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Time-Frequency Morphological Characteristics for Rigid Acoustic Scattering by Underwater Objects
Yang Yang1,2, Xiukun Li1,2     
1. Acoustic Science and Technology Laboratory, Harbin Engineering University, Harbin 150001, China;
2. College of Underwater Acoustic Engineering, Harbin Engineering University, Harbin 150001, China
Abstract: Separation of the components of rigid acoustic scattering by underwater objects is essential in obtaining the structural characteristics of such objects. To overcome the problem of rigid structures appearing to have the same spectral structure in the time domain, time-frequency Blind Source Separation (BSS) can be used in combination with image morphology to separate the rigid scattering components of different objects. Based on a highlight model, the separation of the rigid scattering structure of objects with time-frequency distribution is deduced. Using a morphological filter, different characteristics in a Wigner-Ville Distribution (WVD) observed for single auto term and cross terms can be simplified to remove any cross-term interference. By selecting time and frequency points of the auto terms signal, the accuracy of BSS can be improved. A simulation experimental has been used to analyze the feasibility of the new method, with changing the pulse width of the transmitted signal, the relative amplitude and the time delay parameter. And simulation results show that the new method can not only separate rigid scattering components, but can also separate the components when elastic scattering and rigid scattering exist at the same time. Experimental results confirm that the new method can be used in separating the rigid scattering structure of underwater objects.
Keywords: underwater object     highlight structure     rigid scattering components     image morphology     time-frequency     blind source separation    
1 Introduction

Acoustic scattering characteristics are an important technique for the detection and identification of underwater objects. Object scattering echoes include rigid scattering components and elastic scattering components(Tang, 1994), and these convey the rigid structure and material information of the object, respectively. There is a lot of literature that details the generation mechanism and properties of the underwater acoustic scattering of objects with a simple shape through theoretical calculations and numerical simulations(Pan et al., 2013; Pan et al., 2012; Plotnick et al., 2015; La Follett et al., 2011). However, few studies have been concerned with the separation of highlights. Li et al.(2015)deduced an analytical expression of how rigid scattering components change with the incident angle in the fractional Fourier transform domain. With this in mind, the all-direction model for the optimal fractional Fourier transform domain of scattering waves was established, which provided a theoretical basis for identifying objects under the unknown incident angle. Another study by Li and Xia(2015)proposed a linear signal mapping method. The acoustic scattering signal of an object was mapped to a single frequency signal, and each of the sound scattering components of the echo from that object was separated by narrow band filter. Both of these methods are required to determine information about the transmitted signal. Blind Source Separation(BSS)algorithm is used when little is known about the source. It is based on the separation of observed data or for the recovery of the source signals(Li et al., 2007; Kamran et al., 2000; Liu et al., 2015). Most blind separation algorithms assume each source to be independent, identically distributed and random stationary signals(Holobar et al., 2002). In fact, object acoustic scattering signals are non-stationary, non-overlapping in time or frequency domains, having different distribution in time and frequency domain. When studying such signals the time-frequency BSS algorithm is the most commonly used approach(Zhang et al., 2009; Zhu and Wu, 2009; Li and Xia, 2013). Due to the existence of cross-term interference in Wigner-Ville Distribution(WVD)(Li et al., 2010; Thomas et al., 2012), the spatial time-frequency distribution matrix from observed data is no longer a diagonal matrix(Ma et al., 2007). The presence of cross-term interference seriously affects the accurate separation of the acoustic scattering components of the object. Belouchrani and Amin(1998)obtained the auto-term time and frequency points of the highlights. They achieved this by presetting the parameter values to constrain the trace and characteristic values of the spatial time-frequency matrix. The selection of the parameter value is related to the energy of scattering echoes of the object and the environment interference; the accuracy of the value selection directly affects the performance of BSS.

When active sonar transmits a linear frequency modulated signal, incident sound waves will be scattered by the surface of the object. The observed rigid scatting components of an object echo from the received signal show regular distribution in the time and frequency domain. However, the cross-term interference between the highlights includes an exponential index so that the cross-term spectra are periodically shocked. A morphological filter(Bouaynaya et al., 2008)can be used for removing the cross-term interference between the highlights according to the different morphological characteristics. This is because a preset parameter is not required. The auto-term time and frequency points can be obtained stably and simply, and the time-frequency BSS combined with the image morphology method can be used to realize accurate separation of object rigid acoustic scattering components.

2 Morphological characteristics in the time and frequency domains

Based on the highlight model, the object acoustic scattering components can be seen as linear effects, resulting between the incident sound waves and object acoustic scattering field under high frequency conditions. Due to the different generation mechanism, the scattering components can be divided into rigid highlights and elastic highlights. It should be noted that the highlight model does not correspond to the real rigid highlights, but the equivalent determined points in the light of the wave propagation. The object echo highlight model can be expressed by

$$\begin{array}{l}U(t)= \sum\limits_{m = 1}^N {\{ {A_m}\cos [2{\rm{\pi }}(t - {\tau _m})({f_0} + \frac{M}{2}(t - {\tau _m}))+ {\phi _m}]\} } + \\\begin{array}{*{20}{c}}{}&{}\end{array}{A_e}\cos [2{\rm{\pi }}(t - {\tau _e})({f_e} + \frac{\mu }{2}(t - {\tau _e}))+ {\phi _e}]\end{array}$$ (1)

For the rigid scattering components, ${A_m}$ is the amplitude reflection factor, ${\tau _m}$ is the time delay, ${f_0}$ is the center frequency of the transmitted signal, and ${\varphi _m}$ is the phase transition. For the elastic scattering components, ${f_e}$ is the center frequency of elastic wave, ${A_e}$ is the amplitude, ${\tau _e}$ is the time delay, and ${\varphi _e}$ is the phase transition. Wherein, the transfer function of the rigid highlights is

$$H({\bf{r}}, \omega)= \sum\limits_{m = 1}^N {{A_m}({\bf{r}}, \omega){{\rm{e}}^{j\omega {\tau _m}}}{{\rm{e}}^{j{\phi _m}}}} $$ (2)

The rigid highlights of the object can also be expressed in the frequency domain,

$$X({\bf{r}}, \omega)= S(w)H({\bf{r}}, \omega)$$ (3)

As can be seen in Eq.(2), only the amplitude and time delay of the rigid highlights change with the transmitted signal; the transfer function does not change. The observed object rigid highlights at the sonar receiving end have a uniform distribution with the transmitted signal in the time and frequency domain.

When a linear frequency modulated signal is transmitted, i.e., $s(t)= A \cdot \exp({\rm{j}}2{\rm{\pi }}({f_0}t + m{t^2}/2))$, $m =({f_1} - {f_0})/T$, $t = [- T/2, {\rm{ }}T/2]$ the time-frequency distribution of auto terms for an individual highlight component can be expressed by,

$${W_z}(t, {\rm{ }}f)= {A^2}(T - 2\left| t \right|)\sin c[2{\rm{\pi }}({f_0} + mt - f)(T - 2\left| t \right|)]$$ (4)

The mutual time-frequency distribution between two highlights with relative delay τ and amplitude A, equal to 1 is

$${W_{{z_1}, {z_2}}}(t, f)= \exp({\rm{j}}\omega \tau){W_z}(t - \tau /2, f)$$ (5)
$${W_{{z_2}, {z_1}}}(t, f)= \exp(- {\rm{j}}\omega \tau){W_z}(t - \tau /2, f)$$ (6)

From Eq.(4), the WVD of individual highlight components can be composed by a series of $\sin c$ functions. The individual highlight function is a continuous spectra in time and frequency domain exhibiting the same slope as the transmitted signal. Eq.(5)and Eq.(6)show that the cross-term interference between the highlights includes an exponential index so that the cross-term spectra structure is periodically shocked. This is the basis of restraining the cross-term interference.

3 Time-frequency BSS combined with image morphology

The BSS model can be described as

$${\bf{x}}(t)= {\bf{As}}(t)$$ (7)

A is a m×n mixing matrix and is determined by the complexity of the unknown underwater environment. The purpose of BSS is to build an unmixed matrix W, and moreover to seek the signal estimation of the source based on the observed signal vector x(t). The WVD of pre-whitened signal z(t)is

$${{\bf{D}}_z}(t, f)= {\bf{B}}{{\bf{D}}_x}(t, f){{\bf{B}}^H} = {\bf{U}}{{\bf{D}}_s}(t, f){{\bf{U}}^H}$$ (8)

where ${\bf{U}}$ is an unitary matrix, ${{\bf{D}}_s}$ is a time-frequency matrix of the source signal, and ${\bf{J}}$ is the trace of the ${{\bf{D}}_z}$

$${\bf{J}} = {\rm{trace}}({{\bf{D}}_z}(t, f))= {\rm{trace}}({\bf{U}}{{\bf{D}}_s}(t, f){{\bf{U}}^H})$$ (9)

The time-frequency spectra of Eq.(9)is converted to binarization. The morphology operation is used for the binary result by selecting a linear structure element with the same slope as the transmitted signal. Assuming echo area collection is A, the cross-term and other interference area collection is C, and linear structural elements set is O, the length of collection O needs to be longer than the length of the major axis of collection C. Corrosion and dilation operations are carried out for the collection A and C

$\left\{ \begin{array}{l} A\Theta O = \{ z|{(O)_z} \subseteq A\} = A'\\ C\Theta O = \{ z|{(O)_z} \subseteq C\} = \emptyset \end{array} \right.$ (10)

The auto-term time and frequency points of the highlights are extracted by selecting the points where the image value is greater than the threshold value $\varepsilon $ in Eq.(9), referred to as $\varepsilon = {\bf{\bar J}}$, $(t, {\rm{ }}f)\in {\Omega _s}$.

According to the time and frequency points $(t, {\rm{ }}f)\in {\Omega _s}$, it is possible to obtain W in order to realize the separation of the object echo acoustic scattering components by jointly diagonalizing the refactored spatial time-frequency matrix ${{\bf{D}}_z}(t, {\rm{ }}f)$. Using Eq.(11), source signals in the time domain can be obtained.

$${\bf{\hat S}} = {{\bf{U}}^H}{\bf{WX}}$$ (11)
4 Simulation analysis

The hemisphere head-cylinder model shown in Fig. 1, only considers rigid highlights; performance of time-frequency BSS combined with image morphology is tested by the following simulation analysis.

Fig. 1 Object model

Case 1: A discussion on the influence of the new algorithm when the amplitude, and relative time delay of the highlight are changed. Active sonar transmits a chirp signal, with the normalized frequency ranging from f1=0.05 Hz to f2=0.1 Hz, and a pulse width of the transmitted signal of 2 ms. When the incident angle is θ=6° and θ=8°, respectively, received echoes contain rigid components scattered by edge 1, edge 2 and edge 3. The scattering components from angular 3 can be separated in the time domain from edges 1 and 2 because of the greater time delay. Therefore, only the presence of highlights generated by edges 1 and 2 are considered. It is assumed that the source signals are mixed by random matrix A=rand(2, 2), and that the amplitude is 1 and 0.1 independently. The observed signal and the corresponding WVD are shown in Figs. 2 and 3. The correlation coefficient between the separated signals and the transmitted signal at different incident angles is shown in Figs. 4 and 5.

Fig. 2 Signal waveform
Fig. 3 WVD of the observed signal
Fig. 4 Signal separation under the incident angle θ=6°

In Figs. 2-5, the smaller highlight is submerged in the WVD time and frequency domain, and the time-frequency BSS can recover the smaller energy signals. However, this method is limited by the WVD time and frequency resolution. When the incident angle is 6°, there is a disclosure ingredient in channel 2. However, when the incident angle is 8°, the two rigid highlights can be separated with the correlation coefficient between the separated signal and transmitted signal being equal to 1. In this case, the relative time delay number between the two highlights corresponds to 50 samples. Fig. 6 shows how the correlation coefficient gradually decreases as the time delay between the two highlights increases. It can be seen that the new algorithm can separate the two highlights in the time domain only when the correlation coefficient is less than 0.1.

Fig. 5 Signal separation under the incident angle θ=8°
Fig. 6 Correlation coefficient with different time delays

Case 2: A discussion on the effect of the different pulse width of the transmitted signal to the algorithm. Active sonar transmits a chirp signal, with the normalized frequency ranging from f1=0.05 Hz to f2=0.1 Hz, and the pulse width is 0.5 ms and 2 ms, respectively. According to the object scattering mechanism, when soundwaves enter the nearby beam, object echoes contain rigid scattering components and elastic scattering components. Using Eq.(1)in the free field condition when the incident angle is 110°, the object echoes including three rigid highlights and two elastic highlights with smaller amplitude are simulated. Altogether, there are five channels that are observed signals and five separated channels which are structured correspondingly. The processing results are shown in Figs. 7-10.

Fig. 7 WVD of the observed signal
Fig. 8 Correlation coefficient and WVD
Fig. 9 Time-frequency distribution of the separated signals
Fig. 10 Correlation coefficient of the separated signals

In Figs. 7-10, it can be seen that the WVD of the observed signals has interference from cross terms; only three highlights can be seen because the energy of the elastic highlights are too small. A morphological filter can be used to remove the inference from the cross terms and obtain the single auto term. Time-frequency BSS combined with the image morphology method can realize the separation of object echo highlights. The separation performance is not affected by differing pulse width. That is this algorithm can separate not only rigid scattering components but also elastic scattering components on the condition that the highlights are non-overlapping in the time and frequency domains.

5 Experimental data processing

Experiments were conducted in a free field pool and the sidewall of the tank was a strongly reflective object. The sidewall reflecting components can be removed using geometrical calculations. By doing this, it is possible to analyze only the signal within the time window. A linear frequency modulated pulse is transmitted by a transmitting transducer, with the normalized frequency ranging from 0.02 Hz to 0.08 Hz, and the pulse width of 2 ms. Rotating the object from 0° to 360°, it was possible to observe signals under different incident angles. By extracting the scattering components, the rotating status can be analyzed under conditions where the incident angles are unknown. The measurement object model is shown in Fig. 1; the object echoes are received by uniform linear arrays. Selecting the representative incident angles of 30°, 90°, 180°, respectively, the processing results are shown in Figs. 11-13.

Fig. 11 Signal separation with a grazing angle of 30°
Fig. 12 Signal separation with a grazing angle of 90°
Fig. 13 Signal separation with a grazing angle of 180°

In Figs. 11-13, when the incident angle is 30°, according to the highlight model, there should be three highlights; angle 3 can be separated in the time domain by mathematical calculation. Within the time window we can see that there are two highlights from the WVD of the observed signals, and it can be observed the two highlights have been separated in two channels. When the incident angle is equal to 90°, the object echoes should include edges 1 and 3 and the highlight scattering by a beam theoretically. However, an elastic echo with a complex formation mechanism is difficult to achieve, but only one highlight is observed at the received end. When the incident angle is 180°, edge 3, edge 4 and ball crown should be observed, but we only get a highlight in channel 2. In channel 1 the highlights are still aliasing. This is due to the separability being limited by WVD time and frequency resolution. When the transmitted signal is a linear frequency modulated signal with the pulse width T, equal to 2 ms, frequency resolution is the reciprocal of T, and the time and frequency resolution must satisfy the formula $\Delta t \cdot \Delta f \ge 1/4\pi {\rm{ }}$. And so, the time resolution can be obtained, $\Delta t \approx $0.16 ms. If the time delay between the different highlights is smaller than the time resolution, it will be recognized as the same highlight and cannot be separated.

Through the above analysis, when the incident angles range from 0° to 180°, the algorithm can separate the highlights effectively without nearby 90° and 180°. This is the same as the incident angles ranging from 180° to 360°.

6 Conclusions

The rigid acoustic scattering characteristics are important basis for underwater objects detection and identification. To overcome the problem that rigid scattering components are mixed together, a time-frequency BSS combined with image morphology is used to separate signal highlight structure for active sonar. There are mainly three advantages. Firstly, the method obtains the auto-term time and frequency points just according to the different time-frequency morphological characteristics, and it doesn’t require any channel knowledge. Secondly, the morphological filter could remove the cross-term interference between the acoustic scattering components according to the different morphological characteristics in time and frequency domain and improve the WVD resolution. Thirdly, the waveform of each scattering component is obtained and it is the basis of the characteristics extraction. Simulation and experimental data processing results have shown the feasibility and effectiveness of the new method in separating rigid scattering components by underwater object.

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Article Information

Yang Yang, Xiukun Li
Time-Frequency Morphological Characteristics for Rigid Acoustic Scattering by Underwater Objects
Journal of Marine Science and Application, 2016, 15(02): 201-207.
DOI: 10.1007/s11804-016-1352-z

Article History

Received date: 2015-09-17
Accepted date: 2016-01-22