**1. Introduction**

Synthetic aperture sonar (SAS) is a technique that repeatedly transmits and receives pulses while the sonar is moving and coherently synthesizes the received signals to obtain a high-resolution image [1–3]. By synthesizing multiple pings, it is possible to achieve the effect of a sonar operating with an aperture larger than the actual sonar aperture, therefore called a "synthetic aperture" sonar. Compared to other techniques for obtaining underwater images, such as side-scan sonar, SAS obtains images with a high resolution [4] and is used in various fields such as crude oil exploration, geological exploration, and for military purposes such as in mine detection [5,6].

Conventional SAS methods reconstruct the image by performing Fourier transform and matched filtering in the slant-range or in the azimuth domain. Conventional SAS methods are classified into back-projection in the spatial–temporal domain [1], correlation in the spatial–temporal domain [7], range-Doppler in the range-Doppler domain [8], wavenumber in the wavenumber domain [9,10], and chirp-scaling in the wavenumber domain [11], contingent on whether Fourier transform is performed in the slant-range or in the azimuth domain. To form a synthetic aperture requires sampling following Nyquist theory in the time domain according to the traditional signal processing technique, and dense sampling in the spatial domain alongside the sonar movement is also required. Because conventional SAS signal processing techniques pass through a matched filter, side lobes are generated, resulting in the deterioration of image reconstruction performance [12,13].

This paper proposes SAS imaging algorithms that apply the compressive sensing (CS) framework to compensate for disadvantages associated with conventional SAS signal

**Citation:** Choi, H.-m.; Yang, H.-s.; Seong, W.-j. Compressive Underwater Sonar Imaging with Synthetic Aperture Processing. *Remote Sens.* **2021**, *13*, 1924. https:// doi.org/10.3390/rs13101924

Academic Editor: Vladimir Lukin

Received: 31 March 2021 Accepted: 12 May 2021 Published: 14 May 2021

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processing techniques. CS is a technique used to restore a sparse signal from a small number of measurements [14]. Under suitable conditions, CS obtains a better resolution than conventional signal processing and suppresses side lobes. In addition, admittedly under suitable conditions, CS obtains exact solutions even at a low sampling level, which violates Nyquist theory. In recent years, studies related to the CS framework have been conducted in various fields, such as medical imaging fields—including MRI and ultrasound imaging—and sensor networks [15–20]. In the estimation of the direction-of-arrival (DOA), which is a classical source-localizing method, CS is applied to increase the number of employed sensors or observations, thereby enhancing localization performance [21,22]. Additionally, studies have been conducted on the application of sparse reconstruction to synthetic aperture radar (SAR) [23–28] and SAS [29,30]. Many studies have applied CS to SAR, but as far as could be determined, few studies have been conducted on SAS, especially underwater.

In [29], a method that applies CS to SAS imaging is presented, which estimates the reflectivity function in the area of interest using all the given data. When some of the data were excluded, results were good, showing that large data reduction is possible. However, this study does not show results for actual underwater acoustic conditions; it only shows results for the ultrasonic synthetic aperture laboratory system using assumed point targets. The fact that the laboratory results have not been verified against actual underwater experimental data has significant consequences. Targets in real underwater environments are generally not point targets but targets with continuous characteristics. Therefore, if the reflectivity function is estimated instantly, as in the method proposed in [29], the shape of the target will not be properly revealed and only segments with high reflectivity will be obtained. In [30], CS was applied to SAS to obtain a parsimonious representation to utilize aspect- or frequency-specific information. By way of simulation and employing real underwater experimental data, it was verified that the strategy using aspect- or frequency-specific information was effective. However, the method proposed in [30], which uses an iterative method called the alternating direction method of multipliers (ADMM), has a limitation in that it is unstable because convergence is highly dependent on the regularization parameter. Therefore, we propose a stable method that does not use an iterative method and that expresses the characteristics of a real underwater target by dividing data and repeatedly estimating the reflectivity function of the area of interest.

This study offers three main contributions: First, the proposed method (called the CS-SAS algorithm for simplicity) that shows better reconstruction performance compared to a conventional SAS algorithm. The proposed algorithms are SAS algorithms formulated from the perspective of the CS framework and in accordance with the CS characteristics. Less aliasing occurs and high-resolution results are obtained. Section 3 explains that the proposed method outperforms one of the conventional SAS algorithms, the ω-k algorithm. Second, the proposed algorithms are more robust in the absence of sensor data. Because conventional SAS algorithms require sampling frequency according to Nyquist theory in the time and spatial domains, conventional SAS algorithms are not resistant to sensor failure or data loss in the sonar system. Conversely, the proposed algorithms are robust, as indicated later on in this paper. Third, few studies apply CS to SAS underwater and, therefore, this study is meaningful in that it applies simulation data and actual underwater experimental data.

The remainder of this paper is organized into four sections. In Section 2, the geometry of the SAS system and the ω-k algorithm—which is a representative conventional SAS algorithm—are described. In Section 3, the basic theory of CS is described, and SAS algorithms using CS are proposed. In Section 4, the performance of the proposed method is verified by comparing the results of applying the CS-SAS and the ω-k algorithms to the simulation and experimental data. Finally, conclusions are presented in Section 5.

In the following, vectors are represented by bold lowercase letters, and matrices are represented by bold capital letters. The *lp*-norm of a vector **<sup>x</sup>** <sup>∈</sup> <sup>C</sup>*<sup>N</sup>* is defined as **x***<sup>p</sup>* = - *N* ∑ *i*=1 |*xi*| *p* 1/*<sup>p</sup>* . The imaginary unit √−<sup>1</sup> is denoted as *<sup>j</sup>*. The operators *<sup>T</sup>*, ∗ denote the transpose and conjugate operators, respectively.
