1. Introduction
Due to the all-weather and all-day characteristics of the synthetic aperture radar (SAR), it plays an important role in remote sensing [
1,
2,
3,
4,
5]. Simultaneously, the continuous development of SAR has brought more and more development prospects to interferometric SAR (InSAR). At present, InSAR has a wide range of applications such as surface deformation monitoring and terrain mapping [
6,
7,
8,
9,
10,
11]. The basic principle of InSAR measurement technology mainly extracts the phase difference in the primary and secondary images through the observation angle difference of the primary and secondary antennas, and finally inverts the elevation information of the observation area by using the formula between the phase difference and the height.
In the whole InSAR processing flow, noise is inevitably added to the InSAR phase, which can be divided into three categories: system noise, coherent noise, and noise introduced by signal processing [
12,
13]. The presence of noise severely destroys the follow-up phase unwrapping step, which reduces the accuracy of phase unwrapping and even obtains the incorrect results [
14,
15]. Therefore, interferometric phase filtering is a necessary processing step and has also become a very important technology in InSAR measurement.
Since the invention of InSAR technology, a large number of effective interferometric phase filtering approaches have been developed, and the traditional methods fall into four main categories (i.e., spatial domain local filters [
16,
17,
18,
19,
20], spatial domain nonlocal (NL) filters [
21], transform domain local filters [
22,
23,
24,
25,
26], and transform domain NL filters [
27,
28]). The main idea of spatial domain local filters is to filter out the phase noise in the space domain using a local window with pixels. A well-known spatial domain local filter is the Lee filter [
18]. Unlike spatial domain filters, the transform domain local filters denoise the interferogram in the transform domain such as the Goldstein filter [
26]. However, the above two kinds of filters cannot balance the noise suppression ability and phase detail preservation ability well. In order to further enhance the phase detail preservation capability while ensuring effective noise suppression, the spatial and transform domain NL filters have been proposed, which utilize the patch-by-patch method to measure the patch similarity of the interferogram and the weighted average to restore the interferometric phase [
14] such as NL-InSAR [
21] and InSAR-BM3D [
28]. Although NL filters can consider both noise suppression and phase detail preservation, they suffer from a huge computational cost due to abundant similar block operations. Aiming to bridge this regret, a series of newly advanced filtering algorithms have been proposed [
29,
30,
31,
32,
33,
34].
Over the past few years, deep learning (DL) has been successfully applied to interferogram denoising due to the powerful feature extraction and calculation ability of convolutional neural networks (CNNs) such as Phi-Net [
31] and PFNet [
30]. However, there are two key problems with the vast majority of existing CNNs. On one hand, the underlying structure of the purely data-driven CNN with a black-box nature is difficult to interpret, that is, it lacks interpretability. Of course, interpretability is an important feature in many fields because it relates to conceptual understanding and the development of knowledge frontiers [
35]. On the other hand, most modern CNNs need to learn a large number of parameters, so they excessively depend on huge amounts of data. In other words, a vast majority of CNNs improve the accuracy at the cost of increasing the network complexity. However, in many fields such as in [
36,
37], the performance of the network trained with small training sets will be significantly reduced, and even inferior to the traditional methods.
In recent years, a promising technique that unrolls the SR algorithm into network architectures was developed by Gregor et al. [
38]. Compared to modern CNNs, the unrolled network not only has a sufficient theoretical basis, but also contains fewer layers and parameters, which do not rely on huge training sets. Therefore, some novel networks based on the idea of unrolling the SR algorithm into CNN have been proposed such as in [
39,
40]. However, since SR algorithm unrolling has not been applied to InSAR phase denoising, we attempted to combine this technique into this field. Inspired by [
39,
40,
41,
42], we designed an InSAR phase filtering model and established a model-driven CNN to filter the noisy interferograms.
In this article, we propose a sparse-model-driven network (SMD-Net) for efficient and high-accuracy InSAR phase filtering. In the method, we first establish a SR model for interferometric phase filtering. Then, the SMD-Net is designed as an iteration-based CNN architecture by unrolling each iteration process of the iterative shrinkage-thresholding algorithm (ISTA) [
43] to solve the phase filtering model into a block. In each block, a CNN module with a local block and global context (GC) [
44] block is established to adaptively learn the sparse domain transform of each iteration in ISTA. Finally, due to dealing with complex-valued data, our method is carried out by exploiting the idea of separating the real and imaginary parts of the interferometric phase. In short, the SMD-Net models the interferometric phase filtering process, rather than relying entirely on data fitting as most networks do and its network structure is simple. It thus improves the filtering performance and computational efficiency at the same time. The experimental results on the simulated and measured data demonstrate that the proposed method outperformed the Lee filter [
18], Goldstein filter [
26], InSAR-BM3D filter [
28], ISTA-based filtering method, and the PFNet [
30] in both precision and speed. Furthermore, the filtering performance of the SMD-Net on 10% of the original training samples was also slightly better than that of the PFNet. The main contributions of our work are as follows.
- (1)
We first built an InSAR phase filtering model. Then, the SMD-Net was designed based on the idea of unrolling the ISTA algorithm of solving the model into a simple network architecture, which enhanced the interpretability of the network. Subsequently, the SMD-Net transformed the interferometric phase into a real matrix consisting of the real and imaginary parts of the phase to achieve a complex-valued filtering operation.
- (2)
Unlike the traditional ISTA algorithm setting the sparse transform by handcrafting, the SMD-Net exploits a CNN module to automatically learn the sparse basis operation, which enhances the filtering performance.
- (3)
Plenty of simulated and measured experiments illustrate that the proposed method achieves efficiency and high-precision filtering.
The rest of this article is organized as follows.
Section 2 describes the InSAR phase noise principle and the InSAR phase SR filtering model. We introduce the design of the SMD-Net and loss function in
Section 3. In
Section 4, we describe a method of generating the training and testing data, experimental details, and experimental metrics. Extensive experiments on the simulated and real data are conducted in
Section 5.
Section 6 further discusses the performance of the proposed method under small training samples.
Section 7 presents our conclusions.
6. Discussion
In order to further analyze the performance of the SMD-Net under the small training samples, in the 10 groups of interferogram patches in
Section 4.1, 200 interferogram patches were selected starting from the first interferogram patch in each group at an interval of 10 interferograms as new training sets. Then, the SMD-Net was retrained with the training sets. The indicators of the testing results on the simulated data are listed in
Table 4. As can be seen in
Table 4, the MSE of the proposed method was 9.3% higher, but the MSSIM of the proposed method was equal to that of PFNet and its T was 85.5% faster. Therefore, it can be seen that the performance of the SMD-Net trained with 200 training samples was comparable to the PFNet trained with 2250 training samples. Unlike the PFNet, the performance of the SMD-Net was not constrained by the requirement of the data volume.
Like the simulated data, we processed the measured data utilizing the SMD-Net trained with 200 training samples to analyze the filtering performance of the proposed method. The filtering result of area A (
Figure 9a) is shown in
Figure 12b, and we can see intuitively from the black rectangles in
Figure 12a,b that the phase detail features of the result obtained by our method were better preserved. Next, a flat and low-coherence area B (
Figure 9b) was processed to prove the generalization of the proposed method. The black rectangles in
Figure 12c,d also showed that the proposed method had a stronger phase edge texture preservation capability.
Furthermore, the quantitative indicators of the two areas were calculated and are listed in
Table 5 and
Table 6.
Table 5 and
Table 6, compared with the PFNet, it could be observed that the NORs of the proposed method were higher in both areas, but their metric Qs were higher. This indicates that the PFNet caused the serious loss of phase fringe detail information due to over filtering and its phase detail feature preservation capability was inferior to the proposed method. In addition, we could calculate that the metric Qs of the results obtained by processing area A and area B with the proposed method were 5.6% and 17.1% higher than that of the PFNet, respectively. To sum up, it can be seen that the filtering performance of the SMD-NET trained with 200 training samples outperformed that of the PFNet trained with 2250 training samples.
7. Conclusions
In this article, we propose a sparse-model-driven network (SMD-Net) for efficient and high-accuracy InSAR phase filtering. The SMD-Net was designed by casting the mathematical derivation steps of the traditional ISTA algorithm into the network structure. Unlike the ISTA algorithm, in each block of the SMD-Net, a CNN module was established to adaptively learn the sparse transform instead of the hand-crafted setting. The SMD-Net not only significantly reduced the network complexity, but was also combined with the merit of automatically learning the parameters and sparse transform of CNN. It can thus improve the filtering performance and speed at the same time. Finally, plenty of experiments were performed to validate the proposed method.
We assessed the proposed method qualitatively and quantitatively on the simulated and measured InSAR data. The experimental results on the simulated and measured data demonstrated that the proposed method could better balance the abilities of the noise suppression and phase fringe texture preservation than the several reference filtering methods. In addition, the speed of the proposed method was very fast. Compared with the PFNet, the SMD-Net was 85.5% and 51.9% faster on the simulated and measured data, respectively. Aiming to validate the performance of the proposed method was not limited by the requirement of the number of training samples, so the experiments were carried out again when the number of training samples was decreased to 10%. Compared with the PFNet trained with 2250 samples, the performance of the proposed method was comparable on the simulated data. In the experiments on the real data, the Qs of the results obtained by processing high-coherence and low-coherence areas with the proposed method were 5.6% and 17.1% higher, respectively. This proves that the comprehensive performance of our method outperformed that of the six competitive approaches, even with small samples.