Robust Classification Method for Underwater Targets Using the Chaotic Features of the Flow Field

Journal of Marine Science and Engineering

Article Robust Classiﬁcation Method for Underwater Targets Using the Chaotic Features of the Flow Field

Xinghua Lin 1, Jianguo Wu 1,* and Qing Qin 2

1 School of Mechanical Engineering, Hebei University of Technology, Tianjin 300401, China; [email protected] 2 China Automotive Technology and Research Center Co., Ltd, Tianjin 300300, China; [email protected] * Correspondence: [email protected]

Received: 17 January 2020; Accepted: 10 February 2020; Published: 12 February 2020

Abstract: Fish can sense their surrounding environment by their lateral line system (LLS). In order to understand the extent to which information can be derived via LLS and to improve the adaptive ability of autonomous underwater vehicles (AUVs), a novel strategy is presented, which directly uses the information of the flow field to distinguish the object obstacle. The flow fields around different targets are obtained by the numerical method, and the pressure signal on the virtual lateral line is studied based on the chaos theory and fast Fourier transform (FFT). The compounded parametric features, including the chaotic features (CF) and the power spectrum density (PSD), which is named CF-PSD, are used to recognize the kinds of obstacles. During the research of CF, the largest Lyapunov exponent (LLE), saturated correlation dimension (SCD), and Kolmogorov entropy (KE) are taken into account, and PSD features include the number, amplitude, and position of wave crests. A two-step support vector machine (SVM) is built and used to classify the shapes and incidence angles based on the CF-PSD. It is demonstrated that the flow fields around triangular and square targets are chaotic systems, and the new findings indicate that the object obstacle can be recognized directly based on the information of the flow field, and the consideration of a parametric feature extraction method (CF-PSD) results in considerably higher classification success.

Keywords: object obstacle recognition; chaotic theory; ﬂow sensing; underwater robot; SVM

1. Introduction Vision and sonar are the primary sensory systems available to autonomous underwater vehicles (AUVs) for navigation [1–4]. However, because there are many obstacles near shore and the turbid environment is problematic, vision and sonar systems have many limitations for AUV operation. For example, sonar frequently produces false positives due to the multiple targets and relatively shallow water depth. Similarly, the function of the vision system is directly limited by the reduction of the visibility range due to the turbidity of water [5,6]. In addition to the diﬃculty in functionality, both vision and sonar systems require the acoustic or light energy to be sent into the ﬂuid incessantly; this is named the active sensing system. This leads to substantial power consumption on platforms. Due to these limitations, AUVs are rarely used in the cluttered environment. In order to overcome these limitations, the question of improving the sensing method and adaptive ability is of particular interest to AUVs. According to the research of biologists, many behaviors of fish are associated with their lateral line system (LLS), which is a versatile sensor organ [7]. Although it is unknown what specific information about stimuli is obtained by fish, the information can help fish to identify objects, track prey, and conserve energy while swimming in wakes [8–10]. Since LLS does not use light, the turbid environment has no affection on its function. LLS only focuses on the objects necessary for immediate navigation, and

J. Mar. Sci. Eng. 2020, 8, 111; doi:10.3390/jmse8020111 www.mdpi.com/journal/jmse J. Mar. Sci. Eng. 2020, 8, 111 2 of 17 complicated structures not in the vicinity are disregarded. In other words, LLS naturally emphasizes the closest stimuli, providing both advantages and limitations. Besides, LLS is a passive system, resulting in its power consumption being more than an order of magnitude smaller than that of traditional systems [11]. This can provide detailed aid in navigation with little additional cost. Based on the bionic principle, if this sensory system can be translated to AUVs, it would provide substantial benefits in maneuvering and object detection in difficult environments. In recent years, there has been increasing interest in flow sensing. The research can be divided into two parts. One is the engineering design of sensors based on setting nature as a model, which is called biomimetics [12–14], and the other one is the hydrodynamic mechanism of LLS [15–18]. Biomimetics focuses on natural sensors, which have evolved over billions of years, developing extreme efficiency, compactness, high responsivity, and throughput. Engineers employ a variety of sensing principles to imitate superficial neuromasts and canal neuromasts, such as the piezoresistive [19], capacitive [20], thermal [21], magnetic [22], piezoelectric [23], and optical [24] techniques. However, there is currently no analog to the sensor for AUV, because the type of processing and information extracted by the fish has yet to be discovered. Hydrodynamics mainly focuses on the detection and localization of the dipole source [25–27]. One main way is the detection of Karman vortex streets (KVS), such as the exact location, shedding frequency, and fluctuation range of a vortex [28,29]. Another technique is the study of detection algorithms and nonlinear dynamic models, which are based on the neural network scheme and potential flow theory [30–34]. However, the detection of KVS only focuses on the local information of the flow field, and the environment is different to the natural underwater world. The role of flow measurements in fish responses is also unclear. Without understanding the global information that can be derived via the lateral line organ, it is impossible to transfer LLS to man-made vehicles. The global information measurements in real systems are subject to uncertainties, which can complicate the obstacle recognition problem. To overcome this problem, obstacle recognition can be done based on nonlinear dynamics. Many existing features use the nonlinear characteristics of the flow field to recognize the obstacles. In this situation, the flow sensing problem becomes a problem of nonlinear detection. As is known to all, due to the turbulent wakes, downstream bluff-bodies can be very complex, and strongly intermittent behaviors are produced by large scale and small-scale coherent structures. In the study of automotive aerodynamics, chaotic theory has been used to study the dynamics of a 3D bimodal turbulent wake downstream of a square-back Ahmed body, and the low frequency dynamics of a turbulent 3D wake demonstrated that this is not a purely stochastic process but, rather, a weakly chaotic process exhibiting strange attractors [35]. A synchronization algorithm which uses a hydrodynamic analogy based on the transport of the properties of a particle convected in a fluid flow field was provided by Yechiel Crispin, and the synchronization algorithm was applied to two chaotic systems with variable parameters [36]. Inspired by this, the nonlinear characteristics of flow field can be obtained in the framework of chaotic systems theory. Regarding the flow sensing of object obstacles, two aspects are the most interesting: what the object obstacle is and where it is. In our previous works, the location method was introduced [37]. In the following text, a new flow sensing method for target recognition is presented. We first show how the flow field around different obstacles can be characterized by the chaotic features (CF). In particular, CF includes the largest Lyapunov exponent, the correlation dimension, and the Kolmogorov entropy of the pressure signal time series. The relationship between CF and object shapes is considered primarily, and the family of the flow field to which it belongs is evaluated. After recovering the chaotic pattern, the inner characteristics of the pressure signal are analyzed more thoroughly based on the fast Fourier transform (FFT). The power spectrum density (PSD) is used to distinguish the incidence angle of an obstacle with the same shape. A complete recognizing technique that only uses flow pressure data as the input is provided. In addition, the relationship between the compounded parametric features CF-PSD and the estimation accuracy is examined by the support vector machine (SVM). The results illustrate that the proposed method achieves better detection performance. J. Mar. Sci. Eng. 2020, 8, 111 3 of 17 J. Mar. Sci. Eng. 2020, 8, 111 3 of 18 previousThe results. paper is The organized signal asobtaining follows. method In Section and2, the the background pressure sign of theal around physical different model and targets numerical are shownmethod in isS described,ection 2. In and Section then the3, validationthe novel oftarget the numericalrecognition results strategy is strengthened is described, by previousincluding results. the recognitionThe signal of obtaining shapes and method incidence and the angle pressures, which signal depend around on CF different and PSD targets, respectively. are shown The in support Section2 . vectorIn Section machine3, the (SVM) novel is target selected recognition as the classifier strategy isin described, Section 4, includingand numerical the recognition experiments of shapes are taken and forincidence the evaluation angles, of which the proposed depend onstrategy. CF and In PSD, the experiments, respectively. three The support types of vector underwater machine target (SVM) are is analyzedselected and as the classified classifier under in Section different4, and incidence numerical angles. experiments Finally, concluding are taken for remarks the evaluation are provided of the inproposed Section 5 strategy.. In the experiments, three types of underwater target are analyzed and classified under different incidence angles. Finally, concluding remarks are provided in Section5. 2. Signal Acquisition and Processing Methods 2. Signal Acquisition and Processing Methods 2.1. Physical Model 2.1. Physical Model The flow fields around targets with different shapes and incidence angles are computed based on computationalThe flow fields fluid around dynamics targets (CFD with). differentThree types shapes of underwater and incidence targets angles are are analyzed computed, including based on circularcomputational, triangular fluid, and dynamics square (CFD).. We illustrat Three typese an example of underwater of a square targets target, are analyzed, as seen including in Figure circular, 1; its sidetriangular, length isand H square., and the We illustrate incidence an angle example is  of. T ahe square computational target, as seen domain in Figure  1is; its located side length in a Cartesianis H, and coordinate the incidence with angle a range is θ. of The 30 computationalH 10H . The virtual domain lateralΩ is located line is inunder a Cartesian the center coordinate of the with a range of 30H 10H. The virtual lateral line is under the center of the targets with a distance of d. targets with a distance× of d . The sensors are located at (xi , yi ) , and their range is shown in Equation The sensors are located at (x , y ), and their range is shown in Equation (1). The disturbance of sensors to (1). The disturbance of sensorsi toi the flow field is negligible. the flow field is negligible. ( −10HH  xii 1010H − ≤ ≤ (1)(1) yyi =-dd. .  i − where H is the characteristic length of the target, and d is the detective distance. where H is the characteristic length of the target, and d is the detective distance.

FigureFigure 1. Schematic 1. Schematic diagram diagram of the of the computational computational domain domain for for a square a square target target.. As with our previous works [37], the velocity-inlet and pressure-outlet boundaries are used to As with our previous works [37], the velocity-inlet and pressure-outlet boundaries are used to create a uniform flow environment. The lateral boundaries are described by the symmetry condition, create a uniform flow environment. The lateral boundaries are described by the symmetry condition, and the surface of the targets is seen as a no-slip boundary. A grid increasing function is built to mesh and the surface of the targets is seen as a no-slip boundary. A grid increasing function is built to mesh the computational domain Ω, which is shown in Equation (2). The surface of the target is meshed with the computational domain  , which is shown in Equation (2). The surface of the target is meshed χmin, and the further domain is meshed with χ(i). The ratio of the grid size between two adjacent with min , and the further domain is meshed with (i) . The ratio of the grid size between two layers is δ, and the outside surface of the computational domain Ω is meshed χmax. The square adjacent layers is  , and the outside surface of the computational domain  is meshed max . The target is used to study the grid independence, and three cases with different minimum sizes χmin are squaretaken target into account. is used to When studyδ isthe 1.2, grid the independence, Strouhal number andS threet and cases its percentage with different changes minimum are shown sizes in Tablemin are1. Ittaken is observed into account. that there When is no significantis 1.2, the changeStrouhal between number Grid St and 1 and its Grid percentage 2. However, changes the arerequired shown computerin Table 1. resource It is observed of Grid that 1 is there larger is thanno significant that of Grid change 2. Therefore, between GridGrid 21 isand used Grid in the2. However,investigations the required presented. computer resource of Grid 1 is larger than that of Grid 2. Therefore, Grid 2 is ( used in the investigations presented. χ(i) = δ χ · min . (2) χ χ(i) χmax min ≤ ≤

J. Mar. Sci. Eng. 2020, 8, 111 4 of 18

(i) =   min  . (2) min  (i)  max

J. Mar. Sci. Eng. 2020, 8, 111 4 of 17 Table 1. Analysis of grid independence.

Case min TableNumber 1. Analysis of Elements of grid independence.St Percentage Changes/% Grid 1 0.0001H 8.27 × 106 0.139 \ GridCase 2 0.0χ01minH Number6.26 of × Elements 105 0.S14t 2 Percentage Changes2.1 /% Grid 1 0.00010.01HH 4.828.27 × 101064 0.1390.153 7.2 Grid 3 × \ Grid 2 0.001H 6.26 105 0.142 2.1 × Grid 3 0.01H 4.82 104 0.153 7.2 2.2. Numerical Model and Validation ×

2.2.T Numericalhe flow field Model around and Validation different targets is solved based on the two-step Taylor-characteristic- based Galerkin method (TCBG), which was introduced by Bao et al. [38]. The solution method is shownThe in Eq flowuation field (3) around to Equation different (5) targets. There isare solved three basedsteps onin the the discretizing two-step Taylor-characteristic-based process: the half-time Galerkin methodn+1/ 2 (TCBG), which was introduced by Bao et al. [38]. The solution method is shown in velocity ui can be obtained from Equation (3) firstly, and then from Equation (5), the Equation (3) to Equation (5). There are three steps in the discretizing process: the half-time velocity relationship of p n+1 and u n+1 can be obtained. Finally, Equation (4) can be solved to get the full- un+1/2 can be obtained fromi Equation (3) firstly, and then from Equation (5), the relationship of pn+1 i n+1 n+1 time velocityn+1 ui and pressure p based on the above relationship. n+1 and ui can be obtained. Finally, Equation (4) can be solved to get the full-time velocity ui and n+1 pressure p based on the aboven relationship.n n 2 n n n n+1/ 2 n t n ui p 1  ij t n  n ui p 1  ij ui = ui − (u j + − ) + uk (u j + 2( − )) (3) 2 xn x n Re x n 8 x x n x nRe x n ∆t ∂u j ∂pi 1 ∂τj ij ∆t2 k∂ ∂iu j∂p 1 j∂τij n+1/2 = n ( n i + ) + n ( n i + ( )) ui ui uj uk uj 2 (3) − 2 ∂xj ∂xi − Re ∂xj 8 ∂xk ∂xi ∂xj − Re ∂xj n n+1 n+1/ 2 2 n n+1 n+1/ 2 n+1 n n+1/ 2 ui p 1  ij t n+1/ 2  n+1/ 2 ui p 1  ij ui = ui − t(u j + − n+1/2 ) + uk (u j + − )) n+1/2 (4) n n+1 ∂τ n n+1 ∂τ ∂u ∂xpj xi 1 Re ij x j ∆2t2 xk∂ xi ∂u xi ∂pRe x1j ij un+1 = un t(un+1/2 i + ) + un+1/2 (un+1/2 i + )) i i ∆ j k j (4) − ∂xj ∂xi − Re ∂xj 2 ∂xk ∂xi ∂xi − Re ∂xj 2 n+1 n+1/ 2 n+1/ 2  p 1   u 1  n+1/2 2 n+1 n n+1 n+1/ 2 i n+1/2 ij∂τ ∂ p =1 ∂ (ui −ui ) − ∂ (u j ∂u − 1 ij ) (5) x x= t x( n n+1) x ( n+1/2 xi Re x ) i i iui ui i uj j j (5) ∂xi∂xi ∆t ∂xi − − ∂xi ∂xj − Re ∂xj where u i is the i-component velocity,  is the water density, p is the pressure,  is the where ui is the i-component velocity, ρ is the water density, p is the pressure, τij is theij deviatoric deviatoricstresses, Restresses,is the ReynoldsRe is the number, Reynolds and number,n, n + 1 and/2, andn , nn++1 /denote2 , and then + time1 denote points the of ttimen, tn+ points1/2, and oft nt+n1, , respectively.tn+1/ 2 , and tn+1 , respectively. TheThe accuracy accuracy of ofthe the TCBG TCBG method method was was validated validated by by Bao Bao et etal. al. [3 [838]. ].In Inthis this section, section, the the numerical numerical algorithmalgorithm is isfurther further demonstrated demonstrated by by previous previous results results [37,39 [37,39,40],40 when] when it itis isused used to tosimulate simulate the the flow flow pastpast a asquare square target target with with different different incidence incidence angle angles.s. As As seen seen in in Fig Figureure 22,, thethe StrouhalStrouhal numbers numbers St are arein in accordance accordance with with previous previous reports, reports which, which implies implies that that there there is is reasonable reasonable agreement agreement between between our ournumerical numerical results results and and previous previous data. data.

0.20

Current study Zhao et al.[39] 0.18 Yoon et al.[40]

St 0.16

0.14

0.12 -10 0 10 20 30 40 50 θ FigureFigure 2. 2.SimulatedSimulated results results compar compareded with with previous previous results results..

2.3.2.3. Signal Signal Fusion Fusion M Methodethod A single sensor cannot respond to the flow structure accurately. Thus, multi-sensors are set at the lateral line, and the signal of each sensor is merged with a weighting fusion algorithm to get the final signal time series of the flow structure. The position of the sensors and the signal amplitude are J. Mar. Sci. Eng. 2020, 8, 111 5 of 17

considered in the algorithm. Thus, the position coefficient Si and the amplitude coefficient Ai form the instantaneous weight coefficient Ki, which is expressed by Equation (6).   K = S + A  i i i  ε sin αi  Si =  PN  sin α  i  i=0  (1 ε)Am(s ) = − i (6)  Ai N  P  Am(si)  i=0  N  P  cp(t) = KiSi(t)  i=0

where ε is the weight of position coefficient Si. If ε becomes larger, this implies that the influence of position is more obvious. Am(si) is the amplitude of a signal collected by sensor i. The flow fields around circular, triangular, and square targets are studied, respectively, based on CFD. The lateral line is set at the position of d = 2H, and the pressure signal Cp is collected and shown in Equation (7). pi p Cp = − ∞ , (7) 1 2 2 ρV

J.where Mar. Sci.p Eng.i is the2020 pressure, 8, 111 at the measuring point, p is the pressure at far field, which is 101,3255 Paof 18 in J. Mar. Sci. Eng. 2020, 8, 111 ∞ 5 of 18 this paper, ρ is the density of water, and V is the inlet velocity. 2.3. Signal Fusion Method 2.3. SignalThere Fusion are 21 measuringMethod points on the lateral line, and their signals are weighted and fused to obtainsignal the for time the series, circular which target is shownis periodic in the after Figure five 3seconds,. As seen and in Figurethe signals3, the for pressure the triangular signal for target the signal for the circular target is periodic after five seconds, and the signals for the triangular target andcircular square target target is periodic are aperiodic. after five The seconds, inner characte and theristics signals of for the the flow triangular field are target analyzed and square by chaotic target and square target are aperiodic. The inner characteristics of the flow field are analyzed by chaotic theoryare aperiodic. and fast TheFourier inner transform characteristics in the of following the flow fieldsection. are analyzed by chaotic theory and fast Fourier theory and fast Fourier transform in the following section. transform in the following section.

(a) (a) 0.050.05

0.000.00 p p c c -0.05-0.05

-0.10-0.10

00 5 5 10 10 15 15 20 20 25 25 30 tt

(b)(b) 0.20.2

0.00.0 p p c c -0.2-0.2

-0.4-0.4

00 5 5 10 10 15 15 20 20 25 25 30 tt (c)(c) 1.01.0 Figure 3. Cont.

0.50.5 p p c c 0.00.0

-0.5-0.5

-1.0 -1.0 0 5 10 15 20 25 30 0 5 10 15t 20 25 30 t Figure 3. Waveforms of the three types of signals ((a) circular target; (b) triangular target; (c) square Figure 3. Waveforms of the three types of signals ((a) circular target; (b) triangular target; (c) square target). target).

3. Target Recognition Strategy 3. Target Recognition Strategy 3.1. Feature Selection for Shape Classification 3.1. Feature Selection for Shape Classification 3.1.1. Phase Space Reconstruction 3.1.1. Phase Space Reconstruction The dynamic characteristics of a times series can be revealed by the chaos attractor, and we can The dynamic characteristics of a times series can be revealed by the chaos attractor, and we can use it to estimate whether the system is chaotic or stochastic. Therefore, attractor reconstruction is a use it to estimate whether the system is chaotic or stochastic. Therefore, attractor reconstruction is a key step in chaotic time series analyses. In order to reveal its chaos attractor, a high-dimensional space key step in chaotic time series analyses. In order to reveal its chaos attractor, a high-dimensional space is built to express the time series of the pressure signal, a method which was proposed by Takens is built to express the time series of the pressure signal, a method which was proposed by Takens

J. Mar. Sci. Eng. 2020, 8, 111 5 of 18

2.3. Signal Fusion Method signal for the circular target is periodic after five seconds, and the signals for the triangular target and square target are aperiodic. The inner characteristics of the flow field are analyzed by chaotic theory and fast Fourier transform in the following section.

(a) 0.05

0.00 p c -0.05

-0.10

0 5 10 15 20 25 30 t

(b) 0.2

0.0 p c -0.2

-0.4 J. Mar. Sci. Eng. 2020, 8, 111 6 of 17 0 5 10 15 20 25 30 t (c) 1.0

0.5 p c 0.0

-0.5

-1.0 0 5 10 15 20 25 30 t FigureFigure 3. 3. WaveformsWaveforms of the threethree typestypes of of signals signals ((a (()a circular) circular target; target; (b) triangular(b) triangular target; target; (c) square (c) square target). target). 3. Target Recognition Strategy

3.3.1. Target Feature Recognition Selection for Strategy Shape Classiﬁcation

3.1.3.1.1. Feature Phase Selection Space Reconstruction for Shape Classification

3.1.1. PhaseThe dynamic Space Reconstruction characteristics of a times series can be revealed by the chaos attractor, and we can use it to estimate whether the system is chaotic or stochastic. Therefore, attractor reconstruction is a keyThe step dynamic in chaotic characteristics time series analyses. of a times In orderseries to can reveal be revealed its chaos by attractor, the chaos a high-dimensional attractor, and we space can useis builtit to toestimate express whether the time the series system of the is pressurechaotic or signal, stochastic. a method Therefore, which wasattractor proposed reconstruction by Takens is [41 a ]. key step in chaotic time seriesn oanalyses. In order to reveal its chaos attractor, a high-dimensional space For a given time series cp(t) , the phase space can be expressed as Equation (8) with an optimal delay istime builtτ toand express an optimal the time embedding series of dimension the pressumre. signal, a method which was proposed by Takens

Cp(t ) = [cp(t ), cp(t + τ), cp(t + 2τ), ... , cp(t + (m 1)τ], i = 1, 2, ... , N (m 1)τ, (8) i i i i i − − − where τ is the delay time, m is the embedding dimension, and N is the length of the phase space. The C-C method is used to select the optimal delay time τ and the optimal embedding dimension m, a method which was proposed by Kim H.S. et al. [42]. The method has a strong ability to resist noise with little calculation cost, and the nonlinear characteristics of the time series can be kept. If the parameters meet the requirements of Equation (9), the correlation of pressure vibration intensity series can be reﬂected by S(t) and ∆S(t), which is shown in Equation (10). The solving progress is as follows:

1. The optimal delay time τ will correspond to the ﬁrst local minimum value times of ∆S(t); 2. The embedding dimension m can be obtained from the delay time window τm, where τm = (m 1) τ, and τm correspond to the minimum value of the quantity Scor(t). − ·   2 m 5;  1 ≤ ≤  σ r 2σ; (9)  2 ≤ ≤  N 500 ≥   5 4  ( ) = 1 P P ( )  S t 16 S m, rj, t ;  m=2 j=1   5  ( ) = 1 P ( )  ∆S t 4 ∆S m, t ;  m=2  (10)  Scor(t) = ∆S(t) + ∆S(t) ;   t  ( ) = 1 P [ ( ) m( )]  S m, rj, t t Cs m, rj, t Cs 1, rj, t ;  s=1 −  n o n o  ∆S(m, t) = max S(m, r , t) min S(m, r , t) j − j where r is the radius of time series, and σ is the time series standard deviation. J. Mar. Sci. Eng. 2020, 8, 111 7 of 17

3.1.2. The Largest Lyapunov Exponent (LLE) The Lyapunov exponent λ is an important parameter that is used to characterize chaotic time series, where the exponential rates of divergence or convergence of nearby trajectories in phase space can be reﬂected by λ. If LLE is positive, this implies that the phase plane trajectory is chaotic [43,44]. The Wolf algorithm is used to calculate the LLE [45], which has been proven to be more robust than the least-squares method [46]. In the reconstructed phase space, two adjacent points Cp,n1 and Cp,n1 + δCp are selected to calculate the distance δCp between those two points. Given a residual parameter ζ, if the distance is outside of ζ after one time iteration, new neighbor points will be selected repeatedly until the ﬁducial trajectory has traversed the entire data series. The Lyapunov exponent can be obtained from Equation (11), and LLE is the largest one λ.

 n( )  d f Cp,n1 λ n  δCp(n) = δCp,n1 = e · δCp,n1  dCp · ·  n 1  P d f (Cp) . (11)  λ = lim 1 − ln  n n dCp  →∞ i=0 Cp=Cp(i)

The LLEs of the pressure signal time series around the circular target, triangular target and square target are calculated using the above method. As seen in Figure4, linear regression is conducted to get λ in the range of time steps i from 15 to 25, and the results are shown in Figure5. From Figure5, the LLE ofJ. Mar. the Sci. ﬂow Eng. ﬁeld 2020, 8 around, 111 the circular target is negative, which implies that the time series8 of 18 is regular. However, the results of LLE around the triangular target and square target are positive, which implies J. Mar. Sci. Eng. 2020, 8, 111 8 of 18 their time series are chaotic sequences. 300

300 Triangular target Square target 200 Circular Triangular target target Square target 200 Circular target

y(i) 100

0 0 0 10 20 30 40 i 0 10 20 30 40 i Figure 4. Relation curve between y(i) and i for different targets. Figure 4. Relation curve between y(i) and i for diﬀerent targets. Figure 4. Relation curve between y(i) and i for different targets.

Figure 5. Largest Lyapunov exponent (LLE) of pressure signal time series in different flow fields. Figure 5. Largest Lyapunov exponent (LLE) of pressure signal time series in different flow fields.

3.1.3. TheFigure Saturated 5. Largest Correlation Lyapunov Dimension exponent (LLE (SCD)) of pressure signal time series in different flow fields.

3.1.3.The The fractal Saturated dimension Correlation of a reconstruction Dimension (SCD) attractor in phase space can be reflected by the SCD, which is calculated by the Grassberger–Procaccia (G-P) algorithm in this paper [47,48]. According to The fractal dimension of a reconstruction attractor in phase space can be reflected by the SCD, the algorithm, SCD can be derived from the correlation integral function, which is defined as which is calculated by the Grassberger–Procaccia (G-P) algorithm in this paper [47,48]. According to Equation (12) in the reconstructed phase space. the algorithm, SCD can be derived from the correlation integral function, which is defined as Equation (12) in the reconstructed phase space2 . M M C( ) = ( − dij ), r  0 M (M −1)    2 i=1Mj=iM+1 C( ) =  ( − dij ), r  0 dij = C p (Mti )(M−C−p,1()t j )  i=1 j=i+1 , (12) M = N − (m −1) dij = C p (ti ) −C p,(t j )  , (12)  0, x  0 M(x)==N− (m −1)  1, x  0  0, x  0 (x) =   where C( ) is the correlation integral function1, x  0, and (x) is the Heaviside function.

where C( ) is the correlation integral function, and (x) is the Heaviside function.

J. Mar. Sci. Eng. 2020, 8, 111 8 of 17

Another interesting result is that the LLEs of diﬀerent targets have no overlap. There are obvious characteristic features. This indicates that the shape of underwater targets can be identiﬁed by LLE.

3.1.3. The Saturated Correlation Dimension (SCD) The fractal dimension of a reconstruction attractor in phase space can be reﬂected by the SCD, which is calculated by the Grassberger–Procaccia (G-P) algorithm in this paper [47,48]. According to the algorithm, SCD can be derived from the correlation integral function, which is deﬁned as Equation (12) in the reconstructed phase space.

 M M  2 P P  C(τ) = Θ(τ dij), r > 0  M(M 1) −  − i=1 j=i+1   dij = Cp(ti) Cp,(tj)  , (12)  −  M = N (m 1)τ  −( −  0, x 0  Θ(x) = ≤  1, x > 0 J. Mar. Sci. Eng. 2020, 8, 111 9 of 18 where C(τ) is the correlation integral function, and Θ(x) is the Heaviside function. The SCD can be established by the asymptotic curve of ln C( ) ~ ln( ) . There is a linear The SCD can be established by the asymptotic curve of ln C(τ) ln(τ). There is a linear relationship ∼ betweenrelationship them, between which them, is shown which in is Equation shown in (13). Equation For the (1 pressure3). For the signal pressure of the signal circular of the target, circular the correlationtarget, the correlation integral function integral in function logarithmic in logarithmic scale is shown scale in is Figure shown6. in As Figure seen in 6. FigureAs seen6, thein Figure power 6, 1.0 lawthe power approximation law approximation is τ1.0, which is means  , which that SCD means is 1.0. that The SCD embedding is 1.0. The dimension embeddingm can dimension be obtained m bycan SCD be obtained based on by the SCD Equation based (14),on the and Equation the embedding (14), and dimension the embedding is 3. dimension is 3. ln(C( )) SCD = lim limln(C(τ)) , (13) SCD = limr→lim0 N → ln( ) , (13) r 0N ln(τ) → →∞ m m2SCD2SCD++11 (14)(14) ≥

-2

1.0

C(τ) ∝ τ

-4 ln

-6

-8 -6 -4 -2 0 ln(τ)

Figure 6. Relation curve between ln C(τ) and ln(τ) for the circular target. Figure 6. Relation curve between ln C( ) and ln( ) for the circular target. Using the same method, the SCD values of triangular and square targets can be obtained. The SCD increasesUsing with the an same increase method, in the the embedding SCD values dimension of triangularm, which and is square shown intargets Figures can7 and be 8obtained.. In Figure T7he, theSCD curve increases region with reaches an increas a saturatede in the state embedding gradually dimension as the embedding m , which dimension is shown in increases, Figures 7 and and the 8. convergenceIn Figure 7, value the curve SCD is region 4.715, reaches which means a saturated that the state embedding gradually dimension as the embedding is 11. For the dimension pressure signalincreases of, the and square the convergence target, the value correlation SCD is dimension 4.715, which is 4.826means and that the the embedding embedding dimensiondimension is 11. ThisFor the means pressure that the signal pressure of the intensity square timetarget, series the cancorrelation be completely dimension revealed is 4.826 in 11-dimensional and the embedding space. dimension is 11. This means that the pressure intensity time series can be completely revealed in 11- dimensional space.

-2 m=2

-4

m=12 lnC(τ) -6

-8

-10

-6 -4 -2 0 ln(τ)

Figure 7. Relation curve between and for the triangular target.

J. Mar. Sci. Eng. 2020, 8, 111 9 of 18

The SCD can be established by the asymptotic curve of ln C( ) ~ ln( ) . There is a linear relationship between them, which is shown in Equation (13). For the pressure signal of the circular target, the correlation integral function in logarithmic scale is shown in Figure 6. As seen in Figure 6, the power law approximation is  1.0 , which means that SCD is 1.0. The embedding dimension m can be obtained by SCD based on the Equation (14), and the embedding dimension is 3. ln(C( )) SCD = lim lim , (13) r→0 N → ln( )

m  2SCD+1 (14)

-2

1.0

C(τ) ∝ τ

-4 ln

-6

-8 -6 -4 -2 0 ln(τ)

Figure 6. Relation curve between ln C( ) and ln( ) for the circular target.

Using the same method, the SCD values of triangular and square targets can be obtained. The SCD increases with an increase in the embedding dimension m , which is shown in Figures 7 and 8. In Figure 7, the curve region reaches a saturated state gradually as the embedding dimension increases, and the convergence value SCD is 4.715, which means that the embedding dimension is 11. For the pressure signal of the square target, the correlation dimension is 4.826 and the embedding J.dimension Mar. Sci. Eng. is2020 11., Th8, 111is means that the pressure intensity time series can be completely revealed 9in of 11 17- dimensional space.

-2 m=2

-4

m=12 lnC(τ) -6

-8

-10 J. Mar. Sci. Eng. 2020, 8, 111 10 of 18 -6 -4 -2 0 ln(τ) J. Mar. Sci. Eng. 2020, 8, 111 10 of 18 Figure 7. Relation curve between ln C(τ) and ln(τ) for the triangular target. Figure 7. Relation0 curve between and for the triangular target. m=2 0

-4 m=2 m=11

-4 m=11 lnC(τ) -8

lnC(τ) -8 -12

-12 -6 -4 -2 0 ln(τ)

-6 -4 -2 0 ln(τ) Figure 8. Relation curve between ln C( ) and ln( ) for the square target. Figure 8. Relation curve between ln C(τ) and ln(τ) for the square target. The most interestingFigure 8. Relation result is curve that betweenthe SCD valuesln C( ) of and different ln( ) targetsfor the squarealso have target. no overlap, which is shownThe mostin Figure interesting 9. This resultresult isimplies that the that SCD the values SCD can of di beﬀ erentused to targets distinguish also have the no shapes overlap, of targets. which is shownThe inmost Figure interesting9. This resultresult impliesis that the that SCD the values SCD can of different be used to targets distinguish also have the no shapes overlap of targets., which is shown in Figure 9. This result implies that the SCD can be used to distinguish the shapes of targets. 6

6 4.715 4.826

4 4.715 4.826

4 SCD

2 SCD

2 1

1 0 Circular Triangular Square 0 Figure 9. Saturated CorrelationCircular Dimension (SCD)Triangular of pressure signalSquare time series in different flow fields. Figure 9. Saturated Correlation Dimension (SCD) of pressure signal time series in different flow fields. Figure 9. Saturated Correlation Dimension (SCD) of pressure signal time series in different flow 3.1.4. The Kolmogorov Entropy (KE) fields. The Kolmogorov entropy is an important parameter in a chaotic system. It can describe the rate 3.1.4. The Kolmogorov Entropy (KE) of information loss and is often used to quantify the unpredictability of a chaotic system. In order to measureThe theKolmogorov chaotic systems, entropy if isthe an embedding important dimensionparameter inincrease a chaotics from system. m to It canm + described , the KE the can rate be obtainedof information based losson the and G is-P often algorithm, used to as quantify shown in the Eq unpredictabilityuation (15). of a chaotic system. In order to measure the chaotic systems, if the embedding dimension increases from m to m + d , the KE can be 1 Cm (r) obtained based on the G-P algorithm, asK shown= lim lim in Equationln (15).. (15) r→0 m→ d Cm+d (r) 1 C (r) K = lim lim ln m Based on the above analysis, the KE of a fully predictable system. is zero; for a chaotic system,(15) r→0 m→ d Cm+d (r) KE is a positive and finite number, and KE has no limitation for a stochastic system. As seen in Figure 6, theBased embedding on the dimensionabove analysis, has no the influence KE of a onfully its predictableSCD in the flowsystem field is aroundzero; for the a chaotic circular system, target, whichKE is a implies positive that andK finitecir = 0 number. As seen, and in FigureKE has 10, no forlimitation the triangular for a stochastic target and system. square As target, seen in KE Figure will de6, thecrease embedding sharply with dimension an increase has noin theinfluence embedding on its dimensionSCD in the flowm at field a small around embedding the circular dimension. target, which implies that Kcir = 0 . As seen in Figure 10, for the triangular target and square target, KE will decrease sharply with an increase in the embedding dimension m at a small embedding dimension.

J. Mar. Sci. Eng. 2020, 8, 111 10 of 17

3.1.4. The Kolmogorov Entropy (KE) The Kolmogorov entropy is an important parameter in a chaotic system. It can describe the rate of information loss and is often used to quantify the unpredictability of a chaotic system. In order to measure the chaotic systems, if the embedding dimension increases from m to m + d, the KE can be obtained based on the G-P algorithm, as shown in Equation (15).

1 Cm(r) K = lim lim ln . (15) r 0m dτ C (r) → →∞ m+d Based on the above analysis, the KE of a fully predictable system is zero; for a chaotic system, KE is a positive and finite number, and KE has no limitation for a stochastic system. As seen in Figure6, the embedding dimension has no influence on its SCD in the flow field around the circular target, J. Mar. Sci. Eng. 2020, 8, 111 11 of 18 which implies that Kcir = 0. As seen in Figure 10, for the triangular target and square target, KE will decreaseWhen the sharply embedding with dimension an increase is in high the embeddingenough, KE dimensiontends to bem a atcommon a small value embedding. For the dimension. triangular When the embedding dimension is high enough, KE tends to be a common value. For the triangular target, the common value is Ktri = 4.431 and for the square target, Ksqu = 4.648 . This indicates that target, the common value is K = 4.431 and for the square target, Ksqu = 4.648. This indicates that another chaotic feature, KE, cantri be used to distinguish the shapes of targets. another chaotic feature, KE, can be used to distinguish the shapes of targets.

Triangular target

8 Square target KE

4 2 4 6 8 10 12 m

Figure 10. Relation curve between the Kolmogorov entropy (KE) and embedding dimension for the triangularFigure 10. andRelation square curve targets. between the Kolmogorov entropy (KE) and embedding dimension for the triangular and square targets. 3.2. Feature Selection for Incidence Angle Classiﬁcation

3.2.1.3.2. Feature Analysis Selection of the for Chaotic Incidence Features Angle Classification

3.2.1.The Analysis square of target the C ishaotic chosen Features for analysis with different incidence angles (θ [10,40]). Their chaotic ∈ features are analyzed based on the above method. The LLE of the square target with different incidence The square target is chosen for analysis with different incidence angles (  [10,40] ). Their angles is shown in Figure 11. From Figure 11, we can see that at small incidence angles, the LLE will decreasechaotic features gradually are with analyzed an increase based inon the the incidence above method. angle. The However, LLE of LLE the square increases target sharply with when different the incidence angles is shown in Figure 11. From Figure 11, we can see that at small incidence angles, the incidence angle is 40◦. The result is not wanted, because it implies that the LLE has no monotonicity withLLE thewill change decrease in gradually the incidence with angle, an increas and wee in cannot the incidence use LLE angle. to distinguish However, the LLE status. increases sharply  whenThe the same incidence analysis angle is employed is 40 . The to discuss result theis not SCD wan andted, the because KE. The resultsit implies are that shown the in LLE Figures has 12no andmonotonicity 13. Disappointingly, with the change they have in the overlap incidence when angle, the incidence and we cannot angles use are LLE different, to distinguish which implies the status that. they are also not monotonic. These results prove that the CF cannot be used as features to recognize the incidence angles of one target. We should find new features to complete the goal.

Figure 11. LLE of the pressure signal time series at different incidence angles.

The same analysis is employed to discuss the SCD and the KE. The results are shown in Figures 12 and 13. Disappointingly, they have overlap when the incidence angles are different, which implies

J. Mar. Sci. Eng. 2020, 8, 111 11 of 18

When the embedding dimension is high enough, KE tends to be a common value. For the triangular

target, the common value is Ktri = 4.431 and for the square target, Ksqu = 4.648 . This indicates that another chaotic feature, KE, can be used to distinguish the shapes of targets.

Triangular target

8 Square target KE

4 2 4 6 8 10 12 m

Figure 10. Relation curve between the Kolmogorov entropy (KE) and embedding dimension for the triangular and square targets.

3.2. Feature Selection for Incidence Angle Classification

3.2.1. Analysis of the Chaotic Features The square target is chosen for analysis with different incidence angles (  [10,40] ). Their chaotic features are analyzed based on the above method. The LLE of the square target with different incidence angles is shown in Figure 11. From Figure 11, we can see that at small incidence angles, the LLE will decrease gradually with an increase in the incidence angle. However, LLE increases sharply  J. Mar.when Sci. the Eng. incidence2020, 8, 111 angle is 40 . The result is not wanted, because it implies that the LLE has11 no of 17 monotonicity with the change in the incidence angle, and we cannot use LLE to distinguish the status.

J. Mar. Sci. Eng. 2020, 8, 111 12 of 18

thJ.at Mar. they Sci. Eng. are 2 also020, 8 not, 111 monotonic. These results prove that the CF cannot be used as features12 of 18 to recognize the incidence angles of one target. We should find new features to complete the goal. that they are also not monotonic. These results prove that the CF cannot be used as features to recognize the Figureincidence 11.6 LLEangles of of the one pressure target. signal We should time series find atnew diﬀ featureserent incidence to complete angles. the goal. Figure 11. LLE of the pressure signal time series at different incidence angles. 6 The same analysis is employed to discuss the SCD and the KE. The results are shown in Figures 12 and 13. Disappointingly4 , they have overlap when the incidence angles are different, which implies

4 SCD

SCD 2 2

0 0 0 10 20 30 40 Incidence angle 0 10 20 30 40 Incidence angle Figure 12. SCD of the pressure signal time series at different incidence angles . FigureFigure 12. 12.SCD SCD ofof thethe pressure signal signal time time series series at atdifferent diﬀerent incid incidenceence angles angles.. 6 6

KE KE 2 2

0 0 0 10 20 30 40 0 10 20 30 40 Incidence angle Incidence angle FigureFigure 13.13. KE of the the pressure pressure signal signal time time series series at at different diﬀerent incidence incidence angles angles.. Figure 13. KE of the pressure signal time series at different incidence angles. 3.2.2. Analysis of the PSD Features 3.2.2. Analysis of the PSD Features In order to reveal the inherent features of the flow field with different incidence angles, the In order to reveal the inherent features of the flow field with different incidence angles, the pressure coefficient contours are shown in Figure 14. From Figure 14, we can see that the influence pressure coefficient contours are shown in Figure 14. From Figure 14, we can see that the influence of the incidence angle on the flow structure is obvious. The number and shedding frequency of the of the incidence angle on the flow structure is obvious. The number and shedding frequency of the vortex change with the incidence angle. The results suggest that the PSD may be a distinguishing vortex change with the incidence angle. The results suggest that the PSD may be a distinguishing feature. Therefore, the fast Fourier transform (FFT) is performed on the signal sequence, and the PSD feature. Therefore, the fast Fourier transform (FFT) is performed on the signal sequence, and the PSD withwith different different incidence incidence angles angles is is shownshown inin FigureFigure 15.

J. Mar. Sci. Eng. 2020, 8, 111 12 of 17

3.2.2. Analysis of the PSD Features In order to reveal the inherent features of the flow field with different incidence angles, the pressure coefficient contours are shown in Figure 14. From Figure 14, we can see that the influence of

J.the Mar. incidence Sci. Eng. 2020 angle, 8, 111 on the ﬂow structure is obvious. The number and shedding frequency of the13 vortex of 18 change with the incidence angle. The results suggest that the PSD may be a distinguishing feature. Therefore, the fast Fourier transform (FFT) is performed on the signal sequence, and the PSD with J.di Mar.ﬀerent Sci. Eng. incidence 2020, 8, 111 angles is shown in Figure 15. 13 of 18

Figure 14. Pressure coefficient contours of the flow field around the square target at different incidence angles.

FigureFigure 14. 14. Pressure coefficientcontoursof coefficient contours theflow of the fieldaroundthesquaretargetat flow field around the square different target incidenceangles. at different incidence angles. θ=10° θ=20° 1000 225 ° θ=30 ° ° 150 θ θ=10=40 ° 75 θ=20 1000 225 ° ) 120 θ=30 5 0 ° 150 80 θ=40 1.64 1.68 1.72 1.76

*10 75 40

( 500

) 120

5 0 0 80 -40 PSD 1.64 1.68 1.72 1.76

*10 40 3.0 3.2 3.4

( 500

-40 PSD 0 3.0 3.2 3.4

0 1 2 3 4 Frequency(Hz) . Figure 15. Eﬀects of incidence1 angles of the2 square target on3 the power spectrum4 density. Figure 15. Effects of incidence angles of Frequency(Hz)the square target on the power spectrum density. . As seen in Figure 15, when the incidence angle is 10  , there is one obvious crest at the position Figure 15. Effects of incidence angles of the square target on the power spectrum density. of f10 = 1.5 Hz. At other incidence angles, there are two crests in the curve. The positions of the two   20  30 crestsAs are seen similar in Fig whenure 15 the, when incidence the incidence angles areangle is and10 , there. However, is one obvious when crest the incidence at the position angle is 20 , the first crest ( f = f =1.68 Hz) is stronger than the second crest ( f = f = 3.36 Hz), of f10 = 1.5 Hz. At other20 incidence,1 30,1 angles, there are two crests in the curve. The 20positions,2 30,2 of the two crests are similar when the incidence angles are 20 and 30  . However, when the incidence angle  is 20 , the first crest ( f20,1 = f30,1 =1.68 Hz) is stronger than the second crest ( f20,2 = f30,2 = 3.36 Hz),

J. Mar. Sci. Eng. 2020, 8, 111 13 of 17

As seen in Figure 15, when the incidence angle is 10◦, there is one obvious crest at the position of f10 = 1.5 Hz. At other incidence angles, there are two crests in the curve. The positions of the two crests are similar when the incidence angles are 20◦ and 30◦. However, when the incidence angle is 20◦, the first crest ( f20,1 = f30,1 = 1.68 Hz) is stronger than the second crest ( f20,2 = f30,2 = 3.36 Hz), which is opposite to the situation of θ = 30◦. The first crest of θ = 40◦ coincides with f10, but the intensity is lower than that of f10. Another crest is at the position of f40,2 = 3.04 Hz. The results are exciting, because the PSD values at different incidence angles have no overlap. Therefore, the number, amplitude, and position of wave crests on the PSD curves can be used as the distinguishing features to recognize the incidence angles of one target.

4. Numerical Experiments and Discussion The support vector machine (SVM) was developed from the theory of structural risk minimization and designed for linear classifiers with the linear function $T x + b, where $ is the weight vector and b is · the shift of the separating hyperplane. The margin between the samples and the separating hyperplane is 2/ $ when the labels are +1 or 1. Therefore, the goal of the SVM is to get the minimum value of k k − $ 2. A Lagrangian equation L($, b, α) was built to finish the goal, which is shown in Equation (16). k k Based on Equation (17), interesting results are shown in Equation (18).

n 1 2 X L →ω, b, α = →ω α [y (→ω →x + b) 1], (16) 2k k − i i · i − i=1

 n  ∂L P  = 0 →ω = αi yixi  →  ∂ω ⇒ i=1  Pn  ∂L = y = (17)  ∂b 0 αi i 0  ⇒ i=1   ∂L = 0 α [y (→ω →x + b) 1] = 0 ∂α ⇒ i i · i −   Pn  →ω = α y →x  min i∗ i i  i=1 (18)  h i Pn  ( ) = → → + = [ (→ →) + ]  f x sgn ω x b sgn αi∗ yi x i x b∗  · i=1 · where yi is the corresponding label, xi is a support vector when 0 < αi < C, and C is the penalty factor. For a non-linear classification, the SVM is sufficient with the help of the kernel trick [49]. Compared with the linear SVM classifier, a higher dimensional feature space Ψ is used, which is nonlinearly related to the input space, Φ : n Ψ . In the feature space Ψ, the Euclidean dot-products (→x →y) are < → · transformed to (Φ(→x ) Φ(→y)) based on a kernel function, which is a polynomial kernel in this paper · and is shown in Equation (19). q k(x, y ) = (g (→x →y ) + r) , (19) i × · i where g and r are the parameters, which are 1 in this paper, and q is the order of the kernel function. The grid search algorithm is used to find the optimal C and q. Five-fold cross-validation is used 6 6 10 10 firstly to determine the half-optimal C1/2 in [2− , 2 ] and q1/2 in [2− , 2 ]; then 10-fold cross-validation C1/2 q1/2 is used in [ 4 , 4C1/2] and [ 4 , 4q1/2] to find the optimal Copt and qopt. Three kinds of target with different shapes are considered, including a circular target, triangular target, and square target. The pressure signal on the virtual LLS is selected to build the classification datasets Γ3000 11 = ( Γc 3000 1, Γt 3000 5, Γs 3000 5), and five datasets (η0, η15, η30, η45, η60) and × { } × { } × { } × (γ , γ , γ , γ , γ ) are included in Γt and Γs . The classification strategy is shown in Figure 16. 0 10 20 30 40 { } { } J. Mar. Sci. Eng. 2020, 8, 111 15 of 18 datasets 300011 = ({c}30001,t30005,s30005) , and five datasets (0,15,30,45,60) and J. Mar. Sci. Eng. 8 ( 0,10, 20, 30, 40)2020 are, , 111 included in t  and s . The classification strategy is shown14 in of 17Figure 16.

FigureFigure 16. Workflow 16. Workflow of the of the proposed proposed classification classification strategy. strategy. The chaotic features are used firstly to classify the shapes of targets based on the above SVM; Thethen, chaotic the PSD features features are are used selected firstly to classifyto classify the incidence the shapes angles of [targets50]. For based the first on step, theΓ aboveis used SVM; then, theas PSD the training features data, are andselected a single to individualclassify the from incidence each type angles of signal [50 is]. selectedFor the asfirst the step, testing  data. is used as the trForaining convenient data, and comparison, a singleΓ cindividualis set as label from 1, and each the othertype twoof signal groups, isΓ tselectedand Γs, are as set the as testing label –1. data. The central mark is the recognition rate (RR). If the RR is larger than 0.8, this implies that the testing For convenient comparison, c is set as label 1, and the other two groups, t and s , are set as data is Γc; if not, it implies that the testing data belongs to Γt or Γs. Then, the same process is carried label –1.out The again central until mark the category is the recognition is decided. The rate classification (RR). If the of incidenceRR is larger angles than is the0.8, same this asimplies the above that the testing dataprocedure. is c ; Table if not,2 lists it implies the extracted that the RR testing and corresponding data belongs averaged to valueor (AR).. Then From, the the same table, pro it cess is carriedcan out be seenagain that until the the method category gets a recognitionis decided rate. The of classification 96.73% for shape of recognition,incidence angles and a recognition is the same as rate of 92.48% for incidence angle recognition. Therefore, the proposed method can achieve better the above procedure. Table 2 lists the extracted RR and corresponding averaged value (AR). From performance under these features. the table, it can be seen that the method gets a recognition rate of 96.73% for shape recognition, and a recognitionTable rate 2. Classificationof 92.48% for results incidence of the support angle vector recognition. machine (SVM) Therefore, method based the on proposed chaotic features method can achieve better(CF)-power performance spectrum under density these (PSD) features features. . Triangular Square TableRR 2./% ClassifiCircularcation results of the support vector machine (SVM) method based on chaoticAR/% 0◦ 15◦ 30◦ 45◦ 60◦ 0◦ 10◦ 20◦ 30◦ 40◦ features (CF)-power spectrum density (PSD) features. ε1 100 90.2 100 96.73 ε 95.4 89.7 92.3 95.7 93.8 94.6 88.5 87.3 91.7 95.8 92.48 2 \ Triangular Square RR/% Circular           AR/% 5. Conclusions 0 15 30 45 60 0 10 20 30 40  100 90.2 100 96.73 1 The flow features were analyzed in the framework of chaotic theory, and a new strategy for  underwater target classification using these features was presented, which includes the CF and PSD. 2 \ 95.4 89.7 92.3 95.7 93.8 94.6 88.5 87.3 91.7 95.8 92.48 A two-step SVM was built based on the compounded features CF-PSD, and it was used to classify the 5. Conclusionsshapes and incidence angles of different targets. Some conclusions were obtained from the results, which are as follows: The flow features were analyzed in the framework of chaotic theory, and a new strategy for underwater target classification using these features was presented, which includes the CF and PSD. A two-step SVM was built based on the compounded features CF-PSD, and it was used to classify the shapes and incidence angles of different targets. Some conclusions were obtained from the results, which are as follows:

J. Mar. Sci. Eng. 2020, 8, 111 15 of 17

1. The flow field around the circular target is periodic, and the systems around the triangle target and square target are chaotic systems. The chaotic features of the pressure signal time series in different flow fields have no overlap, which implies that CF can be used to classify the shapes of targets. 2. The relationship between CF and incidence angles is not monotonic, and CF cannot be used to recognize the incidence angles of square targets. However, the number, amplitude, and position of wave crests on the PSD curves have no overlap, and PSD is able to distinguish the incidence angles. 3. A two-step SVM with a polynomial kernel was built. It can achieve better performance under the compounded features CF-PSD and has a recognition rate of 96.73% for shape classification and a recognition rate of 92.48% for incidence angle recognition.

Author Contributions: Conceptualization, X.L. and J.W.; methodology, X.L.; software, X.L. and Q.Q.; validation, J.W., X.L. and Q.Q.; formal analysis, X.L. and Q.Q.; investigation, X.L. and J.W.; resources, J.W.; data curation, X.L.; writing—original draft preparation, X.X.; writing—review and editing, J.W. and Q.Q.; visualization, X.L. and Q.Q.; supervision, J.W.; project administration, X.L.; funding acquisition, J.W. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the NATIONAL NATURAL AND SCIENCE FOUNDATION OF HEBEI, grant number E2018202259. Acknowledgments: The authors would like to thank Xiaoming Wang and Haitao Liu, who provided insight and expertise that greatly assisted in the writing of this manuscript. Conﬂicts of Interest: The authors declare no conﬂict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

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