Micro‐Doppler effect removal for ISAR imaging based on bivariate variational mode decomposition
2017; Institution of Engineering and Technology; Volume: 12; Issue: 1 Linguagem: Inglês
10.1049/iet-rsn.2017.0104
ISSN1751-8792
AutoresWenwu Kang, Yunhua Zhang, Xiao Dong,
Tópico(s)Ultrasonics and Acoustic Wave Propagation
ResumoIET Radar, Sonar & NavigationVolume 12, Issue 1 p. 74-81 Research ArticleFree Access Micro-Doppler effect removal for ISAR imaging based on bivariate variational mode decomposition Wenwu Kang, Wenwu Kang The Key Laboratory of Microwave Remote Sensing, National Space Science Center, Chinese Academy of Sciences, Beijing, People's Republic of China University of Chinese Academy of Sciences, Beijing, People's Republic of ChinaSearch for more papers by this authorYunhua Zhang, Corresponding Author Yunhua Zhang zhangyunhua@mirslab.cn The Key Laboratory of Microwave Remote Sensing, National Space Science Center, Chinese Academy of Sciences, Beijing, People's Republic of China University of Chinese Academy of Sciences, Beijing, People's Republic of ChinaSearch for more papers by this authorXiao Dong, Xiao Dong The Key Laboratory of Microwave Remote Sensing, National Space Science Center, Chinese Academy of Sciences, Beijing, People's Republic of ChinaSearch for more papers by this author Wenwu Kang, Wenwu Kang The Key Laboratory of Microwave Remote Sensing, National Space Science Center, Chinese Academy of Sciences, Beijing, People's Republic of China University of Chinese Academy of Sciences, Beijing, People's Republic of ChinaSearch for more papers by this authorYunhua Zhang, Corresponding Author Yunhua Zhang zhangyunhua@mirslab.cn The Key Laboratory of Microwave Remote Sensing, National Space Science Center, Chinese Academy of Sciences, Beijing, People's Republic of China University of Chinese Academy of Sciences, Beijing, People's Republic of ChinaSearch for more papers by this authorXiao Dong, Xiao Dong The Key Laboratory of Microwave Remote Sensing, National Space Science Center, Chinese Academy of Sciences, Beijing, People's Republic of ChinaSearch for more papers by this author First published: 01 January 2018 https://doi.org/10.1049/iet-rsn.2017.0104Citations: 5AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Micro-Doppler (m-D) effect is caused by vibrations and/or rotations of mechanical components of moving targets. The m-D signatures corresponding to such micro-motions (m-Ms) may significantly degrade the usefulness of synthetic aperture radar/inverse synthetic aperture radar (ISAR) imagery. The strong echo of the main body of a target can make the m-D parameter estimation of the vibrating or rotating parts more difficult. The removal of m-D effect from target's ISAR image is thus very important for realising high-resolution imaging of a complex target involving m-M parts. To treat this problem, the bivariate variational mode decomposition (BVMD) is proposed to get rid of the m-D effect from the image of target's main body. The BVMD method first decomposes the radar echoes of range cells into a series of complex-valued mode functions according to which m-D signatures can be distinguished and removed from common Doppler responses. Finally, a refined ISAR image of the main body is produced using conventional range–Doppler imaging algorithms. Both simulated and real measured data are processed to show the effectiveness of the proposed method. 1 Introduction In inverse synthetic aperture radar (ISAR) imaging, the movement of a target can be decomposed into translational movement and rotational movement. The translational movement results in the same Doppler shift for all subscatterers of a target. So, the translational movement cannot contribute to the radar imaging. Different envelope delay and phase changes of echo are induced by the rotational movement. As the translational movement may cause range migration for a target echo, well-focused imaging cannot be obtained in that case. So, the translational motion compensation including range alignment and autofocusing should be applied. After the ideal turntable model is obtained, a well-focused image can be formed by the traditional range–Doppler (RD) algorithm [1]. In practice, many imaged targets can exist some inner-motion parts which may have some mechanical vibration, rotation, rocking, and coning motions along with the translational and rotational movements of targets as a whole. Those kinds of inner-motions are called as micro-motions (m-Ms) and the resulted Doppler frequencies are known as micro-Doppler (m-D) frequencies [2-5]. Different targets may have their own typical m-D features which can be explored for discriminating them. Owing to the existence of m-D, radar images of targets may be smeared and become difficult to recognise, and thus they should be removed. To achieve the goal, the m-D should be first separated from radar echoes. In recent years, a number of m-D separation algorithms have been proposed [6-10]. In [6], the radar echo was decomposed into a series of chirplet functions, and the signals of the main body and the m-D of the target were separated according to different chirp rates. However, the separation result depends on the selection of the number of chirplets, and the time cost is very high. In [7], the order statistics of spectrogram samples was applied to separate the m-D from the main body. Then, the inverse radon transform was used to estimate the m-D parameter. The Hough transform was utilised to extract the m-D signature of target in [8]. The L-statistics-based method was proposed in [9] and the histogram analysis-based method was applied in [10]. At each frequency bin, the time-frequency distribution was obtained via the short-time Fourier transform. Then, the L-statistics method and the histogram analysis method were applied to separate the main body and the m-D of the target. However, these methods in [7-10] are only applied to simulated data, their effectiveness for real data still needs further validation. The empirical mode decomposition (EMD) was used in [11] to analyse the m-D. In order to apply EMD to complex values, two extensions have been introduced in [12, 13]. Complex empirical mode decomposition (CEMD) was proposed in [12], and bivariate empirical mode decomposition (BEMD) was proposed in [13]. In the CEMD algorithm, the complex-valued signal is first decomposed into positive and negative frequency components using an ideal band-pass filter. Then, EMD is used to their real parts, and intrinsic mode functions (IMFs) are obtained. Finally, the complex-valued IMFs are accomplished by combining the IMFs with their Hilbert transform. In the BEMD algorithm, the complex-valued signal is first projected into different directions. Then, the real-valued EMD algorithm is carried out in each direction. Finally, the complex-valued IMFs are obtained by combining the IMFs of all directions. The CEMD was applied in [14] to separate signals from the main body and from the m-D. In fact, CEMD in [14] is just the BEMD algorithm in [13]. Different from the traditional signal processing algorithms, EMD and CEMD need not any prior information, and thus can both be utilised to analyse non-linear and non-stationary signals. EMD decomposes the signal into a series of IMFs in a recursion fashion with cubic spline interpolation applied, and thus the error will be transferred. The variational mode decomposition (VMD) was proposed in [15]. Just as EMD, VMD also decomposes the signal into a number of mode functions. Although different from EMD method, VMD method adopts variational operation instead of interpolation. Besides, VMD does not involve recursion operation between different mode functions. Compared with EMD, VMD has better performance on dealing with noise and decomposing closely distributed multi-tone signals [16]. The VMD method can only deal with real value (i.e. just amplitude) signals. However, the radar signals are usually of complex values. In order to apply VMD to complex-valued problems, some extensions are needed. Complex VMD was proposed in [17] inspired by the technique proposed in [12]. The BEMD method was applied to separate signals from the main body and from the m-D in [14]. In this paper, inspired by the works of [13, 14], we propose the bivariate variational mode decomposition (BVMD) method to extract the m-D signals so as to remove their responses and get much better radar image. Both simulated and real radar data are processed by the proposed BVMD method to demonstrate its effectiveness and better performance compared with the results by the CEMD method. This paper is organised as follows. The m-D signal model of a rotating target composed of point scatterers is introduced in Section 2. VMD is briefly presented and based on which, the BVMD algorithm is proposed in Section 3. The BVMD algorithm is applied to two simulated data sets in Section 4. BVMD is applied to real radar data, and clear radar images are obtained with m-D effect removed in Section 5. Concluding remarks are finally presented in Section 6. 2 m-D signal model Without loss of generality, the point targets are used to construct the target of m-D features. Fig. 1 shows the ISAR imaging geometry supposing the translational motion has been compensated. As the radar is so distant from the target, the k vector of the radar line of sight is parallel to the X-axis when the transmitted waveform reaches the target. The point Q(, ) of backscatter radar cross-section (RCS) represents the main body of target, while point P(, ) represents the m-M parts of the target, and the origin O is the rotating centre of the turntable model of the main body. The rotating radius of Q is , the initial angle is , and the equivalent rotating angular velocity of the main body is . is the local coordinates for the rotating parts, where is the rotating centre. With backscatter RCS of , P is one of the scatterers of the m-M parts positioning at (, ). The rotating radius of P is and the initial angle is . The scatterer P rotates quickly around , and the rotating angular velocity of the m-M parts is . The initial angle of is in XOY. Fig. 1Open in figure viewerPowerPoint Imaging geometry of a target with m-M scatterers Suppose the pulsed linear frequency modulated signal (chirp) is used, and the waveform is expressed as follows: (1) where (2) denotes the rectangular window; is the fast time, and is the slow time; T, , , and are the pulse repetition interval, the pulse width, the carrier frequency, and the chirp rate of the transmitted signal, respectively. In ISAR imaging, the ideal turntable model is formed after the translational motion is compensated. In Fig. 1, the scatterer point Q is a scatterer of the main body, which rotates a small accumulation angle within the imaging time. The Doppler frequency of the scatterer point Q is (3) where λ denotes the wavelength of the carrier frequency. According to (3), the Doppler frequency of the scatterer from the main body is proportional to the azimuthal Y-axis coordinate in XOY. A two-dimensional ISAR image can be obtained via fast Fourier transform along Y direction after the range profile of each pulse is obtained and aligned. As the scatterer point P rotates rapidly, the assumption of small accumulation angle within the imaging time is not valid anymore and (3) is not appropriate for P. In fact, the correct expression of the Doppler frequency for P should be (4) with (5) where is the initial angle. As does not have any influence on the analysis of the Doppler frequency, it can be omitted for simplicity, i.e. the Doppler frequency of P can be simplified as (6) with (7) represents the frequency component generated by scatterer P as the main body rotates. As shown in (6), except for the Doppler component of the main body, the scatterer point of the rapid rotational parts results in an extra Doppler frequency component, which is a sinusoidal frequency-modulated signal. Under the assumption that the rotation rate of the target's bulk is small and a small accumulation angle is resulted within imaging time, the range migration of the main body can be neglected. After range compression, the echo of P has the following expression [14]: (8) with (9) and (10) where and c denote the range frequency and the speed of light, respectively. After expanding (8) via Bessel functions and then Fourier transforming along the slow time, we get the following spectrum of the m-D [2]: (11) where and denote the azimuth frequency and the whole imaging time, respectively. The coefficient Jl(U) can be further expressed as (12) The coefficient Jl(U) has the following property [18]: (13) According to (11), we can clearly see that the m-D spectrum of the rapid rotating parts consists of pairs of spectral lines on both sides of the central frequency with spacing between adjacent lines. According to the Bessel functions' property, the bandwidth of the m-D is ∼2U. From the expression of U, we know that the larger the rotating radius and the smaller the wavelength of the carrier frequency , the wider the bandwidth of the m-D. When echoes contain m-D responses, conventional RD algorithms produce contaminated ISAR images. If a well-focused image of the target is expected, the influence of the m-D must be removed. 3 m-D effect removal based on BVMD 3.1 Real-valued VMD The VMD algorithm was proposed by Dragomiretskiy and Zosso in 2014 [15]. Different from the traditional signal analysis method, such as Fourier transform, Wavelet transform etc., the VMD method can analyse non-linear and non-stationary signal very well by decomposing a real-valued signal into a number of mode functions and each mode function uk is assumed to be mostly compact around a central frequency . The central frequencies of mode functions are distributed from low to high, and the mode functions reflect the frequency components of a signal. The VMD method forms the following constrained variational problem [15]: (14) where t denotes the time; and denote the Dirac distribution and convolution, respectively, and f denotes the real-valued input signal; {uk}: = {u1, …, uk} denotes the series of all modes, and k is the mode number; denotes the central frequency of mode uk. The summation of all modes uk is exactly the input signal f. The unconstrained formula for obtaining the optimal solution to the constrained variational problem of (14) using a quadratic penalty factor and the Lagrangian multiplier can be expressed as follows: (15) A VMD algorithm mainly includes the following steps: Updating modes as shown in (16) (16) where denotes the frequency script, and is the Fourier transform of . Updating central frequencies by (17) (17) Updating the Lagrangian multiplier by (18) to enforce exact signal reconstruction until (19) is met (18) (19) where is the threshold parameter for convergence control, and the update parameter of . 3.2 Complex-valued VMD As for ISAR imaging, radar system usually adopts the quadrature demodulation technique to get the in-phase and quadrature phase baseband components containing both amplitude and phase information, i.e. the received echo is a complex signal. In the following, we extend the real-valued VMD to complex-valued VMD so as to be applicable to ISAR imaging. In the BVMD algorithm, the complex-valued signal is first projected into different directions. Then, the real-valued VMD algorithm is carried out in each direction. Finally, the complex-valued mode functions are obtained by combining the mode functions of all directions. The complete BVMD algorithm is depicted as follows: Step 1: For a complex-valued signal s(t), initialise the parameter M, let m = 1. Step 2: Calculate (20) where denotes the real part of a complex-valued signal. is the projection of s(t) in the m direction, and it is real valued. is defined as . Step 3: Compute (21) where MF(·) denotes the decomposition of mode functions via the real-valued VMD algorithm. Step 4: If M is even, and m is equal to M/2, or M is odd, and m reaches to M, then carry out step 5, otherwise m = m + 1, and turn back to step 2. Step 5: If M is even, calculate (22) If M is odd, compute (23) When M is even, owing to the symmetry of directions, only M/2 mode functions of directions need to be computed. 3.3 m-D effect removal via BVMD As analysed above the rapid rotating parts will generate a large stripe region in ISAR image, which can seriously smear the image of the main body, and thus make it hard to recognise. In fact, the Doppler frequencies of the main body and the rapid rotating parts can be decomposed into different mode functions via the BVMD algorithm, i.e. we thus can separate the images of the main body and the rapid rotating parts. When a signal has been decomposed by BVMD, we can get the whole spectrum of all mode functions. Usually, the spectrum energy of the main body is concentrated in a small range, while the spectrum energies of the rapid rotating parts distribute in a large range. This means the spectrum energy of the main body mode function is much larger than that of the rapid rotating parts. Provided that a threshold value is properly selected, the echo of the main body and the echoes of the rapid rotating parts can be separated, after which clear radar image of the main body can thus be obtained, and the m-D parameters can be accurately estimated as well. We explain the detailed process of the BVMD algorithm via processing a simple signal as follows. Let us assume the signal is expressed as (24) which is composed of two terms, and the first term is the m-D, and the second term denotes the constant Doppler frequency of the main body. The accumulation time is 1 s, and the sampling rate is 256. Fig. 2a shows the time–frequency (T–F) spectrogram of the signal where the horizontal strip line denotes the T–F spectrogram of the main body, while the blurred sinusoidal curve reflects the T–F spectrogram of the m-D parts. When the number of mode functions is small and the bandwidth of each mode function is wide, then the main body cannot be separated. In this paper, we set the number of mode functions to be 20 and obtain the mode functions of the signal by the BVMD algorithm. Fig. 2b shows the energy ratios of each mode function with respect to the total energy of all mode functions. Just as analysed above, the first mode function of the main body has much larger energy than that of the rest mode functions. According to Fig. 2b, we select 0.1 as the threshold of the energy ratio of mode functions. Fig. 2c shows the T–F spectrogram of the first mode function, which denotes the main body, and Fig. 2d shows the T–F spectrogram of the m-D parts. Obviously, the Doppler frequencies of the main body and the m-D parts have been separated very well. Fig. 2Open in figure viewerPowerPoint Simulation results of a simple target (a) T–F spectrogram of the simulated signal, (b) Energy ratios for each mode function, (c) T–F spectrogram of the first mode function, (d) T–F spectrogram of the rest of the mode functions 4 Simulations In this section, two simulations are carried out by using the BVMD algorithm, and the results are compared with that by the CEMD algorithm. In the first simulation, the signal is composed of three parts and expressed as (25) where the first term is the m-D, the second and the third terms denote the constant Doppler frequencies of the main body. In this simulation, the constant Doppler frequency of the main body is clear from that of the rotating centre. Fig. 3a shows the T–F spectrogram of the signal where the two horizontal strip lines denote the T–F spectrograms of the main body, while the blurred sinusoidal curve is the T–F spectrogram of the m-D parts. Figs. 3b and c, respectively, show the T–F spectrograms of the main body and the m-D parts by the BVMD algorithm, from which one can see that the Doppler frequencies of the main body and the m-D parts are separated well. Figs. 4a and b, respectively, show the T–F spectrograms of the main body and the m-D parts by the CEMD algorithm, from which we can see that the Doppler frequencies of the main body and the m-D parts cannot be well separated. Fig. 3Open in figure viewerPowerPoint Separation of (25) by the BVMD algorithm (a) T–F spectrogram of the simulated signal, (b) T–F spectrogram of the main body, (c) T–F spectrogram of the m-D Fig. 4Open in figure viewerPowerPoint Separation of (25) by the BEMD algorithm (a) T–F spectrogram of the main body, (b) T–F spectrogram of the m-D In the second simulation, the target is composed of five-point scatterers as shown in Fig. 5a. The rotation centre of the main body and the m-D parts are both located at (0, 0). Scatterer 1 rotates fast with a rate of 6.67 Hz and a radius of 0.25 m, and the initial phase of is assumed. The coordinates of the other four scatterers are (25, 0), (−25, 0), (0, −25), and (0, 25), respectively. The carrier frequency of radar is 10 GHz and the system bandwidth is 500 MHz. The rotation rate of the main body is 0.02 Hz. The whole imaging time is 0.256 s, during which 512 pulses are transmitted and their echoes are received with the pulse repetition frequency of 2000 Hz. The back-scattering coefficient of the main body scatterer is two times that of the rotating scatterer. Fig. 5Open in figure viewerPowerPoint Simulation results of a target with five-point scatterers (a) Point scatterers distribution, (b) ISAR imaging without separation of m-D effects, (c) ISAR imaging of the main body with m-D effect removed by the BVMD algorithm, (d) ISAR imaging of the main body with m-D effect removed by the CEMD algorithm Fig. 5b shows the ISAR image of the point target model by the RD algorithm. Scatterers in the range direction can be influenced by the m-M parts depending on the m-M amplitude (e.g. the rotating radius of the m-M parts). In general, the m-M amplitude is small, the m-D effect (shadow) of m-M parts just influences within a small scope in range direction. That is why, as one can see, scatterers 2 and 3 are not shadowed (affected) by the m-D of scatterer 1. Different from the above case, scatterers in azimuth direction can be influenced not only by the m-M amplitude but also by the m-M velocity (e.g. the rotating angular velocity) of the m-M parts. As scatterer 1 rotates quickly, although scatterers 4 and 5 are far from scatterer 1, they are shadowed by the m-D of scatterer 1. We first apply the BVMD algorithm to the azimuthal signals of each range cell, and then separate the mode functions of the main body and the m-D parts, and finally, the ISAR image of the main body is obtained by the RD algorithm. Fig. 5c shows the imaging result of the main body, where the m-D effects are removed, and well-focused imaging is realised. As for comparison, the ISAR imaging result of the main body by the CEMD algorithm is presented in Fig. 5d, from which one can see that the m-D effects cannot be well removed. The entropy and contrast can be used to quantitatively evaluate the focusing level of ISAR imaging [19]. A smaller entropy and a larger contrast mean a better focusing result. The image entropy and contrast of a target of five-point scatterers are shown in Table 1. The image entropy and contrast of the original image obtained by conventional RD algorithm are 4.29 and 9.11, respectively. After the CEMD method applied, the image entropy and contrast become 4.13 and 11.47, respectively, i.e. the image entropy decreases 0.16 while the contrast increases 2.36. After the BVMD method applied, the image entropy and contrast change to 4.00 and 13.81, respectively, i.e. the image entropy decreases 0.29 while the contrast increases 4.70. It is clear that the proposed BVMD method improve the image entropy and contrast, respectively, ∼3.0 and 25.7% more than the CEMD method does. Table 1. Simulation results of a target with five-point scatterers Algorithms Image entropy Image contrast conventional RD 4.29 9.11 CEMD method in [14] 4.13 11.47 proposed BVMD method 4.00 13.81 5 Results of real radar data processing In this section, the BVMD algorithm is applied to the real Ka-band radar data for imaging of a truck [20]. In the experiment, stepped-frequency chirp signal (SFCS) is adopted. The total bandwidth of SFCS is 2 GHz with 20 subchirps and each of which has a bandwidth of 110 MHz. The carrier frequencies increase from 33 to 35 GHz at a step of 100 MHz. The interval between adjoin subchirps is 70 μs, and the burst repetition frequency is 500 Hz. Fig. 6a shows the ISAR image of a moving truck, from which one can see there are two m-D bands around the 95 and 120 range cells, respectively, and they are generated from the front wheels of the truck. Fig. 6b shows the T–F spectrogram of range cell 120, as one can clearly see that the main body of the truck and the m-D parts occupy different Doppler frequency regions. By applying our proposed algorithm to range compressed echoes at each range cell along azimuth direction, we can successfully separate the echoes from the main body and that from the m-D parts, then we can get clear radar image of the main body without influence of m-D effect. Fig. 6c shows the obtained ISAR image of the truck, as can be clearly seen, the m-D effect has been removed very well. For comparison purpose, we also present the result by the CEMD algorithm in Fig. 6d, from which one can see the m-D effect is also not well removed. By comparing Figs. 6c and d, one can clearly see the result of the BVMD algorithm is better than that of the CEMD algorithm. The image entropy and contrast are presented in Table 2. The image entropy and contrast of the original image are 4.08 and 1.88, respectively. After CEMD method applied, the image entropy and contrast become to 3.76 and 2.78, respectively, i.e. the image entropy decreases 0.32, while the contrast increases 0.90. After BVMD method applied, the image entropy and contrast change to 3.61 and 3.46, respectively, i.e. the image entropy decreases 0.47, while the contrast increases 1.58. As one can see, the proposed BVMD method improve the image entropy and contrast, respectively, ∼3.7 and 36.2% more than the CEMD method does. Table 2. Processing results of real data Algorithms Image entropy Image contrast conventional RD 4.08 1.88 CEMD method in [14] 3.76 2.78 proposed BVMD method 3.61 3.46 Fig. 6Open in figure viewerPowerPoint Processing results of real data (a) Original ISAR imaging of a truck, (b) T–F spectrogram for range cell 120, (c) ISAR imaging after the BVMD algorithm applied and m-D effects removed, (d) ISAR imaging after the CEMD algorithm applied and m-D effects removed 6 Conclusion In this paper, a new algorithm for removing m-D effect for ISAR imaging is proposed and called as BVMD. In this algorithm, the echo data is first range compressed and the cross-range signal of each range cell is decomposed into a series of mode functions containing different Doppler frequencies, and then, the mode functions of the main body and the mode functions of the m-D parts are separated. Finally, clearly focused ISAR imaging of the main body is obtained with m-D effect removed. In order to illustrate the effectiveness rigorously, the proposed algorithm is applied to both the simulated and measured data, and the results show that our proposed BVMD outperforms the CEMD algorithm in decomposing the Doppler frequencies of different mode functions. For the simulated data, compared with the CEMD method, the proposed BVMD method achieves ∼3.0 and 25.7% more improvement on the image entropy and contrast, respectively. As for the measured data, the proposed BVMD method achieves ∼3.7 and 36.2% more improvement. We need to point out that due to the image of m-D effect occupies just a small part of the whole image (simulation data) or the selected region (real data), the improvements on entropy are relatively smaller. 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