## 3 Processing Flow
### Dictionary Training  
Compressed sensing denoise depends on the over-complete dictionary. Each atom in the dictionary represents a structural prototype in the original phase. Therefore, the phase has a sparse representation in the dictionary, while the noise component does not. According to this principle, the original phase can be distinguished from noise to achieve the purpose of denoising. In order to ensure that the sparse representation of synchronization phase is sparse enough, the optimal over-complete dictionary is trained according to the characteristics of phase. Considering that the synchronization phase can be decomposed into the linear phase \(\varphi^{l}(t)\) and the higher-order phase \(\varphi^{h}(t)\) as  
\[\varphi(t)=\varphi^{l}(t)+\varphi^{h}(t)=2\pi\Delta f_{c}t+\varphi^{h}(t) \tag{19}\]  
where \(\Delta f_{c}\) is the frequency deviation of USOs between LT-1A and LT-1B, which can be considered to be constant in a short time. After removing the linear term, the characteristics of synchronization phase are mainly reflected by the higher-order phase \(\varphi^{h}(t)\). In order to obtain the characteristic information and eliminate interference item, the higher-order phase \(\varphi^{h}(t)\) separated by the least square method (LSM) is selected for training instead of the whole phase \(\varphi(t)\). The dictionary should represent the structural prototype of the original phase **X** as much as possible. However, the clean phase **X** is unknown. The imaging phase difference \(\varphi_{\text{diff}}(t)\) obtained in the ground validation system is selected to approximate the clean phase. Due to the large time bandwidth product and the ideal transmission medium (optical delay line), the imaging signals exchanged between LT-1A and LT-1B have quite high SNRs. Therefore, such an approximation is reasonable. The higher-order phase of \(\varphi_{\text{diff}}(t)\) is employed for dictionary training. The K-SVD algorithm with an alternating optimization strategy is utilized for dictionary training [40]. The K-SVD algorithm fixes the current dictionary to solve the sparse coefficients, then updates the dictionary atom by atom. After that, new sparse coefficients are solved based on the new dictionary, and the above cycle is repeated. Finally, the appropriate sparse dictionary is obtained by approximating the local optimal to the global optimal. A proper initialization dictionary contributes to a superior reconstruction effect. The Ramanujan Sums (RS) transform matrix, which is widely used in signal processing, time-frequency analysis, and shape recognition, is selected as the initial basis [43]. Compared with the commonly used discrete cosine transform (DCT) dictionary, it can train effectively and reconstruct the information in a better way. After the training by the K-SVD algorithm, the over-complete dictionary **D** contains the characteristics of the clean phase, so as to obtain the prior information of the ideal synchronization phase with noise free. The complete processing flowchart is shown in Figure 4. Since the synchronization frequency is less than PRF, the imaging phase needs to be downsampled to the synchronization frequency through uniform/non-uniform interpolation before dictionary training. After denoising, the synchronization phase should be interpolated to the same length as the imaging phase. Spline interpolation is applied in those processes to keep the phase smooth. In combination with the foregoing, the main steps of the synchronization phase denoising method can be summarized as follows:  
_Step 1_: Extract the imaging phase difference \(\varphi_{\text{diff}}\), synchronization compensation phase \(\varphi_{c}\), and internal calibration compensation phase \(\varphi_{\text{cal}}\), and estimate the SNR of the synchronization signal. _Step 2_: Downsample \(\varphi_{\text{diff}}\) and extract its higher-order phase \(\varphi_{\text{diff}}^{h}\) by the least square method.