**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 [1–5]. 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 [6–11]. 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 intro-

**Citation:** Wang, N.; Zhang, X.; Zhang, T.; Pu, L.; Zhan, X.; Xu, X.; Hu, Y.; Shi, J.; Wei, S. A Sparse-Model-Driven Network for Efficient and High-Accuracy InSAR Phase Filtering. *Remote Sens.* **2022**, *14*, 2614. https://doi.org/10.3390/rs14112614

Academic Editor: Gwanggil Jeon

Received: 28 April 2022 Accepted: 24 May 2022 Published: 30 May 2022

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duced 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 [16–20], spatial domain nonlocal (NL) filters [21], transform domain local filters [22–26], 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 [29–34].

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 [39–42], 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.


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.
