Compressive Underwater Sonar Imaging with Synthetic Aperture Processing

remote sensing

Article Compressive Underwater Sonar Imaging with Synthetic Aperture Processing

Ha-min Choi 1 , Hae-sang Yang 1 and Woo-jae Seong 1,2,*

1 Department of Naval Architecture and Ocean Engineering, Seoul National University, Seoul 08826, Korea; [email protected] (H.-m.C.); [email protected] (H.-s.Y.) 2 Research Institute of Marine Systems Engineering, Seoul National University, Seoul 08826, Korea * Correspondence: [email protected]; Tel.: +82-2-880-7332

Abstract: Synthetic aperture sonar (SAS) is a technique that acquires an underwater image by synthesizing the signal received by the sonar as it moves. By forming a synthetic aperture, the sonar overcomes physical limitations and shows superior resolution when compared with use of a side-scan sonar, which is another technique for obtaining underwater images. Conventional SAS algorithms require a high concentration of sampling in the time and space domains according to Nyquist theory. Because conventional SAS algorithms go through matched filtering, side lobes are generated, resulting in deterioration of imaging performance. To overcome the shortcomings of conventional SAS algorithms, such as the low imaging performance and the requirement for high-level sampling, this paper proposes SAS algorithms applying compressive sensing (CS). SAS imaging algorithms applying CS were formulated for a single sensor and uniform line array and were verified through simulation and experimental data. The simulation showed better resolution than the ω-k algorithms, one of the representative conventional SAS algorithms, with minimal performance degradation by side lobes. The experimental data confirmed that the proposed method is superior and robust with respect to sensor loss. Citation: Choi, H.-m.; Yang, H.-s.; Seong, W.-j. Compressive Keywords: compressive sensing; synthetic aperture sonar; underwater sonar imaging Underwater Sonar Imaging with Synthetic Aperture Processing. Remote Sens. 2021, 13, 1924. https:// doi.org/10.3390/rs13101924 1. Introduction Synthetic aperture sonar (SAS) is a technique that repeatedly transmits and receives Academic Editor: Vladimir Lukin 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 Received: 31 March 2021 the effect of a sonar operating with an aperture larger than the actual sonar aperture, Accepted: 12 May 2021 Published: 14 May 2021 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]

Publisher’s Note: MDPI stays neutral and is used in various fields such as crude oil exploration, geological exploration, and for with regard to jurisdictional claims in military purposes such as in mine detection [5,6]. published maps and institutional affil- Conventional SAS methods reconstruct the image by performing Fourier transform iations. 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 Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. domain. To form a synthetic aperture requires sampling following Nyquist theory in the This article is an open access article time domain according to the traditional signal processing technique, and dense sampling distributed under the terms and in the spatial domain alongside the sonar movement is also required. Because conventional conditions of the Creative Commons SAS signal processing techniques pass through a matched filter, side lobes are generated, Attribution (CC BY) license (https:// resulting in the deterioration of image reconstruction performance [12,13]. creativecommons.org/licenses/by/ This paper proposes SAS imaging algorithms that apply the compressive sensing 4.0/). (CS) framework to compensate for disadvantages associated with conventional SAS signal

Remote Sens. 2021, 13, 1924. https://doi.org/10.3390/rs13101924 https://www.mdpi.com/journal/remotesensing Remote Sens. 2021, 13, 1924 2 of 19

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. Section3 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 Section2, the geometry of the SAS system and the ω-k algorithm—which is a representative conventional SAS algorithm—are described. In Section3, the basic theory of CS is described, and SAS algorithms using CS are proposed. In Section4, 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 Section5. In the following, vectors are represented by bold lowercase letters, and matrices N are represented by bold capital letters. The lp-norm of a vector x ∈ C is defined as Remote Sens. 2021, 13, 1924 3 of 19

Remote Sens. 2020, 12, x FOR PEER REVIEW 3 of 19

⁄N 1/p √ ∗ (∑ |𝑥| ) . Thep imaginary unit √−1 is denoted as 𝑗. The operators , denoteT ∗ the kxkp = ∑ |xi| . The imaginary unit −1 is denoted as j. The operators , denote transpose andi=1 conjugate operators, respectively. the transpose and conjugate operators, respectively. 2. Basic SAS Imaging 2. Basic SAS Imaging 2.1. SAS Geometry 2.1. SAS Geometry The general geometry of SAS is depicted in Figure 1. The direction of 𝑦 along which The general geometry of SAS is depicted in Figure1. The direction of y along which the sonar moves is defined as azimuth or the cross-range axis, the direction 𝑥 perpendic- the sonar moves is deﬁned as azimuth or the cross-range axis, the direction x perpendicular ular to 𝑦 is the slant-range, the size of the sonar is 𝐷, and the synthetic aperture and the to y is the slant-range, the size of the sonar is D, and the synthetic aperture and the distance distance the sonar travels is 2𝐿. The basic concept of SAS is that the sonar moves from −𝐿 the sonar travels is 2L. The basic concept of SAS is that the sonar moves from −L to L along to 𝐿 along the cross-range axis, and then transmits and receives signals to synthesize the the cross-range axis, and then transmits and receives signals to synthesize the received received signals scattered back from the targets to obtain underwater images. signals scattered back from the targets to obtain underwater images.

FigureFigure 1. 1.SASSAS geometry. geometry. One of the standards for evaluating the level of performance of an SAS system is One of the standards for evaluating the level of performance of an SAS system is the the resolution of the reconstructed images. The slant-range resolution ∆ of the SAS was resolution of the reconstructed images. The slant-range resolution ∆ of thex SAS was de- determined by the matched ﬁltering process as follows: termined by the matched filtering process as follows: 𝑐𝜋 𝑐 cπ c ∆ = = , (1) ∆= = x, (1) 𝜔 2𝑓 ωbd 2 fbd

wherewhere 𝜔ωbd isis the the bandwidth bandwidth of the transmitted signal, and c𝑐is is the the sound sound speed. speed. TheThe cross-range cross-range resolution resolution ∆∆y can be derivedderived simplysimply throughthrough thethefollowing following develop- devel- opment.ment. Consider Consider aa singlesingle frequencyfrequency signalsignal asas thethe simplestsimplest signal model. Then, Then, the the −−3 3dB dB mainmain lobe lobe width width of ofan an𝑙-lengthl-length uniform uniform line line array array is 𝜆/𝑙 is ,λ the/l, main the main lobe lobewidth width 𝜃 θofSAS theof syntheticthe synthetic aperture aperture is 𝜆/2𝐿 is λ. /2ForL. the For theslant-range slant-range of the of thetarget target 𝑅, R𝐿≃𝑅𝜃, L ' R andθ and 𝜃=𝜆/𝐷θ = λ/D. . Therefore,Therefore, the the cross-range cross-range resolution resolution ∆∆y cancan be be expressed expressed as ∆ =𝑅𝐷(2𝑅)⁄ =𝐷/2. ∆y = RD/(2R) = D/2.(2) (2) For convenience, a single frequency signal was assumed, but it is known that cross- For convenience, a single frequency signal was assumed, but it is known that cross- range resolution ∆=𝐷/2 even if a signal with bandwidth like LFM signal is used [28,31]. range resolution ∆y = D/2 even if a signal with bandwidth like LFM signal is used [28,31]. From Equation (2), it is clear that the cross-range resolution of the synthetic aperture sonar From Equation (2), it is clear that the cross-range resolution of the synthetic aperture sonar is independent of range and frequency. This independence makes it possible to recon- is independent of range and frequency. This independence makes it possible to reconstruct structhigh-resolution high-resolution images images over over a long a long range range [32,33 [32,33].].

2.2.2.2. Wavenumber Wavenumber Domain Domain Algorithm Algorithm (𝜔 (ω-k-k Algorithm) Algorithm) TheThe wavenumber wavenumber domain domain algorithm, algorithm, a arepr representativeesentative conventional conventional SAS SAS algorithm, algorithm, waswas used used as as a baseline a baseline method. method. The The wavenumber wavenumber domain domain algorithm algorithm is a method is a method that ob- that tains an image using 2-D Fourier transform of recorded signals and is also called the ω-k

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obtains an image using 2-D Fourier transform of recorded signals and is also called the ω-k algorithm because signal processing is performed in the frequency domain [1,8]. The signal of duration Tp transmitted from the sonar, is denoted by p(t), and the signal received by the sonar at position u is denoted by s(t,u). The signal s(t,u) can be expressed as the sum of the signals scattered by the N-targets as follows:  q  q  N 2 2 2 2 2 xn + (yn − u) 2 xn + (yn − u) s(t, u) = ∑ σn pt − 0 < t − < Tp (3) n=1 c c

where σn is the target strength of the n-th target, and xn and yn are the slant-range and cross-range of the n-th target, respectively. The 2-D Fourier transform of Equation (3) using the stationary-phase principle [34] gives

N q 2 2 S(ω, ku) = ∑ P(ω)σn exp −j 4k − kuxn − jkuyn . (4) n=1

where ω is the angular frequency, k is the wavenumber, and ku is the azimuth wavenumber. By changing the coordinates as shown in Equation (5), it can be arranged as in Equation (6). q 2 2 kx = 4k − ku, ky = ku. (5)

N S(ω, ku) = ∑ P(ω)σn exp −jkxxn − jkyyn . (6) n=1 The function of the distribution of the targets is expressed in Equation (7), and its Fourier transform is expressed as Equation (8).

N f0 = ∑ σnδ(x − xn, y − yn). (7) n=1

N F0 kx, ky = ∑ σn exp −jkxxn − jkyyn . (8) n=1 Equation (9) can be obtained by combining Equations (6) and (8).

S(ω, ku) = P(ω)F0. (9)

Therefore, the distribution of the targets can be estimated through the following relationship:

N ∗ 2 F kx, ky = P (ω)S(ω, ku) = |P(ω)| ∑ σn exp −jkxxn − jkyyn . (10) n=1 The mapping—Equation (5) from (ω, ku) to kx, ky , called Stolt mapping [35]— involves interpolation from the data. The interpolation process can be made more rational through the following spatial shift formulation: Fb kx, ky = F kx, ky exp jkxXc + jkyYc (11)

where subscript b indicates baseband conversion, and Xc,Yc are the centers of the area of interest in the slant-range and cross-range, respectively. In Equation (11), the exponential term performs a spatial shift function, which performs a function similar to the carrier removal process in spectrum demodulation. It enables interpolation in a slowly varying region while moving the entire swath down to the origin of the spatial coordinates. Remote Sens. 2021, 13, 1924 5 of 19

Remote Sens. 2020, 12, x FOR PEER REVIEW Because the received signal is scattered by a target with a target strength of 1 in the 5 of 19 center of the area of interest, its Fourier transform can be expressed as Equations (12) and (13), respectively, and Equation (11) can be summarized as follows: 2𝑋 + (𝑌 −𝑢)  q  𝑠(𝑡, 𝑢) =𝑝𝑡− 2 , 2 (12) 𝑐 2 Xc + (Yc − u) s (t, u) = pt − , (12) 0 c

𝑆(𝜔, 𝑘 ) =𝑃(𝜔)exp−𝑗𝑘𝑋 − 𝑗𝑘𝑌, (13) S0(ω, ku) = P(ω) exp −jkxXc − jkyYc , (13) ∗ 𝐹𝑘,𝑘=𝑆(𝜔, 𝑘 )𝑆 (𝜔, 𝑘 ). ∗ (14) Fb kx, ky = S(ω, ku)S0 (ω, ku). (14) TheThe flow ﬂow chart chart of thetheω ω-k-k algorithm algorithm is shownis shown in Figure in Figure2[1]. 2 [1].

FigureFigure 2. 2. FlowFlow chart chart of the ωω-k-k algorithm. algorithm. 3. SAS Algorithm with CS Framework (CS-SAS) 3. 3.1.SAS Compressive Algorithm Sensing with CS Framework (CS-SAS) 3.1. CompressiveCompressive Sensing sensing is a method or framework for solving linear problems, such as yCompressive= Ax for sparse sensing signal isx a[36 method]. x ∈ C orN isframework an unknown for signal solving vector linear that problems, we want to such as 𝐲=𝐀𝐱reconstruct. for sparse The unknown signal 𝐱 signal [36]. vector𝐱∈ℂx is is a ank-sparse unknownvector, signal where vectorx is k-sparse that, we meaning want to re- k k = ∈ M construct.that x 0 Thek, unknown that is, x has signal only vectork non-zero 𝐱 is elements. a 𝑘-sparsey vector,C is where a measurement 𝐱 is 𝑘-sparse vector, mean- consisting of measured values. In many realistic problems, A ∈ M×N—called a sensing ‖𝐱‖ =𝑘 𝐱 𝑘 C 𝐲∈ℂ ing that , that is, has only non-zero elements. is a measurement= matrix—is introduced to represent the problem as a linear relationship, such as×y Ax. vectorWhen consisting the dimension of measured of the measurement values. In vector manyy realisticis smaller problems, than the dimension𝐀∈ℂ of—calledx, a sensingthat is, matrix—isM N, the introducedy = Ax problem to represent becomes the an problem underdetermined as a linear problem relationship, and has such as 𝐲=𝐀𝐱numerous. When solutions, the dimension making it of impossible the measurement to specify x vector. Using the𝐲 is sparse smaller property than the of x ,dimension it is ofpossible 𝐱, that tois, specifyM≪N a, uniquethe 𝐲=𝐀𝐱 and exact problem solution becomes among an countless underdetermined feasible solutions problem of the and has numerousunderdetermined solutions, problem. making The it sparsity impossible is imposed to specify by the 𝐱 sparsity. Using constraintthe sparsel0 -norm.property The of 𝐱, it is lpossible0-norm minimization to specify a problemunique and is formulated exact solu astion follows: among countless feasible solutions of the underdetermined problem. The sparsity is imposed by the sparsity constraint 𝑙 -norm. min kxk subject to y = Ax. (15) N 0 The 𝑙-norm minimization problemx∈C is formulated as follows:

However, theminl0-norm‖𝐱‖ minimizationsubject to 𝐲 problem, = 𝐀𝐱. Equation (15), is an NP-hard problem (15) that is computationally𝐱∈ℂ intractable. To deal with the NP-hard problem, various methods haveHowever, been developed the 𝑙-norm such asminimizationl1-norm relaxation problem, or greedy Equation algorithms (15), is an represented NP-hard byproblem thatorthogonal is computationally match-pursuit. intractable. To deal with the NP-hard problem, various methods One of the most representative methods for solving the compressive sensing problem have been developed such as 𝑙-norm relaxation or greedy algorithms represented by or- thogonalis l1-norm match-pursuit. relaxation, which solves the problem by replacing l0-norm with l1-norm. The l -norm minimization problem, Equation (15), can be relaxed by reformulating as 0 One of the most representative methods for solving the compressive sensing problem is 𝑙 -norm relaxation, which solves the problem by replacing 𝑙 -norm with 𝑙 -norm. The min kxk1 subject to y = Ax. (16) x∈CN 𝑙-norm minimization problem, Equation (15), can be relaxed by reformulating as

min‖𝐱‖ subject to 𝐲 = 𝐀𝐱. 𝐱∈ℂ (16) In the presence of noise, a sparse solution 𝐱 can be obtained by the following equation:

min‖𝐱‖ subject to ‖𝐲−𝐀𝐱‖ <𝜖. 𝐱∈ℂ (17) Equation (16) and Equation (17) are called basis pursuit (BP) and basis pursuit denoising (BPDN) problems, respectively. The larger the hyperparameter 𝜖, the sparser the optimized solution 𝐱. Oppositely, the smaller the 𝜖, the more optimized solution 𝐱 fits

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In the presence of noise, a sparse solution x can be obtained by the following equation:

min kxk1 subject to ky − Axk2 < e. (17) x∈CN Equations (16) and (17) are called basis pursuit (BP) and basis pursuit denoising (BPDN) problems, respectively. The larger the hyperparameter e, the sparser the optimized solution x. Oppositely, the smaller the e, the more optimized solution x ﬁts the data. Therefore, it is important to assume a suitable hyperparameter. However, ﬁnding a suitable hyperparameter is complex and deemed to be outside the scope of this study. In this study, the SAS image was obtained by solving the BPDN problem using the tool provided by CVX [37].

3.2. CS-SAS Algorithm for Single Sensor To handle SAS imaging problems from the perspective of compressive sensing, the problem must first be well defined as the y = Ax problem. To formulate a compressive algorithm in which a single sensor is in linear motion, the signal reflected by the targets and returned to the single sensor needs to be considered first. When a signal p(t) is transmitted from a single sensor located in ru = [x0, y0] and reflected by N-targets, the received signal s(t, ru) can be written as N s(t, ru) = ∑ σk p(t − τ(rk, ru)), (18) k=1 q 2 2 |r − r | (xk − x0) + (yk − y0) τ(r , r ) = 2 k u = 2 , (19) k u c c

where σk, k = 1, ... , N is the target strength of the k-th target, τ(rk, ru) is a function representing travel time, xk is the slant-range of the k-th target, and yk is the cross-range of the k-th target. When p(t) is a continuous wave (CW) signal with a pulse duration of Tp and carrier frequency of fc, Equation (19) can be rewritten as Equation (21).

jωt j2π fct p(t) = e = e 0 ≤ t ≤ Tp , (20)

N s(t, ru) = ∑ σkexp(jω(t − τ(rk, ru))) 0 ≤ t − τ(rk, ru) ≤ Tp . (21) k=1

The above is a formulation of the signal received at one location, ru. This can be expanded to the expression for a single sensor. The operation of a single sensor sonar can be expressed as shown in Figure3. The total number of pings is Np, the single sensor—transmitter and receiver—corresponding

to the m-th ping, is um, and the position of um is rum , m = 1, ..., Np. The center point of the area of interest is (Xc, Yc), the half-size of the area of interest in the range is X0, and the half-size of the area of interest in the cross-range is Y0. By dividing the area of interest into Nx × Ny grids, assuming that there is a virtual target σk at each grid point, the signal

s(t, rum ) received at the m-th ping can be written as follows:

Nx Ny = − ≤ − ≤ s(t, rum ) ∑ σkexp(jω(t τ(rk, ru))) 0 t τ(rk, rum ) Tp . (22) k=1

The signal vector si, which consists of signals received at a speciﬁc time ti at each position of the sonar, can be written as

h iT = ( ) ( ) si s ti, ru1 , s ti, ru2 , . . . , s ti, ruNp . (23) Remote Sens. 2021, 13, 1924 7 of 19

Combining Equations (22) and (23), si can be expressed as the product of the target strength vector σ of the virtual targets and the sensing matrix Ai:

si = Aiσ, (24) Remote Sens. 2020, 12, x FOR PEER REVIEW 7 of 19 h iT σ = σ1,..., σN N , (25) where 𝐀(𝑚, 𝑘) denotes an element correspondingx toy the 𝑚-th row and 𝑘-th column of 𝐀. In the CS system, where the length of 𝐲 is 𝑀 and the length of 𝐱 is 𝑁 and 𝑘-sparse, Ai(m, k) = exp(jω(ti − τ(rk, ru ))) 0 ≤ ti − τ(rk, ru ) ≤ Tp , (26) 𝐱 is successfully recovered when 𝑀≥𝑂𝑘logm (𝑁/𝑘) measurementsm are used [36,38]. In thewhere proposedAi(m, kalgorithm,) denotes anif the element length corresponding 𝑁 of the received to the signalm-th row𝐬 is and took small-th column compared of Ai . y M x N k-sparse x toIn the the length CS system, 𝑁𝑁 whereof 𝛔, an the accurate length ofsolutionis cannotand the be length obtained. of Therefore,is and in the case, is successfully recovered when M ≥ O(k log(N/k)) measurements are used [36,38]. In the where the length of the signal 𝐬 is too small, signals for a total of 𝑁 times are collected toproposed form a long algorithm, signal vector if the length𝐬 and,N similarly,p of the received corresponding signal s matricesi is too small are collected compared to toform the alength long sensingNx Ny of matrixσ, an 𝐀 accurate: solution cannot be obtained. Therefore, in the case where the length of the signal si is too small, signals for a total of Nt times are collected to form a long signal vector s and,𝐬=𝐀𝛔, similarly, corresponding matrices are collected to form a long(27) sensing matrix A: = σ 𝐬=𝐬 ,𝐬 ,…,𝐬 s , A ,(28) (27) h iT T T T s = si , si+1 ,..., si+Nt−1 , (28) 𝐀=𝐀 ,𝐀 ,…,𝐀 , (29) h iT where 𝐬∈ℂ, 𝐀∈ℂ×, andT 𝛔∈ℂT. 𝛔 is estimatedT by solving the following A = Ai , Ai+1 ,..., Ai+Nt−1 , (29) BP or BPDN problems: × where s ∈ CNt Nu , A ∈ CNt Nu Nx Ny , and σ ∈ CNx Ny . σ is estimated by solving the following min ‖𝛔‖ subject to 𝐬 = 𝐀𝛔, (30) BP or BPDN problems:𝛔∈ℂ min kσk1 subject to s = Aσ, (30) Nx Ny min ‖𝛔‖ subjectσ∈C to ‖𝐬−𝐀𝛔‖ <𝜖. (31) 𝛔∈ℂ min kσk subject to ks − Aσk < e. (31) N N 1 2 The 𝛔 value, determinedσ∈C xbyy applying Equation (30) or (31), is a target strength vector obtainedThe σ value, from using determined only the by signals applying received Equation in consecutive (30) or (31), is𝑁 a timetarget snapshots strength vectorfrom 𝑡 . Therefore, to obtain the target strength vector for all the area of interest, the 𝑙 -norm obtained from using only the signals received in consecutive Nt time snapshots from minimization process must be repeated for all time snapshots corresponding to the area ti. Therefore, to obtain the target strength vector for all the area of interest, the l1-norm ofminimization interest. The process final image must bewas repeated compiled for by all timeadding snapshots all the corresponding𝛔 values obtained to the in area each of process.interest. The ﬁnal image was compiled by adding all the σ values obtained in each process.

FigureFigure 3. 3. DescriptionDescription for for a asingle single sensor sensor sonar sonar operation: operation: 𝑢um isis the the single single sensor sensor corresponding corresponding to to thethe 𝑚m-th-th ping, ping, 𝑁Np is is thethe totaltotal numbernumber ofof pings,pings, σ𝜎kis is a a virtual virtual target, target,N 𝑁x is is the the number number of of the the grid grid in inthe thex direction,𝑥 direction,Ny is𝑁 the is numberthe number of the ofgrid the grid in the iny thedirection, 𝑦 direction, (Xc,Yc )( is𝑋 the,𝑌) center is the point center of point the area of of theinterest, area ofX interest,0 is the half-size 𝑋 is the of half-size the area of of the interest area of in interest the x direction, in the 𝑥 Ydirection,0 is the half-size 𝑌 is the of half-size the area of ofinterest the area in of the interesty direction. in the 𝑦 direction.

When obtaining solutions for BP or BPDN, the elements of 𝛔 with a large 𝑙-norm of the corresponding sensing matrix column vector tend to have a non-zero value. To eliminate this bias, each column vector 𝒂 of the sensing matrix 𝐀 and the received signal 𝐬 is normalized by their 𝑙-norms [14].

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When obtaining solutions for BP or BPDN, the elements of σ with a large l1-norm of the corresponding sensing matrix column vector tend to have a non-zero value. To eliminate this bias, each column vector ai of the sensing matrix A and the received signal s is normalized by their l2-norms [14]. s snormal = , (32) ksk2 " #T a1 a2 aNx Ny Anormal = , ,..., . (33) ka1k2 ka2k2 kaNx Ny k2 Therefore, Equations (30) and (31) are modiﬁed, and the ﬁnal solution is obtained by compensating the l2-norms to the σ estimated from Equation (34) or Equation (35) as follows: min kσk subject to snormal = Anormalσ, (34) N N 1 σ∈C x y

min kσk subject to ksnormal − Anormalσk < λ. (35) N N 1 2 σ∈C x y

3.3. CS-SAS Algorithm for Uniform Line Array The algorithm proposed in Section 3.2 is a method used for a single sensor. However, in many cases, a uniform array sonar consisting of one transmitter and multiple receivers is used, and it needs to be extended to the algorithm for a uniform line array. The algorithm for the uniform linear array introduced in this section is the same as the algorithm for a single sensor, except for the travel time function. The transmitter in the m-th ping is

utm, m = 1, ... , Np, and its position vector is rutm . The number of receivers in the physical array is Nu, and the n-th receiver in the physical array is un, n = 1, ... , Nu. Therefore, Equations (22) and (23) can be reformulated as follows:

Nx Ny ( ) = ( ( − ( ))) ≤ − ( ) ≤ s t, run , rutm ∑ σkexp jω t τ rk, run , rutm 0 t τ rk, run , rutm Tp , (36) k=1

|r − r | + |r − r | τ(r , r , r ) = k un k utm . (37) k un utm c

The signal vector composed of measurements received at a speciﬁc time ti of each receiver in the j-th ping is denoted as si,j. Therefore, the signal vectors si,j can be arranged as

si,j = Ai,jσ, (38)

h iT = si,j s ti, ru1 , rutj , s ti, ru2 , rutj , . . . , s ti, ruNu , rutj , (39) Ai,j(m, k) = exp jω ti − τ rk, rum , rutj 0 ≤ ti − τ rk, rum , rutj ≤ Tp . (40) Similar to the previous section, the following formulas are obtained:

s = Aσ, (41)

T h T T T T T Ti s = si,1 , si,2 ,..., si,Np , si+1,1 , si+1,2 ,..., si+Nt−1,Np , (42)

T h T T T T T Ti A = Ai,1 , Ai,2 ,..., Ai,Np , Ai+1,1 , Ai+1,2 ,..., Ai+Nt−1,Np . (43) The CS framework can be applied by formulation as above. Remote Sens. 2021, 13, 1924 9 of 19

Remote Sens. 2020, 12, x FOR PEER REVIEW 9 of 19 4. Results simulationIn this and section, experimental the performance data. In of the the si proposedmulation algorithmsand in the isexperiment, demonstrated the bycarrier com- frequenciesparing the resultsof the CW of applying signal were the ω400-k algorithmand 455 kHz, and respectively, the CS-SAS algorithmswhereas the to sampling both the frequenciessimulation andwere experimental 25 or 50 kHz data. and, Intherefore, the simulation the ω-k andalgorithm in the experiment,included the thebaseband carrier process.frequencies of the CW signal were 400 and 455 kHz, respectively, whereas the sampling fre- quenciesThe werefollowing 25 or 50shows kHz and,that therefore,the CS-SAS the algoω-krithms algorithm exhibit included superi theor baseband performance process. in termsThe of resolution following and shows noise that robustness the CS-SAS and algorithms indicates how exhibit to become superior robust performance when com- in batingterms ofconditions resolution where and noisesensors robustness are not working and indicates or data how are tolost. become robust when com- bating conditions where sensors are not working or data are lost. 4.1. Simulation Results 4.1. Simulation Results 4.1.1.4.1.1. Simulation Simulation Results Results for for Single Single Sensor Sensor ω TheThe ω-k-k and and CS-SAS CS-SAS algorithms algorithms were were compared compared for for various various cases. cases. For For the the singlesingle- sensorsensor SAS, SAS, five ﬁve cases cases were were simulated. Th Thee basic basic simulation simulation environment environment was was a a singlesingle- sensorsensor sonar sonar operated operated at at 0.02 0.02 m m intervals intervals from from −5− to5 5 to in 5 the in thecross-range cross-range axis, axis, that thatis, 𝑁 is, = 501. The signal 𝑝(𝑡) was a CW signal with carrier frequency 𝑓 = 400 kHz and pulse Np = 501. The signal p(t) was a CW signal with carrier frequency fc = 400 kHz and pulse duration 𝑇 = 0.1 ms. The center point of the area of interest was [𝑋 ,𝑌] = [250,0], the duration Tp = 0.1 ms. The center point of the area of interest was [Xc,Yc] = [250, 0], the 𝑋 half-sizehalf-size of of the the area area of of interest interest in in slant-range slant-range was was X0 = 0.5 m, the half-size of the area of 𝑌 𝑓 interestinterest in cross-rangecross-range waswasY 0= = 1 1 m, m, the the sampling sampling frequency frequency was wasfs = 50 = kHz, 50 kHz, and and the areathe areaof interest of interest consisted consisted of N Xof= 𝑁 101 = and 101N andY = 101𝑁 =uniform 101 uniform grid points. grid points. The sound The sound speed speed was c was= 1500 𝑐 =m/s. 1500 Twelve-point m/s. Twelve-point targets targets wereplaced, were placed, as described as described in Figure in Figure4. 4.

FigureFigure 4. 4. LocationsLocations of of the the 12 12 targets. targets.

AllAll simulation simulation environments environments for for single-sen single-sensorsor sonar were were fundamentally the same as thethe basic basic simulation simulation environment environment described described above above and and were performed by changing the noise level, Xc, fs, and sonar interval, as shown in Table1. noise level, 𝑋, 𝑓, and sonar interval, as shown in Table 1.

TableTable 1. Simulation Case for Single Sensor.

Case Noise Existence Xc (m) fs (kHz) Sonar IntervalSonar (m) Case Noise Existence 𝑿 (m) 𝒇 (kHz) 1 X 250𝒄 50𝒔 Interval 0.02 (m) 21 XX 250250 2550 0.020.02 3 X 150 50 0.2 42 XX 150250 5025 0.40.02 53 OX 250150 5050 0.020.2 4 X 150 50 0.4 5 O 250 50 0.02 • Case 1

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Remote Sens. 2020, 12, x FOR PEER REVIEW 10 of 19 • Case 1 As shown in Figure 5, it was confirmed that the images obtained through the CS-SAS As shown in Figure5, it was confirmed that the images obtained through the CS-SAS algorithm accurately distinguished 12 targets and had a good azimuth resolution. In ad- algorithm accurately distinguished 12 targets and had a good azimuth resolution. In dition, it was confirmed that there was no performance degradation caused by the side addition, it was confirmed that there was no performance degradation caused by the side lobes. However, the result for the ω-k algorithm is unable to distinguish between targets lobes. However, the result for the ω-k algorithm is unable to distinguish between targets adjacent to each other in the center of the area of interest. In addition, much aliasing oc- adjacent to each other in the center of the area of interest. In addition, much aliasing occurs, curs, especially in the azimuth direction. On account of the influence of matched filtering especially in the azimuth direction. On account of the influence of matched filtering and Fourier transform,and Fourier the exacttransform, position the of exact the pointposition targets of the cannot point betargets obtained, cannot resulting be obtained, in a resulting blurred result.in a blurred result.

(a) (b)

Figure 5. SimulationFigure results 5. Simulation for Case 1 results for single for sensor: Case 1 for(a) result single for sensor: the ω (a-k) resultalgorithm; for the (b)ω result-k algorithm; for the CS-SAS (b) result algorithm for single sensor. for the CS-SAS algorithm for single sensor. • • Case 2 Case 2 The results of the simulation are shown in Figure 6. In the case of the ω-k algorithm, The results of the simulation are shown in Figure6. In the case of the ω-k algorithm, because the sampling frequency is reduced by half compared to Case 1, the resolution in because the sampling frequency is reduced by half compared to Case 1, the resolution the slant-range direction is reduced. The aliasing at the center of the area of interest has a in the slant-range direction is reduced. The aliasing at the center of the area of inter- larger value than the values at the four target locations, [250 ± 0.02, 0] and [250 ± 0.04, 0]. est has a larger value than the values at the four target locations, [250 ± 0.02, 0] and Nonetheless, the image obtained using the CS-SAS algorithm yielded an accurate target [250 ± 0.04, 0]. Nonetheless, the image obtained using the CS-SAS algorithm yielded an image. accurate target image.

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Remote Sens. 2020, 12, x FOR PEER REVIEW 11 of 19 Remote Sens. 2020, 12, x FOR PEER REVIEW 11 of 19

(a(a) ) (b(b) ) Figure 6. Simulation results for Case 2 for single sensor: (a) result for the ω-k algorithm; (b) result for the CS-SAS algorithm Figure 6. SimulationFigure results 6. Simulation for Case results2 for single for Casesensor: 2 for (a) singleresult sensor:for the ω (a-k) result algorithm; for the (bω) result-k algorithm; for the CS-SAS (b) result algorithm for single sensor. for single sensor.for the CS-SAS algorithm for single sensor. • • Case 3• CaseCase 3 3 The results are shown in Figure 7. The CS-SAS algorithm accurately fetches the im- The results areThe shown results in are Figure shown7. The in Figure CS-SAS 7. algorithm The CS-SAS accurately algorithm fetches accurately the images fetches the images of 12 targets. However, the ω-k algorithm requires a Fourier transform in the space of 12 targets.ages However, of 12 targets. the ω-k However, algorithm the requires ω-k algorithm a Fourier transformrequires a in Fourier the space transform domain, in the space domain, which violates Nyquist theory because the sampling level in the space domain is which violatesdomain, Nyquist which theory violates because Nyquist the sampling theory because level in the the sampling space domain level in is reducedthe space domain is reduced to 1/10. Therefore, aliasing occurred, and the image of the target could not be to 1/10. Therefore,reduced aliasingto 1/10. occurred,Therefore, and aliasing the image occurred, of the and target the couldimagenot of the be properlytarget could not be properly obtained. The eight target points in the center were not distinguishable, and the obtained. Theproperly eight target obtained. points The in eight the center target were points not in distinguishable, the center were and not the distinguishable, remaining and the remaining four points were difficult to determine. four pointsremaining were difﬁcult four to points determine. were difficult to determine.

(a(a) ) (b(b) ) Figure 7. Simulation results for Case 3 for single sensor: (a) result for the ω-k algorithm; (b) result for the CS-SAS algorithm Figure 7. Simulation results for Case 3 for single sensor: (a) result for the ω-k algorithm; (b) result for the CS-SAS algorithm for single sensor.Figure 7. Simulation results for Case 3 for single sensor: (a) result for the ω-k algorithm; (b) result for single sensor. for the CS-SAS algorithm for single sensor. • • CaseCase 4 4

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Remote Sens. 2020, 12, x FOR PEER REVIEW 12 of 19 • Case 4 The resultsThe are results depicted are depicted in Figure in8. Figure Even when 8. Even the when spatial the sampling spatial sampling is reduced is toreduced a to a level of 1/20level and of 1/20 when and some when side some lobes side occur, lobes the occur, CS-SAS the algorithmCS-SAS algorithm still fetches still the fetches image the image Remote Sens. 2020, 12, x FOR PEERof REVIEW 12 targets, of 12 whereastargets, whereas the ω-k algorithmthe ω-k algorithm fails to depict fails to the depict proper the image proper of image the12 targets. of 19of the targets.

The results are depicted in Figure 8. Even when the spatial sampling is reduced to a level of 1/20 and when some side lobes occur, the CS-SAS algorithm still fetches the image of 12 targets, whereas the ω-k algorithm fails to depict the proper image of the targets.

(a) (b)

Figure 8. Simulation results for Case 4 for single sensor: (a) result for the ω-k algorithm; (b) result for the CS-SAS algorithm Figure 8.(a)Simulation results for Case 4 for single sensor:(b) ( a) result for the ω-k algorithm; (b) result for single sensor. for the CS-SAS algorithm for single sensor. Figure 8. Simulation results for Case 4 for single sensor: (a) result for the ω-k algorithm; (b) result for the CS-SAS algorithm for single sensor. • • Case 5 Case 5 Noisy conditions with signal-to-noise ratios (SNRs) of 20, 10, and 5 dB were simu- • NoisyCase conditions 5 with signal-to-noise ratios (SNRs) of 20, 10, and 5 dB were simulated. lated. The results in Figure 9 indicate that the CS-SAS algorithm applied to an environ- The resultsNoisy in conditions Figure9 indicate with signal-to-noise that the CS-SAS ratios algorithm (SNRs) of applied20, 10, and to an5 dB environment were simu- with ment with SNRs of 20 dB and 10 dB, obtained an almost accurate image of the targets. SNRslated. of 20The dB results and 10in dB,Figure obtained 9 indicate an almostthat the accurate CS-SAS imagealgorithm of theapplied targets. to an However, environ- when However, when the SNR was 5 dB, the values at the grid point between [250, ±0.04] and thement SNR with was SNRs 5 dB, of the 20 values dB and at 10 the dB, grid obtained point betweenan almost [250, accurate±0.04] image and of [250, the targets.±0.08] were [250, ±0.08] were greater than the values at [250, ±0.04] and [250, ±0.08], and an accurate greaterHowever, than when the values the SNR at [250,was 5 ±dB,0.04] the andvalues [250, at the±0.08], grid point and anbetween accurate [250, image ±0.04] couldand not [250, ±0.08]image were could greater not thanbe obtained. the values As at [250,the noise ±0.04] becameand [250, louder, ±0.08], anda degree an accurate of degradation oc- be obtained. As the noise became louder, a degree of degradation occurred. Nevertheless, image curred.could not Nevertheless, be obtained. Asthe the CS-SAS noise becamealgorithm louder, was a stilldegree superior of degradation to the ωoc--k algorithm in the CS-SAS algorithm was still superior to the ω-k algorithm in terms of resolution and curred.terms Nevertheless, of resolution the CS-SAS and sidelobe algorithm suppression. was still superior to the ω-k algorithm in sidelobeterms of suppression. resolution and sidelobe suppression.

(a) (b) (c) (a) (b) (c) FigureFigure 9. Simulation 9. Simulation results results of of CS-SASCS-SAS algori algorithmthm forfor Case Case 5 for 5single for singlesensor: sensor:(a) SNR = ( a20) SNRdB; (b) = SNR 20 dB;= 10 (dB;b) ( SNRc) SNR = = 10 dB; 5 dB. (c) SNR =Figure 5 dB. 9. Simulation results of CS-SAS algorithm for Case 5 for single sensor: (a) SNR = 20 dB; (b) SNR = 10 dB; (c) SNR = 5 dB.

Remote Sens. 2021Remote, 13 ,Sens. 1924 2020, 12, x FOR PEER REVIEW 13 of 19 13 of 19

4.1.2. Simulation Results for Uniform Line Array 4.1.2. SimulationThe Results CS-SAS for Uniformand ω-k Linealgorithms Array for a uniform line array were simulated in two cases. The first simulation case is as follows: The simulation environments of sampling The CS-SAS and ω-k algorithms for a uniform line array were simulated in two cases. frequency 𝑓 and sound velocity 𝑐, excluding sonar configuration and 𝑋 , are the same The first simulation case is as follows: The simulation environments of sampling frequency as those of simulation Case 1 for a single sensor. The array has 20 receivers, with a 0.04 m fs and sound velocity c, excluding sonar configuration and Xc, are the same as those of simulationspacing Case 1 forbetween a single receivers, sensor. as The shown array in has Figure 20 receivers, 10a. By withmoving a 0.04 0.4 m m spacing between pings, a between receivers,total of 25 as pings shown were in Figureshot. The 10a. source By moving position 0.4 of m the between uniform pings, line array a total is of 0.1 m away 25 pings werefrom shot. the first The sensor source in position the cross-range of the uniform direction. line Conditions array is 0.1 for m awayother fromsimulations the are the first sensorsame in the as cross-rangefor Case 1, except direction. that Conditionsthe sensor spacing for other of simulationsthe array is different. are the same The array has as for Casetwo 1, exceptreceivers, that with the sensor0.4 m spacing spacing between of the array receivers, is different. as shown The in array Figure has 10b. two By moving receivers, with0.4 m 0.4 between m spacing pings, between a total receivers,of 25 pings as were shown shot. in FigureThe source 10b. Byposition moving of the 0.4 muniform line between pings,array ais total 0.1 m of away 25 pings from were the shot.first sensor The source in the position cross-range of the direction. uniform line array is 0.1 m away from the first sensor in the cross-range direction.

(a)

(b)

Figure 10. DescriptionFigure 10.of lineDescription array in: of (a line) the array first in: case (a) of the simulation; ﬁrst case of ( simulation;b) the second (b) case the second of simulation; case of simulation; the red box is a transmitter, and the black boxes are receivers. The black-dash boxes in (b) are actually empty but marked for comparison the red box is a transmitter, and the black boxes are receivers. The black-dash boxes in (b) are actually with (a). empty but marked for comparison with (a). • • Case 1 Case 1 The results for Case 1 are shown in Figure 11. The CS-SAS algorithm accurately The resultsfetched for images Case 1 areof the shown 12 targets. in Figure Contrarily, 11. The CS-SAS the result algorithm for the accurately ω-k algorithm fetched showed ali- ω images of theasing. 12 In targets. particular, Contrarily, the total the number result for of thesensors-k algorithmused were showed500 = 20 aliasing.× 25, which In compares × particular,favorably the total number to the simulation of sensors usedenvironment were 500 of = a 20 single25, sensor; which however, compares the favorably interval between to the simulationthe sensors environment doubled of to a 0.04, single and sensor; the spatial however, sampling the interval interval between also thedoubled, sensors resulting in doubled to 0.04, and the spatial sampling interval also doubled, resulting in aliasing near aliasing near [250, ±0.07]. [250, ±0.07].

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RemoteRemote Sens.Sens. 20202020,, 1212,, xx FORFOR PEERPEER REVIEWREVIEW 1414 ofof 1919

((aa)) ((bb)) FigureFigure 11.11. SimulationSimulationFigure resultsresults 11. Simulation forfor CaseCase 11 for resultsfor uniformuniform for Case lineline array: 1array: for uniform ((aa)) resultresult line forfor array:thethe ωω-k-k (a algorithm;)algorithm; result for ((b theb)) resultresultω-k algorithm;forfor thethe CS-SASCS-SAS algorithm for uniform line array. algorithm for uniform(b) result line forarray. the CS-SAS algorithm for uniform line array. • • Case• 2 CaseCase 22 The results for Case 2 are shown in Figure 12. The synthetic aperture is the same as The resultsThe for results Case 2for are Case shown 2 are in shown Figure in12 .Figure The synthetic 12. The synthetic aperture isaperture the same is asthe same as in Case 1, but the sensor spacing is increased 10 times. Severe aliasing occurred in the ω- in Case 1,in but Case the 1, sensor but the spacing sensor spacing is increased is increased 10 times. 10 Severetimes. Severe aliasing aliasing occurred occurred in the in the ω- k results as well as the inability to properly identify the targets. However, the CS-SAS ω-k resultsk results as well as as well the inability as the inability to properly to properly identify theidentify targets. theHowever, targets. However, the CS-SAS the CS-SAS algorithm was significantly better distinguished. algorithmalgorithm was signiﬁcantly was significantly better distinguished. better distinguished.

((aa)) ((bb)) Figure 12. Simulation results for Case 2 for uniform line array: (a) result for the ω-k algorithm; (b) result for the CS-SAS Figure 12. SimulationFigure results 12. Simulation for Case 2 resultsfor uniform for Case line 2array: for uniform (a) result line for array:the ω-k (a )algorithm; result for ( theb) resultω-k algorithm;for the CS-SAS algorithmalgorithm forfor aa uniformuniform lineline array.array. (b) result for the CS-SAS algorithm for a uniform line array. The performances of the CS-SAS and ω-k algorithms were compared using simula- The performancesThe performances of the CS-SAS of the and CS-SASω-k algorithms and ω-k werealgorithms compared were using compared simulation using simulation results for a single sensor and uniform line array. In the case of the ω-k algorithm, results fortion a single results sensor for a andsingle uniform sensor line and array. uniform In the line case array. of theIn theω-k case algorithm, of the ω even-k algorithm, even under the most naïve simulation conditions, adjacent targets could not be distin- under theeven most under naïve the simulation most naïve conditions, simulation adjacent condit targetsions, adjacent could not targets be distinguished could not be distinguished and aliasing occurred, whereas in the case of the proposed algorithm, because CS and aliasingguished occurred, and aliasing whereas occurred, in the case whereas of the in proposed the case of algorithm, the proposed because algorithm, CS was because CS was applied and sidelobes were rather suppressed, high-resolution results were obtained. applied andwas sidelobes applied and were sidelobes rather suppressed, were rather high-resolutionsuppressed, high-resolution results were results obtained. were In obtained. InIn effect,effect, CS-SASCS-SAS hashas clearlyclearly distinguisheddistinguished tatargetsrgets underunder harsherharsher conditionsconditions byby increasingincreasing

Remote Sens. 2020, 12, x FOR PEER REVIEW 15 of 19

Remote Sens. 2021, 13, 1924 15 of 19 spatial sampling or reducing 𝑓, and has obtained accurate locations and shown robustness in noisy situations. This is possible because the measured signal can be expressed in Remote Sens. 2020, 12, x FORa PEER sparse REVIEW representation for a certain domain, and CS can significantly lower15 the of 19 sampling effect, CS-SAS has clearly distinguished targets under harsher conditions by increasing rate and has robust resistance to noise [36,39]. spatialspatial sampling oror reducingreducingf s𝑓, and, and has has obtained obtained accurate accurate locations locations and and shown shown robustness robust- nessin noisy in noisy situations. situations. This This is possible is possible because because the the measured measured signal signal can can be be expressed expressed in in a 4.2. Experimentalasparse sparse representation representation Data Results for for a a certain certain domain, domain, and and CSCS cancan signiﬁcantlysignificantly lowerlower thethe sampling Thisraterate andwas and has a waterrobust resistance tank experiment to noise [[36,39].36 conducte,39]. d by SonaTech Inc. As shown in Figure 13a, an4.2. experiment Experimental Experimental Data Datawas Results Resultsperformed to obtain images of two rings in a water tank. As shown in Figure 13b, the sonar has one transmitter and 32 receivers. After transmitting This was a water tank experiment conducte conductedd by by SonaTech Inc. As (Santa shown Barbara, in Figure CA, and receiving13a,USA). an As experiment shownthe signal in Figure was once, performed 13a, the an experiment transmitter to obtain was images moves performed of two616.5 to rings obtain mm in images anda water then of twotank. sends rings As and re- ceivesshown inthe a waternext in Figure tank.signal. As13b, This shown the processsonar in Figure has was one 13b, transmrepe the sonarateditter hassevenand one 32 timesreceivers. transmitter to receiveAfter and transmitting 32 receivers.signals from 224 After transmitting and receiving the signal once, the transmitter moves 616.5 mm and then locations.and receivingThe ping the signal signal 𝑝once,(𝑡) the is a transmitter CW signal moves with 616.5 carrier mm frequencyand then sends 𝑓 and = 455 re- kHz and sends and receives the next signal. This process was repeated seven times to receive signals ceives the next𝑇 signal. This process was repeated seven times𝑓 to receive signals from 224 pulse durationfrom 224 locations. = 0.3 The ms, ping the signal samplingp(t) is a frequency CW signal with is carrier = 50 frequency kHz, andfc = the 455 sound kHz speed locations. The ping signal 𝑝(𝑡) is a CW signal with carrier frequency 𝑓 = 455 kHz and 𝑐 and pulse duration T = 0.3 ms, the sampling frequency is f = 50 kHz, and the sound speed = 1480pulse m/s. duration There 𝑇 are = p0.3 two ms, ring-shaped the sampling frequency targets of is approximate𝑓 s = 50 kHz, and length the sound of majorspeed axis 1.5 c = 1480 m/s. There are two ring-shaped targets of approximate length of major axis 1.5 m m each𝑐 in = 1480 the m/s.slant-range There are of two 7 toring-shaped 10 m and targets a cross-range of approximate of −2.4385 length ofto major2.4385 axis m. 1.5 each in the slant-range of 7 to 10 m and a cross-range of −2.4385 to 2.4385 m. m each in the slant-range of 7 to 10 m and a cross-range of −2.4385 to 2.4385 m.

(a) (b) Figure 13. Water tank experiment(a) overview: (a) water tank structure; (b) synthetic aperture(b) line-array sonar.

Figure 13. Water Figuretank experiment 13. Water tank overview: experimentThe raw ( overview:a )data water recorded (tanka) water structure; in tankthe slant-range structure; (b) synthetic (b )ar synthetice shown aperture aperture in Figure line-array line-array 14a. The sonar.sonar. CS-SAS result was derived by dividing the area of interest into a uniform grid of 𝑁 = 101 and 𝑁 = The raw data recorded in the slant-range are shown in Figure 14a. The CS-SAS result The101. raw The dataresults recorded of the ω-k in and the CS-SAS slant-range algorithms are areshown shown in in Figure Figure 14a. 14b andThe 14c, CS-SAS re- result was derived by dividing the area of interest into a uniform grid of NX = 101 and NY = 101. spectively. was derivedThe results by ofdividing the ω-k and the CS-SAS area of algorithms interest are into shown a uniform in Figure grid14b,c, of respectively. 𝑁 = 101 and 𝑁 = 101. The results of the ω-k and CS-SAS algorithms are shown in Figure 14b and 14c, respectively.

(a) (b) (c)

Figure 14. (a) RawFigure data; 14. (b )(a result) Raw for data; the (ωb)-k result algorithm; for the ( cω)-k result algorithm; for the CS-SAS(c) result algorithm. for the CS-SAS algorithm.

From the raw data, the targets can be seen in the form of rings, but the shape appears thick, and it is difficult to accurately determin e the location of the targets. In the results of (a) the ω-k algorithm, the shapes(b) are slightly thinner, but aliasing is severe(c) in the azimuth Figure 14. (a) Raw data; (b) result for the ω-k algorithm; (c) result for the CS-SAS algorithm.

From the raw data, the targets can be seen in the form of rings, but the shape appears thick, and it is difficult to accurately determine the location of the targets. In the results of the ω-k algorithm, the shapes are slightly thinner, but aliasing is severe in the azimuth

Remote Sens. 2021, 13, 1924 16 of 19

From the raw data, the targets can be seen in the form of rings, but the shape appears Remote Sens. 2020, 12, x FOR PEER REVIEWthick, and it is difﬁcult to accurately determine the location of the targets. In the results16 of 19 of the ω-k algorithm, the shapes are slightly thinner, but aliasing is severe in the azimuth direction,direction, and and it it appears appears that that there there are severa severall rings. The The results results of the CS-SAS algorithm constructconstruct tworing-shaped tworing-shaped targets. targets. Because Because the the CS-SAS CS-SAS algorithm algorithm attempts attempts to bring to bring an im- an ageimage with with as aslittle little target target distribution distribution as as possi possible,ble, the the side side lobes lobes are are suppressed suppressed to to obtain thinthin ring-shaped ring-shaped targets. targets. ToTo examine whether the CS-SAS algorithm is robust under conditions where some ofof the the sensors sensors are are broken, broken, the the result was derived by assuming a situation in which data fromfrom some some sensors sensors were were lost. lost. The The experiment experiment was was divided divided into into two two cases: cases: one one case case where where thethe sensor sensor failed failed uniformly uniformly (Sensor (Sensor Loss: Loss: Un Uniform,iform, SLU) SLU) and and another another case casewhere where the sen- the sorsensor failed failed randomly randomly (Sensor (Sensor Loss: Loss: Random, Random, SLR). SLR). The Thepercentages percentages of sensors of sensors that thatdid not did malfunctionnot malfunction and andoperated operated normally normally are also are alsoindicated indicated in the in results. the results. The results The results are dis- are playeddisplayed in Figure in Figure 15. 15.

(a) (b) (c)

(d) (e) (f)

Figure 15. Results for some sensor loss conditions. Uniformly Uniformly brok brokenen case case (SLU) (SLU) where where the pe percentagercentage of the remaining sensors are: (a) 75%; (b) 50%; (c) 25%. Randomly broken case (SRU) where the percentage of the remaining sensors are (d) sensors are: (a) 75%; (b) 50%; (c) 25%. Randomly broken case (SRU) where the percentage of the remaining sensors are 75%; (e) 50%; (f) 25%. (d) 75%; (e) 50%; (f) 25%.

InIn addition, addition, peak signal-to-noisesignal-to-noise ratioratio (PSNR) (PSNR) and and structural structural similarity similarity index index measure measure(SSIM), (SSIM), which which are are representative representative image image quality quality measurement measurement indicatorsindicators [40,41], [40,41], were calculated for quantitative comparison. PSNR is an index that evaluates loss information for the quality of the generated or compressed images and is expressed by the peak signal/mean square error (MSE) term. It has a unit of dB, and a higher value indicates less loss, i.e., higher image quality. SSIM is an index designed to evaluate differences in human visual quality rather than numerical errors. SSIM quality is evaluated in three aspects:

Remote Sens. 2021, 13, 1924 17 of 19

calculated for quantitative comparison. PSNR is an index that evaluates loss information for the quality of the generated or compressed images and is expressed by the peak signal/mean square error (MSE) term. It has a unit of dB, and a higher value indicates less loss, i.e., higher image quality. SSIM is an index designed to evaluate differences in human visual quality rather than numerical errors. SSIM quality is evaluated in three aspects: luminance, contrast, and structural. However, since the correct answer image is not known, PSNR and SSIM were calculated for all sensor loss situations using Figure 14c as a reference image. The calculation results are shown in Table2.

Table 2. PSNR and SSIM.

SLU 75% SLU 50% SLU 25% PSNR (dB) 33.7691 29.2229 28.1871 SSIM 0.8850 0.6947 0.6029 SLR 75% SLR 50% SLR 25% PSNR (dB) 32.9409 30.6011 28.4189 SSIM 0.8609 0.7324 0.6178

In the case of some conventional methods such as ω-k, a Fourier transform in space is performed. Therefore, it is difficult to obtain results freely in the form of an array because linear sampling is not possible in space when some sensors in the array fail. However, the CS-SAS algorithm does not perform Fourier transform in space and has a formulation that is free in the form of an array and, therefore, it is easy to obtain a result in a sensor loss situation. In addition, it is difficult to detect significant performance degradation of up to 75% for both SLU and SLR, and 50% of the SLR show particularly good results; note that CS has the best performance for random array or random down sampling [42,43]. Table2 also shows that the random array results are generally better. In Table2, it can be seen that the image quality of SLR is high for both 50% and 25%. At 75%, the indicators of SLR are worse than at 75% of SLU. Because there is little deterioration in image quality in 75% of cases, it can be seen that the PSNR and SSIM simply show how similar the reconstruction results are to Figure 14c, rather than showing the results of image quality deterioration. When it reaches approximately 25%, both SLU and SLR seem to blur to some extent, but in terms of side lobe suppression, it still shows no inferiority over the ω-k algorithm. Using the CS-SAS algorithm in this study made it possible to obtain a higher resolution image than when using the conventional synthetic aperture sonar algorithm—the ω-k algorithm—and made it possible to reduce the problem of aliasing which also occurs in the conventional method. In addition, even with less spatial sampling, better results were obtained than compared to the conventional algorithm, and it was confirmed to be robust even when some sensors failed. Good results can be expected even if the number of sensors are reduced during actual sonar operation, and as a consequence, cost reduction is possible. Moreover, it is durable because it presents robust characteristics in failure situations. Results of actual experimental data were also observed, and it is expected that satisfying results will be obtained in the event that the CS-SAS algorithm is applied to a natural underwater environment.

5. Conclusions In this paper, we proposed an algorithm that applies compressive sensing (CS) to a synthetic aperture sonar (SAS) under the assumption that the target distribution in water is sparse. Through simulation, it was conﬁrmed that the proposed algorithm produces images with better resolution than the conventional SAS algorithm, the ω-k algorithm. In addition, because images obtained by the proposed method present very few and small side lobes, no deterioration of imaging performance occurs. Furthermore, even in the case of sampling at a low level that violates Nyquist theory in the time and space domain, a higher quality target image was obtained than with the ω-k algorithm. Remote Sens. 2021, 13, 1924 18 of 19

Real environment applicability was revealed for the proposed method when comparing the results with actual experimental data. The results conﬁrm that aliasing is reduced and side lobes are suppressed when applying the compressive sensing method. Contrarily, the ω-k algorithm does not obtain accurate target images due to aliasing. Importantly, it was conﬁrmed that the proposed method is robust in the event of some sensors of the sonar system failing or when some data are lost.

Author Contributions: Conceptualization, H.-m.C. and H.-s.Y.; methodology, H.-m.C.; software, H.- m.C.; validation, H.-s.Y. and W.-j.S.; formal analysis, H.-m.C.; investigation, H.-m.C.; resources, H.-s.Y. and W.-j.S.; data curation, H.-m.C.; writing—original draft preparation, H.-m.C.; writing—review and editing, H.-s.Y. and W.-j.S.; visualization, H.-m.C.; supervision, W.-j.S.; project administration, H.-s.Y.; funding acquisition, W.-j.S. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Data Availability Statement: The data presented in this study are not publicly available because these data belong to SonaTech Inc. Acknowledgments: This research was supported by the Agency for Defense Development in Korea under Contract No. UD190005DD and the SonaTech Inc under Contract No. C180005. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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