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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
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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
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