IET Radar, Sonar & Navigation Research Article 3D reconstruction of high-speed moving targets based on HRR measurements ISSN 1751-8784 Received on 24th August 2016 Accepted on 20th November 2016 E-First on 18th January 2017 doi: 10.1049/iet-rsn.2016.0426 www.ietdl.org Bi Yanxian1, Wei Shaoming1, Wang Jun1 1School of Electronic and Information Engineering, Beihang University, Beijing 100191, People's Republic of China E-mail: wangj203@buaa.edu.cn Abstract: Three-dimensional (3D) target reconstruction from inverse synthetic aperture radar (ISAR) data has a wide application in target scattering modelling, detection, and identification. In ISAR imaging of targets with complex motions such as the non-cooperative manoeuvring targets, the scattering centres on the target may rotate slowly in 3D space during the observation time. In this study, the authors have developed a new formulation for 3D target geometry reconstruction from the scattering centres high-resolution range (HRR) measurements, based on target motion features. First, after the translation compensation, the multi-view HRR of the scattering centres is extracted by HR spectral estimation technique. Then, the multiview measurements data without correspondence information are associated using the multiple hypotheses tracking algorithm. Finally, the 3D target geometry and motion are reconstructed from the singular value decomposition of the correlated HRR data matrix. The effectiveness of the proposed algorithm is demonstrated by both simulated and real data experiment results. 1 Introduction For many application areas such as target recognition and classification, it is becoming increasingly important to extract three-dimensional (3D) scattering centres from inverse synthetic aperture radar (ISAR) data. This is because 3D scattering centre model can provide geometrically correct 3D shape and size information about the radar target. In the past decades, the study and development of extracting the target 3D geometry via selective dominant scatterers has attracted much attention [1–3]. More recently, there are a few algorithms on the 3D target imaging [4–7]. In [4], the 3D reflectivity of the target is retrieved from the data in both azimuth and elevation aperture by using 3D Fourier transform with the knowledge of the target motion. The algorithms presented in [5–7] are based on interferometric ISAR techniques. The interferometric ISAR system employs multireceivers to obtain multiple 2D range–Doppler (RD) images. The height information of each scatterer is extracted using the interferometric phase difference among these 2D images. Therefore, the key step of interferometric approaches is to form the well-focused 2D RD images as well as preserve the phases. However, in realistic applications, most of the observed targets are non-cooperative and in the complex motions, which can be decomposed into two components: the translation of the centre of the gravity, and the rotation with pitch, roll, and yaw in 3D space. Owing to the time-varying Doppler frequencies of the scattering centres, the ISAR image achieved by standard RD algorithm is blurred. Thus, some more complex algorithms that preserve the phase information maybe used to form the 2D images [8, 9]. In the aforementioned approaches, it is notable that multisensors are used to receive the target echo signal and the scattering centre 3D positions are reconstructed by multiple 2D images. On the other hand, Stuff [10] and Stuff et al. [11] propose a general approach to recover rigid target geometric information with arbitrary unknown 3D rotational motion from 1D range-only measurements based on the invariant theory of rigid body. Ferrara et al. [12] present a modification to this work with eliminating the invariant constants constraints and reducing the computational cost. Both the two algorithms obtain the shape and motion estimation by singular value decomposition (SVD) of the measurement matrix, which we call SVD method. Although partial theoretical unification of the SVD method has been outlined in [11, 12], nevertheless the works solely focus on the data matrix decomposition problem, which is nearly a computer vision IET Radar Sonar Navig., 2017, Vol. 11 Iss. 5, pp. 778-787 © The Institution of Engineering and Technology 2016 problem. The discussion and analysis of the 3D reconstruction algorithms using 1D multi-view high-resolution range (HRR) data are all based on the following assumptions: (i) the 1D multi-view HRR data has been precisely extracted and (ii) the 1D multi-view HRR data has been correctly associated. Particularly, for the real application scenario about high-speed moving target, the complete reconstruction procedure using the 1D multi-view HRR data has not been put forward. In this paper, we propose a complete 3D reconstruction process for the space high-speed moving target using the 1D multi-view HRR data. After the translational motion compensation of the raw data, the HRR data extraction is carried out by applying modern spectral estimation technique [13]. Comparing with the conventional peak-finding method based on Fourier transform [14– 16], this technique offers two advantages: (i) the number of scattering centres of the target is correctly determined by the minimum description length or Akaike information criterion [17]. (ii) A significant resolution improving close to Cramer–Rao bound [18] is obtained, while the conventional Fourier-based technique is easily confused by the noise in the case of low signal-to-noise rate (SNR) and blind for the too closely distributed scattering centres. After the range data extracted by the spectral estimation routine, the multi-view HRR measurements need to be associated with their corresponding 3D scattering centres correctly. Here, the multi-view scatterers range data association problem is similar to the multitarget tracking problem. Therefore, multiple hypotheses tracking (MHT) algorithm [19, 20] is used to correlate the HRR measurements without knowing HRR correspondence information. MHT algorithm has been proven to be suitable for multi-target tracking in the dense clutter or false alarms (FAs) environment. It enumerates the feasible candidate measurement-to-track association hypotheses with a certain ‘time depth’, and performs multi-target tracking using track evaluation and management techniques such as track initialisation, maintenance, and termination. The classical measurement-to-track data association algorithms including the global nearest neighbour (NN) algorithm and joint probabilistic data association are regarded as a subset [21]. The final 3D ISAR reconstruction result is achieved by applying the SVD method to the associated range data matrix. In addition, the effects of wrongly range data association and the rotational angle diversity on the SVD method are investigated. It should be emphasised that the objective of this paper is to provide a practical unification for 3D target geometry reconstruction from 778 Fig. 1 3D image reconstruction geometry 1D multi-view HRR measurements paying particular attention to such above-mentioned problems. where V T is the translational velocity. For a short observation time, the influence of high-order components can be ignored. Therefore, we assume 2 System and measurement model V̇ T = V̈ T = ⋯ = 0 2.1 System model In this section, the geometry and signal models for 3D reconstruction of targets with complex motion will be derived and analysed in detail. The 3D image reconstruction geometry is shown in Fig. 1. For the applications considered here, the rigid target undergoes the translational motion and complicated 3D rotational motion. To model the problem, the orthogonal coordinate system Q − XYZ is defined as the radar reference system or the inertial coordinate system. The target translational velocity vector is V O and O is the gravity centre of the target. Coordinate system O − xyz is the target centred to describe the target 3D rotational motion. Basically, the 3D rotational motion is decomposed into three rotations (i.e. pitch, roll, and yaw) around the three coordinate axes. We assume the target consists of S non-coplanar prominent point scatterers, which are denoted as Ps = (xs, ys, zs), s = 1⋯S . The radar line-of-sight (RLOS) direction in the O − xyz coordinate system at time tm can be given by dm = sin θ cos ϕx^ + sin θ sin ϕy^ + cos θz^ where θ = θ tm , ϕ = ϕ tm , and x^ , y^ , z^ are unit vectors corresponding to the axes x, y, z, respectively. The instantaneous slant range between scattering centre Ps and the radar can be modelled by Rs tm = RO tm + rs tm (2) where RO tm and rs tm denote the instantaneous slant range induced by the translational and rotational motions, respectively. Obviously, scattering centres on the target share the same RO tm , which can be written in a polynomial form as 2 RO tm = RO0 + V Ttm + V̇ Ttm + V̈ Tt3m + ⋯ IET Radar Sonar Navig., 2017, Vol. 11 Iss. 5, pp. 778-787 © The Institution of Engineering and Technology 2016 According to Fig. 1, rs tm is the projection of 3D scattering centre Ps(xs, ys, zs) onto the RLOS, we have rs tm = dm, Ps (3) (5) dm, Ps where denotes the inner products and Ps = xs x^ + ys y^ + zs z^. The reconstruction problem here is to estimate the spatial coordinate of the scattering centres from (5). For the convenience of further derivation, we put (5) into the vector/matrix form rm = dmS, m = 1, …, M ϕM × S = DM × 3S3 × S where (1) (4) ϕ = [r1, …, rM]T with (6a) (6b) rm = [r1m, r2m, …, rSm], S = [s1, …, sS] with ss = [xs, ys, zs]T, rm is the 1D HRR sequence at the mth viewing angle, ϕ is the 1D range matrix, and S is the shape matrix. In principle, (6b) can be directly inverted to obtain the three unknowns xs, ys, zs with the accurate knowledge of the viewing angles [22, 23]. However, generally, in real scenario, we do not have any independent prior information of the target rotational motion. Therefore, in the following sections, we are focusing on estimating the shape matrix S from the associated 1D ^ HRR metric matrix ϕ according to (6b) with the constraints ∥ dm ∥ = 1. Remark 1: The translational motion reflects the entire target motion and carries no identity information. However, the 3D rotational motion is encoded with the target structural information. To reconstruct the scattering centre 3D coordinates information, the instantaneous range rs induced by rotational motion should be extracted accurately at every scan time tm. 779 2.2 HRR extraction S ∑ As rect s=1 tm f 4π rect exp − j f + f Rs tm B Ta c c (7) 1, u ≤ 1/2 ~ , B = γt is the signal bandwidth, γ is 0, u > 0 ~ the frequency modulation rate, t is the fast time, f c is the carrier frequency, T a is the observation time window, tm is the slow time, c is the light speed, and As is the well-known reflectivity coefficient. Generally, the scattering coefficient is a complex function of frequency and radar look-angle. However, considering the narrowangle measurement here, we assume the backscattered intensity of each scatterer does not vary with the frequency and the view angle. For the targets in high-speed motion, the stop–go model does not hold and the variation of RO in the fast time, together with the residual video phase terms, results in distorted and widened HRR profiles (HRRPs). Therefore, coherent pre-processing is carried out to eliminate their effects. After the coherent pre-processing method [24] is applied to the ISAR raw data, according to (2) and (3) we can have where rect u = E1 tm, f = S ∑ As rect s=1 tm f 4π rect exp − j f + f rs tm B Ta c c (8) The instantaneous range induced by 3D rotational motion could then be obtained using Fourier transform on (8). However, enhanced resolution is realised when the spectral estimation algorithm is applied to the measurement data set. We use the 2D spectral estimation algorithm based on state-space approach presented in [13] to extract the range data. Therefore, (8) could be rewritten in discrete form as follows: E1 m, n = c α^ 4π f 0Δt s (11) r^s = c ^ β 4πΔ f s (12) ^ In this section, we will discuss the feature extraction process. Suppose the radar transmits wideband linear frequency modulation (LFM) signal and echo signal can be written as E tm, f = ṙs = S ∑ As exp − j4π s=1 f 0 + nΔ f rs + ṙsmΔt /c (9) where Δt is the pulse repetition time, rs and ṙs are the range and range rate of the scattering centre Ps at time zero, respectively, n = 0⋯N − 1, N is the sample number of the LFM pulse in fasttime domain, Δ f = B/N, f 0 = f c − B/2, m = 0⋯Mhit − 1, Mhit is the total number of the coherent process hits. One can note that the cross-product j4πmnΔ f Δtṙs /c of the exponent in (9) keeps it from confirming to the state-space signal structure ∑s A′s exp jmαs + jnβs .To remove the cross-product j4πmnΔ f Δtṙs /c, we choose a new time-sampling interval Δt′ satisfying f 0 + nΔ f Δt = f 0Δt′, then the data matrix can be expressed as E2 m, n = ∑ A′s exp jmαs exp jnβs s (10) where αs = 4π f 0ṙsΔt/c, βs = 4πrsΔ f /c, As′ = As exp j4π f 0rs /c , where now Δt is the original time-sampling interval and the ranges of both m and n over the rows and columns of E2 are centred on zero. After transforming the measured data array E2 in a form of the output of two coupled eigenvalue problems, the estimation ^ results α^ s and βs of the parameters can be straightforward determined. Moreover, the amplitude estimation can be evaluated by a ‘least squares’ procedure applied to (10), the αs and βs now being known quantities. Then, the pairs provide a direct extraction of the range rate and the range Remark 2: The translational motion reflects the entire target motion and carries no identity information. However, the 3D rotational motion is encoded with the target structural information. To reconstruct the scattering centre 3D coordinates information, the instantaneous range rs induced by rotational motion should be extracted accurately at every scan time tm. 3 Measurement data association After the extraction of the range data, the association process to ensure the scattering centres HRR data set with occlusions and clutters associated correctly is of ultimate importance for the subsequent SVD step. The effect of the wrongly associated range data on the 3D reconstruction will be discussed in the next section. Here, the MHT approach using K-best hypotheses [25] is used to deal with the unknown correspondence in HRR measurements with occlusions and FAs. The processing loop for K-best hypotheses MHT method is shown in Fig. 2. For convenience, the details are briefly reported from the filtering and prediction phase as follows: i. Filtering and prediction: The standard Kalman filter (KF) is used to estimate the scattering centres motion state vector and covariance from the measurement histories. Since the motion filter is independent of the number of scattering centres, we can dynamically change the number of scattering centres to update the motion filter when there are occlusions and FAs in the HRR measurements. ii. Associated matrix generation: As described in Section 2, the target of interest is assumed to have S scattering centres. However, for actual measurements, the measurements come from three sources: existed track, a new target, and FAs and the number of the measurement points N m at time tm is generally not equal to S. The associated matrix, i.e. cost matrix N × N + 2N Ω ∈ ℝ m l m can be calculated with the three kinds of source, where N l is the number of existed tracks. iii. Associated hypotheses generation and management: Unlike NN method choosing the nearest measurement to update the track, MHT algorithm contains several hypotheses, i.e. a set of compatible tracks. This results in an exponential growth in the number of hypotheses. For reducing the computational cost, two strategies are used as follows: K-best technique is exploited to generate the first K hypotheses on the basis of auction assignment method [25] in the hypotheses generation phase. N-scan pruning technique is employed for hypotheses pruning, which limits the number of hypotheses by controlling the depth of track tree. While the depth of track tree is larger than N, the approach will search for the highest node in current scan within the track tree, and then keep the root node branch which contains the highest node and delete the others. 4 3D reconstruction 4.1 3D reconstruction On the basis of the above development, the following discussion will introduce the steps in detail. Fig. 3 presents the flowchart for the 3D shape reconstruction of the target based on the HRR measurements. The steps can be summarised as follows: Step 1: Compensating translational motion for raw data based on the coherent pre-processing method [24] as follows: 780 IET Radar Sonar Navig., 2017, Vol. 11 Iss. 5, pp. 778-787 © The Institution of Engineering and Technology 2016 Fig. 2 Data flow of the association process based on MHT range RΔro tm = Rref tm − Ro tm ) via envelope alignment ^ method [27] and apply the estimate RΔro construct the compensation function H 2 in the fast-time domain ~ H 2 t , tm = exp − j 4π ~ ^ f + γt RΔro ⋅ exp c c (17) 2 4πγ ^ 2 RΔro c iii. Compress the range with fast Fourier transform (FFT), then construct the compensation function in the fast-time frequency ^ ^ domain with the estimates of V T and RΔro −j ~ ~ ~ H 3 f , tm = exp − j Φ31 f , tm + Φ32 f , tm Fig. 3 Flowchart of 3D reconstruction of high-speed target i. where Estimate the target velocity in (3) from the raw data [26] and ^ then apply the velocity estimate V T and the dechirping reference range Rref to construct the phase compensation function H 1 in the fast-time domain ~ ~ ^ VT ~ Φ31 f , tm = 2π − f c ~ ^ c + VT γ ^ +2π ~ H 1 t , tm = exp − j Φ11 t , tm + Φ12 t , tm + Φ13 ~ (13) t , tm where VT ^ c + VT 2 ~ ~ Φ11 t , tm = 2π f c ^ c c + VT ^2 −γ −2V T ^ c2 c + V T 2 (14) 2 Rref ~ −2V T ^ c + VT ^ 6Rref V T ^ c + VT ~ ^ 2 ^ ^ c c + VT RΔro ^ −2V T ^ c c + VT ^ 2 ~2 ~ 2RΔro f +f 2γ c RΔro −2Rref (19) ^ (20) iv. Take inverse FFT over range and we obtain the coherent signal in the form of (8). Step 2: Extract the scattering centres HR instantaneous range induced by 3D rotational motion as follows: i. ^ Φ12 t , tm = 2π f c −γ −Rref 2V T f −2Rref − γ ~ ^ ^ f − fc Φ32 f , tm = 2π 2V T (18) + 2γRref Φ13 t , tm = 2π γ V 2T + 2V Tc ^ c c + VT ^ −2cV T ^ c + VT 2 2 ~ t− ~ t− 2Rref c 2Rref c (15) 2 (16) ii. Estimate the reference range displacement (i.e. between the reference range of each pulse and the target rotation centre IET Radar Sonar Navig., 2017, Vol. 11 Iss. 5, pp. 778-787 © The Institution of Engineering and Technology 2016 Rearrange the coherent data obtained in step 1 into the state-space structure as (10) by resampling technique. ii. Apply the 2D-state-space spectral estimation method [13] to the data array E2 in (10) and obtain the range estimate r^s ^ and range rate estimate ṙs at each scan time tm. Step 3: Associate the HRR measurements data set by K-best hypotheses MHT method [25]. KF is used as the motion filter to predict the scattering centres rotational motion state. K-best technique [25] and N-scan pruning technique [20] are used to reduce the computational burden. The associated HRR measurements matrix can be rewritten as 781 Fig. 4 Shape and motion estimation result of the uncorrelated data (a) Range data of the five scatterers are in different colours. We switch some part of range history 1, 2, and 4 with range history 3, 4, and 5, respectively. The circles mean where the rearrangements happen, (b) Singular value distribution of the uncorrelated data, (c) Motion estimation from the uncorrelated data, (d) 3D shape reconstruction from the uncorrelated data v. Take the SVD of W , W = OΣO−1, to obtain the shape estimate S = UΣ−1/2 = VOΣ−1/2 vi. Estimate look-angles (up to an arbitrary rotation) via ϕ = UΣ−1/2d, with ∥ d ∥2 = 1 Table 1 Parameters of scattering centres Index x y z Backscattering coefficient #1 #2 #3 #4 #5 1 0 0 −1 0 0 0 1 0 0 −1 0 0 0 4 1 1 1 1 3 ^ ϕ= 4.2 Analysis for the SVD method (numerical examples) r^11 r^12 ⋯ r^1S r21 ^ r22 ⋯ r^2S ⋮ ⋮ ⋯ ⋮ ⋯ r^MS ^ r^M1 r^M2 (21) where M is the number of the successful association times. ^ Step 4: After we obtain the associated HRR set ϕ from step 3, the SVD method [12] is taken as: Compute the reduced SVD of the M × S measurement ^ ^ matrix ϕ to obtain ϕ = QAV T ii. Form the 3 × M matrix B = AQT iii. Form the M × 6 matrix Ξ from the rows of B iv. Solve the M × 6 system Ξw = 1 to obtain the elements of the 3 × 3 symmetric matrix W i. 782 4.2.1 Effect of uncorrelated range measurements: Here, the effect of uncorrelated range measurements on the target 3D reconstruction is investigated by numerical examples. For simplicity, we consider a target consisting of five point scatterers. The coordinates of the five scattering centres are listed in Table 1. The target is observed from 1000 different 3D viewpoints on a path as shown in Fig. 4c. In this experiment, we first calculate the exact range data according to the motion history. To obtain the uncorrelated range data, some of the data matrix rows are randomly mixed up as shown in Fig. 4a. Fig. 4b shows the uncorrelated range data matrix has four nonzero singular values. Figs. 4c and d give the shape and motion reconstruction from the uncorrelated data, respectively. The reconstructed motion path is skewed and does not coincide with the true motion path. Moreover, the motion history ends up with the three corresponding jumps. In this experiment, the Euclidean distance between the reconstructed scattering centres and the real ones is 5.41. The large error indicates the reconstruction is useless. Remark 3: As shown in the experiment, the wrongly associated 1D range data matrix always has at least four non-zero singular values. IET Radar Sonar Navig., 2017, Vol. 11 Iss. 5, pp. 778-787 © The Institution of Engineering and Technology 2016 Fig. 5 Mean Procrustes’ error for varying angular apertures and range noise levels (a) Reconstruction error with the variation of the angle in the case without noise, (b) Reconstruction error with the variation of the angle when the noise standard derivation is half times of the range resolution, (c) Reconstruction error with the variation of the angle when the noise standard derivation is one times of the range resolution, (d) Reconstruction error with the variation of the angle when the noise standard derivation is one and half times of the range resolution This will result in a meaningless shape and motion reconstruction. Therefore, it is most desirable to achieve right range data association after the range data acquisition. 4.2.2 Effect of rotational angle diversity: Here, the effect of the rotational angle diversity on the target 3D reconstruction is investigated by numerical examples. In this experiment, assume the signal bandwidth is 2 GHz (corresponding to a range resolution of 0.075 m). Given the initial elevation angle θ0 = 10∘, initial azimuth angle φ0 = 20∘, and both the angles increments vary from 1∘ to 30∘. For every angular aperture, the target is observed from 1000 different 3D viewpoints. Gaussian noise with the standard deviation of 0.5, 1, and 1.5 times of the range resolution is added to the initial 1D range matrix. Moreover, 500 Monte Carlo simulations are performed per noise level. At each aperture extent/ noise level pair, the reconstruction performance was measured by the Procrustes distance from the true point configuration, which accounts for object scale, rotations, and translations [28]. The experimental results are presented in Fig. 5. Fig. 5 shows the reconstruction performance becomes better with the increasing angular diversity. Figs. 5b–d show that the relationship between reconstruction performance and the angle diversity is similar to the case without noise as shown in Fig. 5a. Fig. 5a shows the Procrustes error is acceptable when angle variation ranges meet the condition Δϕ > 3∘, Δθ > 5∘. Otherwise, though one of the angles has a wide variation, the reconstruction error is still very big. As shown in Figs. 5b–d, in the case of Δϕ > 10∘, Δθ > 10∘, the reconstruction error is acceptable and the reconstruction performance is better when the two variation ranges IET Radar Sonar Navig., 2017, Vol. 11 Iss. 5, pp. 778-787 © The Institution of Engineering and Technology 2016 are approximately equal to each other, that is, Δϕ ≃ Δθ. In the case of Δϕ < 10∘, Δθ < 10∘, SVD method is less robust to noise and the reconstruction performance degrades quickly with the higher-level noise. Remark 4: As shown in this experiment, the SVD method actually obtains better reconstruction with higher 1D range data accuracy and larger variations of the rotational angles. 5 Experiment results In this section, both simulated and real experiment data are provided to test the effectiveness of the proposed 3D target reconstruction method. 5.1 3D reconstruction for simulated data Here, we use the simulated echo data of the target consisting of five scattering centres listed in Table 1 to demonstrate the effectiveness of the proposed method. The centre frequency of the signal is 10 GHz; the signal bandwidth is 2 GHz (corresponding to a range resolution of 0.075 m); the pulse repetition frequency is 1000 Hz; the pulse time width is 0.3 ms; and the dechirping signal sampling rate is 5 MHz. The range tracking error is within ±10 m, which means that the reference range displacement RΔro is within ±10 m. Furthermore, the complex Gaussian noise is added to the echoes and the SNR is 15 dB. As shown in Fig. 6a, the target is flying along a path with a radial speed around 4000 m/s. The distance of the target to the radar is 400 km. At the same time, the target undergoes the 3D rotational motion as shown in Fig. 4c. 783 Fig. 6 Flight trajectory and translational velocity estimation (a) Flight trajectory of the target, (b) Translational velocity estimation of the target Fig. 7 HRRP series before and after the translational motion compensation (a) 1D HRRP series of the scattering centres before the translational compensation, (b) 1D HRRP series of the scattering centres after the translational compensation The performance of the velocity estimation by using method in [26] is illustrated in Fig. 6b. To remove the high-order component in the estimation result, the estimation is further smoothed using a first-order polynomial fit for the translational compensation. Fig. 7a shows the HRRPs of the scattering centres before the translational motion compensation, where the HRRP series is disorderly and unsystematic because of the reference range displacement. Fig. 7b shows the HRRP series after the translational motion compensation via the pre-coherent method [24], from which it can be noted that the translational motion is removed and the 1D range profile is encoded with the target structural and rotational motion information. Fig. 8a shows the HRR measurements extracted by the spectral estimation method [13]. Each scattering centres has been uniquely associated with a radial range. The agreement between the extracted trajectories and the real ones is seen to be very good. It should be noted that because of the presence of noise, there are many extraneous FAs as shown in Fig. 8a. If these individual estimates are applied to the next 3D reconstruction process, one obtains a noisy image. In this experiment, the threshold of the SVD process of the Hankel matrix in the extraction process [13] is set to be very low, so some extraneous FAs as shown in Fig. 8a can be removed by improving the value of the threshold in the process of SVD of the Hankel matrix. Moreover, other extraneous FAs can be deleted in the next process of the HRR measurements association by applying the MHT algorithm [25]. Fig. 8b illustrates the association result by applying the MHT technique to the measurements set. One can note that the 1D range data has been correctly associated and the FAs have been deleted. The output of 784 the association process provides a foundation for the following SVD process for achieving a consistent 3D reconstruction result. The resultant 3D target geometry and rotational motion reconstruction are illustrated in Figs. 9a and b, respectively. The reconstructed 3D scattering centres coordinates are listed in Table 2. From Figs. 9a and b, it can be clearly noted that the target 3D scattering centres and rotational motion are correctly reconstructed. Fig. 10a depicts the effect of the noise on the Euclidean distance between the reconstructed scattering centres and the real ones. Fig. 10b shows the root-mean-square error (RMSE) of the reconstructed scattering centres. The process of the experimenting is implemented with 50 Monte Carlo runs for each SNR. As Fig. 10 shows, it can be found that the Euclidean distance decreases when the SNR increases, which means the reconstruction accuracy improves with the increase of the SNR. 5.2 3D reconstruction for civil aircraft B737-800 In the following, we apply the proposed 3D geometry reconstruction method to the real data set of civil aircraft B737-800 to demonstrate the validity of the method. The civil aircraft is at the take-off stage. During this phase, the attitude change of the aircraft is relatively large comparing with that of steady flight. Moreover, the rotational motion can be thought as a 3D motion, which is the requirement of the application of the SVD method. Some important parameters are given as follows: the distance between the radar and the aeroplane is about 29 km, the aircraft flies away from the radar with a radial velocity around 37 m/s, the echo is dechirping signal IET Radar Sonar Navig., 2017, Vol. 11 Iss. 5, pp. 778-787 © The Institution of Engineering and Technology 2016 Fig. 8 High-quality HRR measurements and association result (a) HRR extraction by using the 2D spectral estimate method, (b) Scattering centres trajectories after the association Fig. 9 3D reconstruction result (a) 3D reconstruction of the target geometry, (b) 3D reconstruction of the target rotational motion Table 2 Coordinates of reconstructed scattering centres Index x y z #1 reconstructed #1 #2 reconstructed #2 #3 reconstructed #3 #4 reconstructed #4 #5 reconstructed #5 1 0.9981 0 0.0001 −1 −0.9969 0 −0.0005 0 0.0607 0 0.0020 1 1.0351 0 −0.0036 −1 −1.0343 0 −0.1034 0 −0.0376 0 0.0225 0 0.0333 0 −0.0225 4 4.0473 with the bandwidth of 1 GHz, and the width of the signal received window is 150 m. The translational motion of the civil aeroplane can be regarded as uniform during the short observation window. After the translational motion compensation, the spectral estimation method is applied to the data to extract the scattering centres range induced by the rotational motion. As shown in Fig. 11a, it can be seen that there are ten HRRP trajectories curve of ten scattering centres and many FAs. Fig. 11b shows the range data association result using K-best hypotheses MHT method. It should be noted that the associated scattering centres range trajectories end at the 250th pulse. This is because there exist too many FAs and the data association fails after the 250th pulse using the MHT method. Fortunately, we just need more than 45 1D projections of the target to reconstruct the ten scattering centres 3D coordinates according IET Radar Sonar Navig., 2017, Vol. 11 Iss. 5, pp. 778-787 © The Institution of Engineering and Technology 2016 to the invariant theory [11]. So, these associated 1D HRR measurements are enough to reconstruct the ten scattering centres. It can also be seen that the trajectories of the scattering centres are almost smooth lines. This is because the aeroplane has a slight 3D rotational motion. Fig. 12 illustrates the 3D reconstruction result of the aeroplane. One can note that the reconstructed 3D aeroplane model is consistent with the parameters in Table 3. In a summary, the experimental results demonstrate the effectiveness of the proposed method. 6 Summary and conclusion For the ISAR target, particularly, the non-cooperative manoeuvring targets, 1D instantaneous range of the scattering centres induced by the 3D rotational motion is encoded the target 3D geometry information. In this paper, after the translational compensation, scattering centres 1D HRR data is extracted by the spectral estimation method. Then, the data set is associated via MHT technique. Finally, the target 3D geometry reconstruction is obtained from the associated scattering centres 1D HRR data matrix using SVD method. The effect of imperfect range data association on the target reconstruction is also investigated. Simulated data and real experiment data have confirmed the effectiveness of the proposed method. 7 Acknowledgment This work was supported by the National Natural Science Foundation of China under grant nos. 61671035, 61501012, 61501011, and 61302166. 785 Fig. 10 3D reconstruction Euclidean distance error and RMSE (a) 3D reconstruction Euclidean distance error, (b) 3D reconstruction Euclidean distance RMSE Fig. 11 High-quality HRRP series and association result of the B737-800 data (a) HRR extraction from B737-800 data using the 2D spectrum estimate method, (b) Scattering centres trajectories of the B737-800 after the association Fig. 12 3D reconstruction of the B737-800 (a) 3D view of the reconstructed scattering centres of B737-800, (b) 2D view of the reconstructed scattering centres of B737-800 Table 3 Parameters of civil aircraft B737-800 [29] Index Wing span Length Height 38.50 786 39.50 12.50 IET Radar Sonar Navig., 2017, Vol. 11 Iss. 5, pp. 778-787 © The Institution of Engineering and Technology 2016 8 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] Ausherman, D.A., Kozma, A., Walker, J.L., et al.: ‘Developments in radar imaging’, IEEE Trans. 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