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Abstract
A new signal-processing method to realize blind source separation (BSS) in an underwater lidar-radar system based on complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) and independent component analysis (ICA) is presented in this paper. The new statistical signal processing approach can recover weak target reflections from strong backward scattering clutters in turbid water, thus greatly improve the ranging accuracy. The proposed method can overcome the common problem of ICA, i.e. the number of observations must be equal to or larger than the number of sources to be separated, therefore multiple independent observations are required, which normally is realized by repeating the measurements in identical circumstances. In the new approach, the observation matrix for ICA is constructed by CEEMDAN from a single measurement. BSS can be performed on a single measurement of the mixed source signals. The CEEMDAN-ICA method avoid the uncertainty induced by the change of measurement circumstances and reduce the errors in ICA algorithm. In addition, the new approach can also improve the detection efficiency because the number of measurement is reduced. The new approach was tested in an underwater lidar-radar system. A mirror and a white Polyvinyl chloride (PVC) plate were used as target, respectively. Without using the CEEMDAN- Fast ICA, the ranging error with the mirror was 12.5 cm at 2 m distance when the attenuation coefficient of the water was 7.1 m-1. After applying the algorithm, under the same experimental conditions, the ranging accuracy was improved to 4.33 cm. For the PVC plate, the ranging errors were 5.01 cm and 21.54 cm at 3.75 attenuation length with and without the algorithm respectively. In both cases, applying this algorithm can significantly improve the ranging accuracy.
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1. Introduction
Sonar detection and ranging has long been dominant in the field of marine survey and under water target detection, especially in long-distance deep-sea detection [1–4]. However, sonar systems must be installed on ships, rafts, or other fixed platforms where the sound wave must directly contact water. The working platforms of sonar limit its application in large-area searching and detection. In contrast, lidar system is very compact and the working platforms of lidar system is more flexible, it can be shipborne, airborne, or even spaceborne. Lidar technology has applications in many fields. In 2015, Adrian J. Brown from SETI proposed to apply polarization-sensitive lidar to the analysis of the atmosphere and surface of Mars. This proves that lidar has important development prospects in the field of outer space. [5,6]. Moreover, because of the narrow beam divergence of the laser, lidar imaging has a much higher spatial resolution than sound-wave detection. The above characteristics of lidar make it very suitable for applications such as shallow seabed mapping, river cross-section measurement, and large-area searching. lidar is an effective supplementary means of sonar system in underwater detection, surveying, and mapping [7–8].
Absorption and scattering of light in water are two biggest challenges facing underwater laser detection and ranging (LiDAR). Absorption causes signal attenuation and shortens the detection distance. Scattering not only causes attenuation but also adds strong clusters to the optical returns [9–12]. Strong backscattering can completely swamp the signal, cause false targets to be detected.
Lidar-radar technology was applied to suppress scattering clusters in underwater ranging [13–14]. In this technology, the intensity of the carrier is modulated at the radar frequency. It uses the different responses of the target and water scattering to the modulated signal to separate the signal from the scattered noise and to improve the signal-to-noise ratio. However, in turbid water, even after coherent demodulation, the scattering clutters by the water body may still account for most of the received echoes [15]. Therefore, with only carrier modulation technology, scattering clusters cannot be removed efficiently, and the remaining scattering clusters can deteriorate the ranging accuracy of underwater lidar [16–18]. To solve this problem, blind source separation (BSS) has been applied to separate backward scattering from target reflection, because they are statistically independent signals [19–22].
BSS is a promising signal processing method. It has broad applications in many fields, such as wireless communication, sonar, biomedical science, optical fiber communication, and neural networks [23–26]. It can separate the source signals from mixed observations without prior knowledge of the source signals and signal mixing parameters. Independent component analysis (ICA) is an important algorithm to realize BSS. It is based on the assumption that the mixed signals are statistically independent, therefore ICA can decompose the mixed signals into independent components.
In ICA algorithm, the number of independent observations must be equal to or greater than the number of sources that need to be separated. In underwater target detection experiments, more than two observations are required to separate the target reflection and the scattering clusters. Normally, the two observations are realized by repeating the measurements in an identical circumstance. However, owing to the flow of water, the small movement of the target, and the shift of smapling clock et al., it is difficult to ensure that multiple measurements of the same target are in the same state and condition. The change of measurement situations will lead to errors in ICA algorithm. To solve this problem, we propose a single-time observation blind source separation algorithm based on complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN).
Empirical mode decomposition (EMD) is the foundation of CEEMDAN. It is an adaptive time-frequency signal processing method proposed by NASA scientist Huang in 1998 [27]. EMD can adaptively decompose a signal into Intrinsic Mode Functions (IMFs) according to the characteristics of the input signal without any prior knowledge [28–32]. In 2011, Torres proposed the CEEMDAN method [29–33]. This adds adaptive noise at each stage of EMD decomposition. Compared with traditional EMD, the errors of the reconstructed signal of each component of CEEMDAN were significantly reduced, and the decomposition efficiency increased [34–36].
In this paper, a new signal processing algorithm based on CEEMDAN and Fast ICA in an underwater lidar-radar system was presented. The new algorithm used CEEMDAN to decompose the detected signal into a series of IMFs, then the IMFs which are closely correlated to the original detected signal and the original signal were used to construct a new observation matrix for ICA. The new approach used a single measurement to realize BSS, thereby avoid the uncertainties of multiple measurements induced by the change of experimental conditions. Compared with only ICA algorithm, the application of CEEMDAN and Fast ICA algorithm was tested being able to improve the ranging accuracy in an underwater lidar-radar system.
The paper is laid out as follows: For the sake of readability and completeness, in Section 2 we introduce the procedures and applicable conditions of Fast ICA for BSS and CEEMDAN method, respectively. In Section 3 the underwater lidar experiments are presented, and the ranging results with and without CEEMDAN-ICA data processing are compared. Finally, in Section 4 we summarize our work and discuss possible improvements in future research.
2. Model description
2.1 Fast ICA
In this paper, ICA is used to separate the target echo from the scattering clusters. Statistical independence between independent elements is not only a prerequisite for the ICA algorithm, but also the basis for judging the convergence of the algorithm. According to the central limit theorem [37–39], the aliased signals are the superposition of multiple independent components, and are more in line with the Gaussian distribution. In the ICA algorithm, the Gaussianity of the aliasing observations by sensors is stronger than that of the source signals. Therefore, non-Gaussianity can be used as a criterion for judging the independence of the components to be separated.
In this study, kurtosis was used as the separation metric to indicate the degree of non-Gaussianity. The kurtosis is a fourth-order statistic. For a zero-mean random variable x, the kurtosis is defined as,
where $E\{{\cdot} \}$ is the operator that calculates the expectations. Using kurtosis, the Gaussianity of the components may be weakened or removed, and ICA separates the independent components. Suppose X is the observation matrix $(N \times M)$. The number of rows in X matches the number of observations N (or the number of sensors), and the number of columns equal to number of data bins of the signal M. then, X can be expressed as, where S is the source matrix $(Y \times M)$, It consists of the independent components to be separated, where A is the ‘mixed’ matrix $(N \times Y)$. The number of rows in A is the number of independent observations N, and the number of columns equal to number of sources Y. One row of the source matrix S corresponds to one source signal, and one column corresponds to a set of one observed data. In order to obtain the source matrix S from the given observation matrix $X$. Based on the Gaussianity we mentioned before, ICA calculate the “weight matrix” to maximize the non-Gaussianity of each independent sources:Once the algorithm converges, the source signals are successfully separated, and the estimate of the signal matrix can be expressed as,
Fast ICA was first proposed by A. Hyvarinen in 1997 [23]. It is an improved algorithm based on ICA. Fast ICA added whitening process to reduce the amount of computation of ICA. A linear transformation was carried out to the observation matrix, after the transformation, one obtained ICA was then performed to the matrix. The process was called prewhitening, which reduced the dimension of the data and also had the effect of reducing noise. Fast ICA can effectively reduce the complexity of the algorithm and improve the speed of calculation. More detailed procedure can be found in many literature and text books [40–43].
The learning rule of this algorithm is to find the iterative matrix W such that the separated signal has the biggest non-Gaussianity. The full technical details of Fast ICA procedure can be found in many previous publications [40–43].
Traditionally, the application of ICA needs to meet two prior conditions:
- a) The components to be separated are independent of each other.
- b) The number of independent observations must be equal to or greater than the number of components to be separated.
The echo signal detected by underwater lidar is usually composed of a variety of independent aliasing components, and these signals may overlap in time and frequency. When the target is detected, the signals received mainly originate from three sources:
- a) The light directly reflected by the target.
- b) The light that touches the target and undergoes forward scattering.
- c) The backscattered light that is directly generated by the water body without contact with the target.
The backscattered light is not in contact with the object, therefore, it is independent of the light directly reflected by the target and the forward-scattered light. Thus, Fast ICA can be used to separate scattering clusters from the target reflection and forward scattering.
However, traditional ICA algorithm requires the number of observation must be equal or bigger than the number of signals to be separated. In case to separate backward scatterings from the rest of underwater lidar echo, at least two observations are needed. The two observations in ICA can be realized by two measurements in exactly same condition. But in reality, it is very difficult to have two measurements under identical condition, any small turbulence or even the clock shift of the sampling can induce errors in the ICA algorithm.
2.2 CEEMDAN
In order to solve the multi-channel problem, we proposed to combine CEEMDAN with ICA. By using CEEMDAN, one can construct multiple data sets as observations for ICA from one measurement.
EMD is the foundation of CEEMDAN. It is an adaptive time-frequency signal processing method. It can adaptively decompose a signal into IMFs according to the characteristics of the input signal without any prior knowledge. The main steps of EMD are as follows:
- a) Find all the extreme points of the original signal $x(t)$, and obtain the upper and lower envelopes by cubic spline interpolation. Take $k = 1$, calculate the mean of the upper and lower envelopes as the mean envelope of the original signal ${m_k}$. Calculate the difference between $x(t)$ and ${m_k}$:
Use ${r_k}$ as the new signal $x(t) = {r_k}$ and repeat steps (1) and (2) to obtain other IMFs. Until the remaining term ${r_k}$ stops oscillating and converges to a monotonic function of t, then, the loop of EMD ends. The original signal is decomposed as follows:
Although the IMFs obtained by EMD can be used for subsequent calculations, there are also problems with EMD, such as mode mixing and redundant false components. This means that when the signal is mixed with high-frequency noise and intermittent oscillation components, the decomposed IMFs exhibit waveform aliasing and are difficult to distinguish.
In order to solve a series of problems existing in EMD, Torres proposed CEEMDAN in 2011 [44]. In this method, a Gaussian white noise was added to each residual component after decomposition, which simplified the calculation and avoided mode aliasing problem. The main steps of CEEMDAN are as follows:
- a) Suppose $x(t)$ is the original signal, $\omega (t)$ is the Gaussian white noise, and ${\omega ^i}(t)$ is the noise added at the $i$-th time. The Gaussian white noise is added to the original signal to form a new signal ${x_i}(t)$ for the subsequent decomposition. ${x_i}(t) = x(t) + {\sigma _0}{\omega ^i}(t)$, $i = (1, \cdot{\cdot} \cdot ,N)$, and ${\sigma _0}$ is the standard deviation of noise.
- b) Decompose ${x_i}(t)$ N times using EMD. Each time, the first decomposed IMF is maintained. After n times of composition, calculate the average value of these remaining IMFs and take this average value as the first IMF of the entire CEEMDAN calculation. where ${M_k}[{\cdot} ]$ denotes the operator to obtain the $k$-th IMF by EMD, ${\sigma _k}$ is the amplitude of the noise added by the $k + 1$-th IMF. Subtract $IM{F_1}$ from the original signal $x(t)$ to get the first residual ${r_1}$.
- c) Perform EMD decomposition on the Gaussian white noise ${\omega ^i}(t)$ to obtain the adaptive noise. $k = 2$, add adaptive noise to the residual obtained in step (2) to construct the new signal $x(t)^{\prime}$.
Perform the same decomposition and calculation on the new signal $x(t)^{\prime}$ as in steps (1) and (2), and calculate the $k$-th IMF, $IM{F_k}$ and $k$-th residual ${r_k}$.
- d) Judge whether the number of extreme points of ${r_k}$ is greater than two, this shows that the CEEMDAN decomposition is not completely completed, then it is necessary to repeat step (3) to continue the decomposition of ${r_k}$. If not, it proves that the entire CEEMDAN is completed, and the final K IMFs are obtained ($k = 1,2,\ldots K$).
2.3 Signal processing using CEEMDAN-fast ICA
We consider the data measured in a certain experiment as an example, and apply CEEMDAN-Fast ICA to verify the feasibility of the algorithm. Figure 1 shows the entire calculation process of CEEMDAN-Fast ICA
Figure 2 shows the waveform of the signal detected by the PMT when the attenuation coefficient was 1.5 m-1, and the detected target was the PVC plate. It was placed 1 m away from the incident window. The attenuation coefficient refers to the attenuation value of the optical signal every one meter in an aquatic environment. Assuming that the target is placed at a distance of ${x_1}$ and ${x_2}$ from the zero point, the detected intensity echoes are $I{}_1$ and $I{}_2$ respectively.
where ${I_0}$ is the intensity of the incident light, $\alpha$ is the transmittance of the entrance window of the water tank, and $\beta$ is the reflectivity of the target. The attenuation coefficient of water is then calculated byFigure 2 shows a waveform of a detected signal. The target was a diffuse reflection target, the received signal was weak and contained a lot of scattering clusters.
We performed CEEMDAN on the detected signal and decomposed it into several IMFs as shown in Fig. 3. The separated IMFs were arranged from high to low frequency from top to bottom. (The number of IMF finally decomposed is 12, then the number of extreme points of the residual was less than 2, and the calculation process ended.)
The correlation coefficients between the IMF and the original signal were calculated as, 0.3088, 0.3300, 0.5197, 0.6047, 0.5488, 0.5081, 0.1983, 0.0057, 0.0031, 0.0424, 0.0117, and 0.0148, respectively. From the waveforms shown in Fig. 3, one can see that when the correlation coefficient is less than 0.3 (IMF7, 8, 9, 10, 11, 12), the IMFs have lost all the high-frequency components and have low correlations with the original signal. Therefore, they were abandoned, and we subtracted them from the original signal to obtain the denoised signal. We then used the remaining IMFs with high correlation (IMF1, 2, 3, 4, 5, and 6) and the denoised signal to reconstruct the new observed signal matrix for the subsequent ICA algorithm.
We performed the Fast ICA on the newly constructed observation matrix. Figure 4 shows the waveforms and the spectra of the separated target reflection and scattering clusters after Fast ICA. This clearly shows that the algorithm can successfully separate the target reflected signal from the scattering clusters. The separated target reflected (plus forward scattering) signal can be used for the subsequent ranging calculation.
Fig. 4. Time domain and frequency domain waveforms of target echo and backscatter signal obtained by decomposition (a) Time-domain waveform of the target signal (b) Frequency waveform of the target signal (c) Time domain waveform of the scattering clusters (d) Frequency diagram of the scattering clusters.
3. Underwater ranging experiments
3.1 Experimental setup
The light source was a lab-built radio-frequency (RF) intensity-modulated 532 nm laser [18]. A single-frequency laser output at 1064 nm from a NPRO was coupled to a Mach-Zehnder electro-optic modulator (EOM). The modulation signal of the EOM was generated by a RF signal source (Siglent, SSG3021), the modulation frequency can be tuned from 10 MHz to 2.1 GHz. The intensity modulated laser output was amplified with a 2-stage ytterbium-doped fiber amplifier (YDFA) and then frequency doubled with A 15 mm long magnesium oxide doped periodically-poled lithium niobate (MgO: PPLN) nonlinear crystal. The maximum output power was 2.56 W at 532 nm. The modulation depth was 0.76.
Figure 5 shows the schematic of the experimental setup. The 532 nm laser was incident into a water tank through a window and reached the target. The reflected light from the target left the water through the same window and was received by the detector after passing through the a focusing lens and a 532 nm fliter. The focusing lens has a diameter of 2 inches and a focal length of 200 mm. The water tank was an acrylic water pipe with an inner diameter of 0.4 m and a length of 3 m. The incident window was a quartz glass with a thickness of 10 mm. We placed a track that could move back and forth inside the tank to fix the target so that it could be placed anywhere in the tank. We used a mirror and a diffuse PVC plate as the test targets. A photomultiplier tube (PMT, Hamamatsu Company H10720) was used as the detector. The effective receiving area of the PMT was 0.5 cm2. A digital oscilloscope was used to sample the output of the PMT and the reference signal from the RF signal generator for later cross correlation. The amplifier module of the PMT we used has a bandwidth of 300 MHz. Therefore, the modulation frequency of the laser was set to 250 MHz.
Mg(OH)2 powder was added to change the turbidity of water. A submerged water pump was used to circulate the water and keep the Mg(OH)2 powder suspending in the water as homogeneous as possible.
3.2 Ranging method
Cross-correlation of the detected signal and the reference signal added to the modulator was performed to obtain the target ranging information. The target range was determined by the location of the cross-correlation maxima.
To establish the origin of the range, we first placed the mirror close to the glass window of the water tank and performed distance measurements. We took this value as the origin. The other measured distances were calculated from this point. A millimeter-scale tape was attached to the left side of the tank for reference. In order to avoid the interference of stray light on the PMT, we conducted a shading treatment, and wrapped the whole optical path with a baffle made of black aluminum alloy. The results of the cross-correlation of the origin are shown in Fig. 6 (a), and the peak position corresponds to a time delay of -0.38 ns.
Fig. 6. The correlations of the echo and reference signal, and the target was placed at 1 m from the window, the attenuation coefficient was 1.5. (a) Range of the origin. (b) Target range without CEEMDAN-ICA. (c) Target range with CEEMDAN-ICA.
Then we put Mg(OH)2 into the water, and changed the attenuation coefficient. We used a PVC plastic plate as the detection target and placed it 1 m away from the starting point in the water tank.
We performed the cross-correlation on the data processed in Section II-D. Figure 6 (b) shows the results without applying ICA, the peak position corresponds to a time delay of 9.1702ns. Taking into account the refraction index of water, the lidar measured distance of the object to the origin was 107.44 cm, the ranging error was 7.44 cm. After applying CEEMDAN- Fast ICA to the data, we performed the same correlation calculations on the processed data. The correlation results are presented in Fig. 6 (c). The peak position corresponds to a time delay of 8.6973 ns. The measured distance was 102.12 cm, and the ranging error was reduced to 2.12 cm.
3.3 Ranging results
We tested the algorithms using experimental data. Firstly, a mirror was used as the target. The target was placed at four positions, 0.5, 1, 1.5, and 2 m from the origin, under two water turbidity conditions.
To prove the validity of the CEEMDAN- Fast ICA method, we applied ICA without CEEMDAN to the signal matrix obtained with multiple measurements. The results obtained by using different algorithms are shown in Fig. 7. The nominal range was measured with a millimeter resolution. When ICA and CEEMDAN- ICA were not performed, the received signal contained a lot of backward scattering clutters. As we can see from Fig. 7 (a)- (d), under the same attenuation coefficient, when the distance increased, the measured results deviated further, and the ranging errors increased. After ICA and CEEMDAN- ICA were applied, the influence of the scattering clusters was greatly reduced. The ranging accuracy was significantly improved, and the improvement was more pronounced in more turbid water. ICA and CEEMDAN- ICA gave very similar results. However, because CEEMDAN- ICA overcame the uncertainty caused by the change in measurement conditions in multiple measurements, the results of CEEMDAN- ICA were slightly better than those of traditional ICA.
Fig. 7. Ranging results and ranging errors of specular reflection target under different attenuation coefficients. (a) (a) Ranging results of a target with specular reflection at turbidity of c = 4.1 m-1 (b) Ranging results of a target with specular reflection at turbidity of c = 5.2m-1 (c) (c) Ranging results of a target with specular reflection at turbidity of c = 6.0 m-1 (d) Ranging results of a target with specular reflection at turbidity of c = 7.1 m-1 (e) Ranging errors as functions of attenuation length
To summarize the measured data under different attenuation coefficients, Fig. 7 (e) shows the ranging errors at different attenuation lengths. Without the ICA algorithm, the ranging error increases sharply with the increase in the attenuation length. At 14.2 AL, the ranging error reached 12.5 cm. After Fast ICA was applied, the ranging error was reduced to less than 7.46 cm. After CEEMDAN- ICA was applied, the ranging error was reduced to less than 4.33 cm. The results show that CEEMDAN- ICA algorithm greatly improved the ranging accuracy of underwater lidar in turbid water. However, as shown in Fig. 7 (e), at AL= 10.4, the experimental point shows a larger ranging error, which we believe was due to an error during the measurement. Similar results are shown in Fig. 8 (e), at AL= 1.5. We still placed these two points on the final curve to ensure the authenticity of the experimental data.
Fig. 8. Ranging results and ranging errors of diffuse reflection target under different attenuation coefficients. (a) Ranging results of a target with diffuse reflection at turbidity of c = 0.9 m-1 (b) Ranging results of a target with diffuse reflection at turbidity of c = 1.5 m-1(c) Ranging results of a target with diffuse reflection at turbidity of c = 2.1 m-1 (d) Ranging results of a target with diffuse reflection at turbidity of c = 2.5 m-1 (e) Ranging errors as functions of attenuation length.
In real applications, the target of lidar is often not a target with specular reflection, but a diffuse-reflection target with low reflectivity. The target was then replaced with a white PVC plate. The measurements were repeated in different attenuation coefficients. The results are shown in Fig. 8.
With the diffuse PVC plate, the largest measured AL was 3.75. When the ICA algorithm was not used, the ranging error increased sharply with an increase in the attenuation length. At 3.75 AL, the ranging error reached 21.54 cm. After ICA was applied, the ranging error was reduced to less than 8.09 cm. After CEEMDAN-Fast ICA was applied, the ranging error was reduced to less than 5.01 cm. It can be clearly seen that the newly proposed CEEMDAN-Fast ICA algorithm was very efficient in suppressing the negative influence of backward scattering clusters on ranging accuracy in the underwater lidar-radar system.
4. Conclusion
A new signal processing algorithm based on CEEMDAN and Fast ICA in an underwater lidar-radar system was presented in this paper. It was applied to separate the backward scattering clusters from the received signal, thereby improving the ranging accuracy of the underwater lidar system. This new approach overcame the problem in ICA algorithm where the number of independent observations must be equal to or larger than the number of sources to be separated, which required multiple measurements to construct the observation matrix under same circumstances. The CEEMDAN- Fast ICA algorithm can use a single measurement to realize BSS, thereby avoiding the uncertainties of multiple measurements induced by the change of experimental conditions and increasing the efficiency of the measurement. The new signal-processing approach was tested in underwater ranging experiments with both a mirror and a diffuse PVC plate as targets. The results showed that both ICA and CEEMDAN- Fast ICA could improve the ranging accuracy of lidar in a turbid medium. With CEEMDAN- Fast ICA, the ranging accuracy was better than that with only ICA. The ranging results proved the validity of the CEEMDAN- Fast ICA algorithm. The improvement in ranging accuracy with CEEMDAN was not very large in the laboratory environment where the conditions of multiple measurements were easily controlled and almost unchanged. However, in real lidar applications, the water flow speed, temperature distribution, waves will all induce changes of the condition of measurements. Therefore, it is impossible to ensure that multiple measurements of the same target are done under the same condition, which means the observation matrix rows cannot be obtained at same condition, which will induce errors in ICA algorithm. Thus, the advantage of the new approach with CEEMDAN- Fast ICA algorithm will be more pronounced in suppressing the effect of backward scattering clusters in a real application scenario
Funding
National Natural Science Foundation of China (61835001).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available from the corresponding authors upon reasonable request.
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