1. Introduction
Synthetic aperture radar interferometry (InSAR) is playing an increasingly important role in the field of surface deformation monitoring and topographic mapping [
1,
2,
3]. The InSAR system uses two co-registered complex images from different viewing angles to obtain the two-dimensional interferometric phase images. Due to the trigonometric function in the transmitting and receiving models, the obtained interferometric phase is wrapped—that is, its range is in
[
4,
5]. In order to obtain an accurate elevation measurement of the surveying area, the unwrapped phase must be obtained by adding the correct wrap count to each pixel of the wrapped phase, which is called phase unwrapping. Therefore, in the InSAR data processing pipeline, the measurement accuracy of elevation level is highly correlated with the accuracy of phase unwrapping.
Since phase unwrapping is an ill-posed problem, the phase continuity assumption is usually considered in the process of phase unwrapping: the absolute values of the gradients in the two directions of the unwrapped phase are less than
[
6]. Under this assumption, many kinds of phase unwrapping methods have been presented in recent decades, and they can be divided into two categories: path following [
5,
7,
8] and optimization-based methods [
9,
10,
11,
12,
13,
14,
15]. A path following method selects the integration path for integrating the estimated phase gradient through the residue distribution or the phase quality map, so as to avoid the local error from being propagated globally. Examples are the branch-cut method [
5] and the quality-guided method [
7]. An optimization-based method minimizes the difference between the estimated gradient and the unwrapping phase gradient through the objective function to obtain the optimal unwrapped phase. Examples are the least squares (LS) method [
13] and the statistical-cost, network-flow algorithm for phase unwrapping (SNAPHU) method [
15], and phase unwrapping max-flow/min-cut algorithm (PUMA) [
10]. Both types of method need to obtain the estimated value of the phase gradient through the phase continuity assumption before unwrapping. Due to the presence of noise and abrupt terrain changes, the phase continuity assumption is not always satisfied—that is, the unwrapped phase may jump above
, which may cause local errors in the unwrapping process. This local error may produce a global error along the integration path, so the estimated phase gradient information will directly affect the final unwrapping accuracy. Therefore, it is a valuable aim to seek a more accurate estimation method of phase gradient information instead of directly relying on the traditional estimation method based on the phase continuity assumption.
In recent years, the deep learning-based phase unwrapping methods have attracted significant interest [
16,
17,
18,
19,
20,
21,
22,
23,
24]. Most of these methods [
16,
17,
18] convert the unwrapping problem into a classification problem of the wrap count, and their effectiveness is verified using optical images. In the field of InSAR, the unwrapping problem becomes more difficult because of two characteristics: the complex wrapped phase caused by topography features and the low coherence coefficient. Therefore, combining traditional phase unwrapping methods with deep learning, instead of relying solely on deep learning, is a promising development trend [
19,
20,
21,
22,
23]. In [
19], a modified fully convolutional network was first applied to classify the wrapped phase into normal pixels and layover residues, which can suppress the error propagation of layover residues during the phase unwrapping process. Additionally, a CNN-based unwrapping method was proposed in [
20], which feeds the wrapped phase and coherence map into the network at the same time for training to obtain the wrap count gradient. In this method, the wrap count reconstruction is necessary for obtaining the final unwrapping result. A deep learning-based method combined with the minimum cost flow (MCF) unwrapping model was proposed in [
21]. In this method, the phase gradient is discretized to match the MCF unwrapping model and treated as a three-classification deep learning problem, but the number of categories may need to change according to the terrain changes, because the three categories cannot cover all situations. In addition, the ambiguity gradient [
23] is taken as ground truth for network training, and the MCF model is used as the postprocessing step for final unwrapped phase reconstruction. However, the MCF unwrapping model is usually very complex computationally and requires numerous computational resources [
25].
The LS phase unwrapping method is widely used in practical applications and converges quickly [
9,
26,
27]; therefore, we considered combining it and deep learning to improve the unwrapping accuracy while retaining the advantages of the LS method. In the traditional LS method, estimating the phase gradient according to the phase continuity assumption (PGE-PCA) is an essential step. Recent studies [
28,
29,
30,
31,
32] have indicated that the encoder–decoder architecture based on deep convolutional neural networks (DCNN) can learn the global features from a large number of input images with different levels of noise or other disturbances, which is useful for obtaining the robust phase gradient from noisy wrapped phase images.
In this paper, we propose a robust LS InSAR phase unwrapping method that works via a phase gradient estimation network (PGENet-LS). In this method, we transform the phase gradient estimation into a regression problem and design a phase gradient estimation network based on the encoder–decoder architecture (PGENet) for InSAR. From lots of wrapped phase images with topography features and different levels of noise, PGENet can extract global high-level phase features and recognize the phase gradient between adjacent pixels, so the more accurate and robust phase gradient can be estimated by PGENet than that obtained by PGE-PCA. Finally, the phase unwrapping result is obtained by using the least squares solver to minimize the difference between the gradient obtained by PGENet and the gradient of the unwrapped phase. The phase gradient estimated by PGENet is used to replace the PGE-PCA in the traditional LS unwrapping method. As the accuracy of the phase gradient estimated by PGENet is significantly higher and more robust than that of the phase gradient estimated by PGE-PCA, the proposed method has higher accuracy than the traditional LS phase unwrapping method. A series of experimental results of simulated wrapped phase and real InSAR data demonstrate that the proposed method outperforms the other five well-established phase unwrapping methods and is robust to noise.
This paper is organized as follows.
Section 2 introduces the principles of phase unwrapping, problem analysis, PGENet, and the proposed method. In
Section 3, the data generation method, loss function, performance evaluation index, and experiment settings are described. In
Section 4, a series of experimental results using simulated and real InSAR data are presented.
Section 5 and
Section 6 present the discussion and conclusions of the paper, respectively.
6. Conclusions
In this paper, a robust InSAR phase unwrapping method combining PGENet and the least squares solver was proposed to improve the accuracy of phase unwrapping. We designed PGENet to estimate the horizontal and vertical gradients first, and then the phase unwrapping result is obtained by using the least squares solver to minimize the difference between the gradient obtained by PGENet and the gradient of the unwrapped phase. The horizontal and vertical gradients estimated by PGENet are used to replace the gradients estimated by PGE-PCA in the traditional LS unwrapping method. PGENet can extract global high-level phase features and recognize the phase gradient between adjacent pixels from lots of wrapped phase images with topography features and different coherence coefficients. Therefore, compared with the phase gradient obtained by PGE-PCA, the more accurate and robust phase gradient can be estimated by PGENet. This is the reason why the proposed method has higher precision and better robustness than the traditional LS unwrapping method. The experimental results on simulated data showed that the proposed method has the highest unwrapping accuracy among six widely-used unwrapping methods and is robust to noise. Furthermore, when processing the real Sentinel-1 InSAR data, the proposed method had the best performance among these six unwrapping methods.
The proposed method successfully combines deep learning and the traditional LS method for InSAR phase unwrapping. The core of this method is the accurate and robust phase gradient estimation based on PGENet, which makes the proposed method have high accuracy and robustness. In future work, to achieve more accurate unwrapping, we will make targeted modifications to PGENet to match more traditional InSAR phase unwrapping methods. In addition, we will use the proposed phase unwrapping method to process more real InSAR data.