# Improved Least Squares Phase Unwrapping Method Based on Chebyshev Filter
## 1 Introduction  
Interferometric synthetic aperture radar (InSAR) is a remote sensing technique that combines electromagnetic wave interferometry with synthetic aperture radar (SAR) and extracts information on ground elevation or surface deformation, facilitating ground surveillance and measurement, by processing SAR image data through interferometry [1; 2]. In recent years, with the rapid advancement of remote sensing technology and the widespread use of high-resolution satellites, spaceborne InSAR has increasingly become crucial in applications such as ground elevation measurement, surface deformation studies, and atmospheric research [3; 4; 5; 6]. InSAR has many advantages in terrain mapping and surface deformation monitoring, such as being robust to time and weather, but the InSAR processing process can be affected by many factors, such as interference decorrelation, atmospheric phase, and phase unwrapping. These factors could cause reduced measurement accuracy or measurement failure, especially for single-baseline InSAR systems. Among these factors, phase unwrapping is one of the key steps to obtaining high-precision terrain and surface deformation using InSAR [7; 8; 9]. The phase unwrapping method used in InSAR processing is directly related to the accuracy of the solved terrain and surface deformation information [10; 11]. There still does not exist a suitable phase unwrapping method for all cases, as especially in the case of large phase noise and steep terrain, many phase unwrapping methods will produce large errors. So far, many phase unwrapping methods have been proposed, which can mainly be divided into the following three categories, path-following methods [12; 13; 14], optimal estimation methods [15], andminimum norm methods [16; 17; 18; 19], as well as other methods [20; 21; 22; 23; 24; 25], such as the Kalman filtering methods [20] and machine learning methods [21; 22]. The first category, path-following methods, is represented by the Goldstein branch-cutting method [12; 14] and the quality map guidance method [13]. The Goldstein branch-cutting method identifies positive and negative residual points, connects the residual points according to the nearest neighbor principle, and generates \"branch cuts\". It prevents the propagation of errors by following the principle that the integral path does not cross the \"branch cuts\". The Goldstein branch-cutting method has the obvious advantages of fast speed and high accuracy under the condition of high signal-to-noise ratio (SNR) and low phase gradient. However, when the noise is large and the phase gradient is steep, it is difficult for this method to accurately connect branch tangents, which often leads to errors in integration path selection and introduces errors into the unwrapping stage [14]. The quality map guidance method does not identify residual points and does not set branch tangents. It is a method that defines the wrapping phase quality through the phase quality map, controls the integration path to advance along the high-quality pixels to the low-quality pixels, and sequentially unwraps the wrapping phase. However, its accuracy depends largely on the reliability of the quality map; when the noise is large and the phase gradient is steep, the residuals will be distributed in the high-quality area, and the quality-map guidance method will not work properly [13]. The second category, optimal estimation methods, such as the network flow method, is also commonly used for phase unwrapping. Unlike the methods mentioned earlier, the network flow method transforms the phase problem into a network flow problem with the concept of minimum cost flow, and it limits the phase error transmission in the low-mass region and obtains a globally optimal solution by minimizing the difference between the phase gradient of the unwrapped phase and the discrete partial derivatives of the unwrapped phase. However, for low SNR and steep phase gradient, the error can still be significant.