proposed a refined two-stage programming-based (TSPA) MB approach using a local plane model. In summary, MBPU can obtain better unwraped results, even in areas with complex terrain, but its robustness and efficiency still need further improvement. The Kalman filter is a better filtering method based on a Bayesian framework.