Ship recognition via its radiated sound - Acoustical Society of America

Ship recognition via its radiated sound: The fractal based approaches Su Yanga) State Key Laboratory of Modern Acoustics & Institute of Acoustics, Nanjing University, Nanjing 210093, People’s Republic of China

Zhishun Lib) National Key Laboratory of Underwater Information Processing and Control & College of Marine Engineering, Northwestern Polytechnical University, Xi’an 710072, People’s Republic of China

Xinlong Wangc) State Key Laboratory of Modern Acoustics & Institute of Acoustics, Nanjing University, Nanjing 210093, People’s Republic of China

共Received 7 July 2000; revised 18 April 2002; accepted 18 April 2002兲 Due to the complexity of its radiated sound, ship recognition is difficult. Fractal approaches are proposed in this study, including fractal Brownian motion based analysis, fractal dimension analysis, and wavelet analysis, to augment existing feature extraction methods that are based on spectrum analysis. Experimental results show that fractal approaches are effective. When used to augment two traditional features, line and average spectra, fractal approaches led to better classification results. This implies that fractal approaches can capture some information not detected by traditional approaches alone. © 2002 Acoustical Society of America. 关DOI: 10.1121/1.1487840兴 PACS numbers: 43.60.Lq 关JCB兴

I. INTRODUCTION

Ship recognition via its radiated sound has attracted attention. Many efforts1–14 have been made to seek effective features that can lead to good classification of oceanic signals. Due to the complexity of ship sound, it is not easy to obtain the regularities that are essential to ship recognition. The experimental results achieved here show that fractal approaches are effective for improving ship recognition. These approaches include analysis based on fractional Brownian motion, fractal dimension analysis, and wavelet analysis. In the classification experiments, each approach was independently tested to show how it contributed to classification. Each classification experiment was reported 10 times with randomly selected training and testing samples so that fortuitous conclusions could be excluded. In comparison with the two traditional features, line spectrum and average spectrum, the fractal approaches led to better classification results. This implies that some information that cannot be detected by using traditional approaches can be obtained using fractal approaches. A major concern about studies based on fractal approaches is whether ship sounds are fractal signals. As practical situations are complex, in a rigorous mathematical sense, ship sounds might not behave exactly as 1/f signals. But from an engineering viewpoint, a visible resemblance exists between the spectra of ship sound and those of 1/f signals. Through extensive experiments, Urick15 found that the power spectra of ship sounds decrease 6 decibels per octave at high frequencies. This property is similar to that of a兲

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b兲 c兲

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J. Acoust. Soc. Am. 112 (1), July 2002

1/f signals.16 For the purpose of classification, it is enough as long as the spectra of ship sounds are similar to those of 1/f signals. As shown by the experiments, ship sounds can be effectively classified using fractal approaches. In contrast, active sonar signals cannot be well classified using fractal features because their spectra are distinctly different from those of 1/f signals. The fractal approaches described in this study 共1兲 allow the regularities in ship sound to be viewed in a new way; 共2兲 provide some information not provided by traditional features; and 共3兲 improve ship recognition. Because no single approach can capture all useful information, different features working in collaboration with each other can provide complementary information. As shown in this study, some classes that cannot be well classified using traditional features can be well classified using fractal features.

II. CLASSIFIER

It is assumed that there are M classes H 1 ,H 2 ,...,H M , and that each class is composed of 2N feature vectors. Let T i ⫽ 兵 U i,1 ,U i,2 ,...,U i,N 其 and S i ⫽ 兵 V i,1 ,V i,2 ,...,V i,N 其 , respectively, denote N training and testing samples that are randomly selected from H i , where T i艚 S i ⫽ ␾ and T i艛 S i ⫽H i . The mean vector of T i is defined as 1 ␮ i⫽ N

N

兺

j⫽1

U i, j .

共1兲

In the training stage, 兵 ␮ i 兩 i⫽1,2,...,M 其 should be calculated. In the testing stage, 兵 D(V i, j , ␮ k ) 兩 k⫽1,2,...,M ; j ⫽1,2,...,M ; j⫽1,2,...,N 其 should be computed, where D(V i, j , ␮ k ) represents the Mahalonobis distance17 between 0001-4966/2002/112(1)/172/6/$19.00

© 2002 Acoustical Society of America

FIG. 1. 共a兲 Power spectrum of ship A sample in log–log plot. 共b兲 Power spectrum of ship B sample in log–log plot. 共c兲 Power spectrum of ship C sample in log–log plot. 共d兲 Power spectrum of ship D sample in log–log plot. 共e兲 Power spectrum of ship E sample in log–log plot. 共f兲 Power spectrum of ship F sample in log–log plot.

V i j and ␮ k . If l⫽arg mink 兵 D(V i, j , ␮ k ) 其 , it can be decided that V i, j belongs to H l . III. DATA

The data set used in the experiments contains the sound samples radiated from six ships. With regard to each ship, 100 samples were collected. The data length of each sample is 3264. In order to compensate for the distance variation between ship and receiver, each sample was normalized to possess unit energy by dividing the square root of its energy. The power spectra of the representative samples for each ship are illustrated, respectively, in log–log plots in Fig. 1. Figure 2 illustrates the power spectrum of a fractional Brownian motion 共fBm兲 sample selected from 100 fBm samples that were generated using a Fourier filtering method.18 Figures 1 and 2 show that the spectra of the ship sound samples are visibly similar to that of the fBm sample. IV. FRACTIONAL BROWNIAN MOTION BASED ANALYSIS

The power spectra of ship sound, which decrease 6 decibels per octave at high frequencies,15 are similar to those of fractional Brownian motion 共fBm兲.19 This motivates us to apply the tools that are useful in analyzing fBm to ship sound analysis. Let B H (t) denote a fBm with parameter H. The differential sequences of B H (t) satisfy the T H law,19

␴ k ⫽ 冑var关 B H 共 t⫹k⌬ 兲 ⫺B H 共 t 兲兴 ⫽C H 共 k⌬ 兲 H ,

共2兲

where ⌬ represents the sampling interval, k is the step to form the differential sequence B H (t⫹k⌬)⫺B H (t), and C H can be deemed as a constant if H is given. Let H k ⫽ 关 log ␴ k⫹1 ⫺log ␴ k 兴 / 关 log共 k⫹1 兲 ⫺log k 兴 .

共3兲

By substituting Eq. 共2兲 into Eq. 共3兲, it follows that H k ⫽H. Different H corresponds to different fBm model. In correspondence with H, 兵 H k 其 and 兵 ␴ k 其 are taken as two kinds of feature vectors in this study. In practice, ship sounds could not be ideal fBm signals. Thus, H k could not be constant at each step k, and ␴ k might not satisfy T H law exactly. Even so, as confirmed by the experiments, 兵 H k 其 and 兵 ␴ k 其 provide essential information for ship recognition. Supposing s i : i⫽1,2,...,N is a ship sound sequence, the mean value and the standard deviation of its differential sequence s i⫹k ⫺s i : i⫽1,2,...,N⫺k are, respectively, 1 ␮ k⫽ N⫺k

␴ k⫽

冑

N⫺k

兺

i⫽1

1 N⫺k

共 s i⫹k ⫺s i 兲 ,

共4兲

N⫺k

兺

i⫽1

共 s i⫹k ⫺s i ⫺ ␮ k 兲 2 .

共5兲

By substituting Eq. 共5兲 into Eq. 共3兲, the two feature vectors

兵 ␴ k 其 and 兵 H k 其 can be calculated. Figures 3 and 4, respectively, show 兵 ␴ k 其 and 兵 H k 其 of the sound samples from the six ships. Both the consistency between the samples from each identical ship, and the divisibility between the samples from every two different ships, can be observed in Figs. 3 and 4. V. FRACTAL DIMENSION ANALYSIS

FIG. 2. Power spectrum of fBm sample in log–log plot. J. Acoust. Soc. Am., Vol. 112, No. 1, July 2002

As a geometrical tool, a fractal dimension can be used to characterize the waveform details of a signal. Among the many definitions of fractal dimension, the ‘‘blanketcovering’’ dimension is suitable for feature extraction20,21 beYang, Li, and Wang: Ship recognition

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FIG. 3. 共a兲 ␴ k of ship A samples, 共b兲 ␴ k of ship B samples, 共c兲 ␴ k of ship C samples, 共d兲 ␴ k of ship D samples, 共e兲 ␴ k of ship E samples, 共f兲 ␴ k of ship F samples.

cause its computation is invariant to signal shifting in the coordinates of time and amplitude.22 For a time sequence f (n): n⫽0,1,...,N, choosing a scale r to form an upper envelope U r (n) and a lower envelope L r (n), where U r 共 n 兲 ⫽max兵 U r⫺1 共 n⫺1 兲 ,U r⫺1 共 n 兲 ⫹1,U r⫺1 共 n⫹1 兲其 , 共6兲 L r 共 n 兲 ⫽min兵 L r⫺1 共 n⫺1 兲 ,L r⫺1 共 n 兲 ⫺1,L r⫺1 共 n⫹1 兲兴其 , U 0 共 n 兲 ⫽L 0 共 n 兲 ⫽ f 共 n 兲 ,

共7兲共8兲

the fractal measurement can be calculated by L共 r 兲⫽

1 2r

N

兺

n⫽0

关 U r 共 n 兲 ⫺L r 共 n 兲兴 .

共9兲

In ideal conditions, log L(r) and log r should satisfy the following linear relation:

log L 共 r 兲 ⫽ 共 1⫺D 兲 log r⫹log K,

共10兲

where D is the so-called ‘‘blanket-covering’’ dimension and K is a constant. Then, D can be acquired by fitting log L(r) with respect to log r in a least-square sense. In practice, since log L(r) versus logr is not a straight line but a smooth curve, the fractal dimension is not constant at each scale r. Accordingly, the fractal dimension at a given scale r is computed via D r ⫽1⫺ 关 log L 共 r⫹1 兲 ⫺log L 共 r 兲兴 / 关 log共 r⫹1 兲 ⫺log r 兴 , 共11兲 which can be derived from Eq. 共10兲. Here 兵 D r 其 is used as one kind of feature vector in this study. Figure 5 shows 兵 D r 其 for the samples from six ships. The divisibility between every two different classes, and the consistency within each identical class, visibly exist in Fig. 5. Also, 兵 D r 其 for the 100

FIG. 4. 共a兲 H k of ship A samples, 共b兲 H k of ship B samples, 共c兲 H k of ship C samples, 共d兲 H k of ship D samples, 共e兲 H k of ship E samples, 共f兲 H k of ship F samples. 174

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Yang, Li, and Wang: Ship recognition

FIG. 5. 共a兲 D r of ship A samples, 共b兲 D r of ship B samples, 共c兲 D r of ship C samples, 共d兲 D r of ship D samples, 共e兲 D r of ship E samples, 共f兲 D r of ship F samples.

fBm samples that were generated using the Fourier filtering method are illustrated in Fig. 6. In both Figs. 5 and 6, D r versus r ascends at small scales and is nearly constant at big scales. It indicates that log L(r) versus log r is not a straight line in practice, even for fBm samples. Therefore, it is reasonable to utilize 兵 D r 其 for the purpose of classification. VI. WAVELET ANALYSIS

Wavelet analysis is a useful tool for analyzing 1/f signals.23 As the spectra of ship sounds are similar to 1/f signals, wavelet analysis is appropriate. The synthesis and analysis of a signal n(t) using a wavelet transform can be expressed, respectively, as n共 t 兲⫽

兺m 兺j n mj ␺ mj 共 t 兲 ,

冕

共12兲

2 eter set 兵 ␴ m 其 for each model. The maximum likelihood esti2 is23 mator of ␴ m

␴ˆ m2 ⫽

1 N共 m 兲

兺j 共 n mj 兲 2 ,

共14兲

where N(m) represents the number of wavelet coefficients at 2 scale m. Because the estimated parameter set 兵 ␴ˆ m 其 is the key to identify the signal model that generates a given sample n(t), it is used as one kind of feature vector in this study. VII. LINE SPECTRUM

The line spectrum is a widely used traditional feature in ship sound recognition. The feature extraction method is as follows. First, the frequencies at which line components frequently occur should be selected through the following procedure.

where 兵 ␺ mj (t) 其 can represent any orthogonal wavelet basis, and 兵 n mj 其 denote the corresponding coefficients at scale m. In the following computations, Daubechies wavelets24 共4 taps兲 are used. It has been demonstrated that the wavelet coefficients of any given 1/f signal satisfy a zero-mean Gaussian 2 ). It means that the distinctions distribution,23 n mj ⬃N(0,␴ m between different 1/f models consist in the param-

共1兲 Compute the power spectrum of each sample. 共2兲 In the power spectrum of each sample, locate the points that have local minimum amplitudes and denote them as valleys. 共3兲 Within the region between each pair of adjacent valleys, select the point that has the maximum amplitude and denote such a point as a peak. 共4兲 Regarding the samples in each class, if the probability that a peak occurs at a given frequency exceeds 60%, this frequency is selected as a line spectral frequency.

FIG. 6. D r of fBm samples.

Second, for each sample, let the amplitude of the peak at each line spectral frequency be a feature value and take the feature values at all the line spectral frequencies to construct the feature vector of the line spectrum. In view of possible frequency shifting of line components, if there are several peaks within a small range around a line spectral frequency, the maximum amplitude of these peaks is selected as the feature value at this line spectral frequency. If no peak occurs within the small region around a given line spectral frequency, let the feature value at this line spectral frequency be

n mj ⫽2 j/2

⫹⬁

⫺⬁

n 共 t 兲 ␺ mj 共 t 兲 dt,


共13兲


175

TABLE I. Classification results using different features.

cases. As the spectra of ship sound are close to those of 1/f signals, the performance of FBM1 and WA is thus better than that of AS.

Classification accuracy 共%兲 Feature

A

B

C

D

E

F

Total

FBM 1 FBM 2 FD WA LS AS

99.6 96.8 97.4 100 94.4 100

100 100 100 100 93.8 97.8

100 100 100 100 89.8 94.4

98.2 99.8 99.8 100 88.6 81.6

93.2 94 96.6 95.6 78.6 80.2

100 100 100 99.8 72 100

98.5 98.43 98.97 99.23 86.2 92.33

0. Here, the half-width of such a small region is allowed to be at most one interval in the frequency axis. VIII. AVERAGE SPECTRUM

Unlike line components, which are mainly located at low frequencies, the average spectrum reflects the characteristics of a signal over the entire frequency domain. Computation of average spectrum is as follows. First, compute the power spectrum of a given signal using Welch’s averaged periodogram method,25 where a Hanning window is used, and the computation is based on a four-segment average without overlay. Second, divide the whole frequency axis into several equal parts without overlay. Third, let the mean amplitude of each part be a feature value, and take the feature values of all the parts to construct the feature vector of the average spectrum. In the following classification experiments, the number of points included in each part is 50. IX. CLASSIFICATION EXPERIMENTS

Here, the two fBm feature vectors 兵 ␴ k 兩 k⫽1,2,...,10其 and 兵 H k 兩 k⫽1,2,...,10其 , the fractal dimension feature vector 兵 D r 兩 r⫽1,2,...,10其 , the wavelet feature vector 兵 ␴ˆ m2 其 , the line spectrum feature vector, and the average spectrum feature vector, which are denoted as FBM1, FBM2, FD, WA, LS, and AS, respectively, are used to classify the data set. Each single feature is used independently to classify the data set. The classification experiment using the same feature is repeated 10 times, in each of which 50 samples are randomly selected from the 100 samples of each ship to construct the training set and the other 50 samples of each ship are used to form the testing set. To compare the performance of each feature, the average classification accuracy of the 10 tests with regard to each ship and the overall classification accuracy regarding the six ships are summarized in Table I. The performance of LS is obviously inferior to that of any other feature. Besides, AS is apparently inferior to the fractal features in distinguishing ships D and E. That is the major reason why the overall performance of AS is worse than that of any fractal feature in this study. The above comparison shows that fractal features can capture information no detected by traditional features. From anther viewpoint, ␴ k in FBM1 can be understood as the variance of a ship sound passing through a filter whose transfer function is H( ␻ )⫽(1⫺e ⫺ jk ␻ ). Meanwhile, WA concentrates on constant-Q bands while AS focuses on bands with equivalent width. The band partition of FBM1 and WA aims at 1/f signals, while that of AS is for more general 176


X. SUMMARY

Due to the complexity of ship sound, ship recognition via its radiated sound has been a difficult problem to date. This paper concerns a new approach that can improve ship recognition. The contribution of the fractal based feature extraction methods to ship recognition is confirmed by this study. The essential points in this study are as follows: 共1兲 Four fractal features, which include two fBm based features, the fractal dimension, and a wavelet feature, are proposed to augment the two traditional features, line and average spectra. 共2兲 The following regularities can be observed in the experiments. The fractal features of the samples from every identical ship are similar to each other while the distinctions among different ships are obvious. 共3兲 In view of the classification results, the contribution of fractal features to ship recognition is shown to exist. Since each approach has been tested independently through 10 experiments with randomly selected training and testing samples, the effectiveness of each fractal feature is substantially confirmed. 共4兲 According to the experimental results, some samples that cannot be well classified by using traditional features can be satisfactorily classified using fractal features. Fractal features thus provide essential information not provided by traditional features. 共5兲 Fractal approaches allow the characteristics of ship sound to be viewed in a new way compared to traditional methods and then augment existing ways for ship recognition. ACKNOWLEDGMENT

The authors are very grateful to Professor John C. Burgess, the associate editor, for his beneficial comments in improving the technical presentation and great help in improving the language of this paper. This work is partially supported by the National Science Foundation of China for Distinguished Young Scholars under Grant No. 19925414 and the National Science Foundation of China under Grant No. 19834040. 1

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