1. Introduction
Tomography is a general imaging technique that is based on lower-dimensional projections of an object from different spatial aspects, which are then processed using the projection-slice theorem [
1] to reconstruct an image of the object. Radar tomography uses reflective scattering phenomenology and radar waveforms for the measurements, which may be wideband or narrowband. Wideband waveforms exploit
spectral diversity as system resources to facilitate radar imaging and have probably been the most exploited resources in practical applications in the last few decades. The well-known synthetic aperture radar (SAR) and inverse SAR (ISAR) imaging techniques may be described as two special forms of wideband tomography, in which another system resource—
spatial diversity—is exploited only minimally [
2]. Range-Doppler ISAR imaging, and stripmap SAR in particular, typically involve aspect angle changes of a few degrees [
3,
4,
5]. This constraint of small rotation angles in the linear phase regimes allows the image inversion processing to take advantage of the computationally efficient fast Fourier transform (FFT) without needing signal interpolation onto rectangular grids.
Spotlight SAR makes use of wider angles [
6], while circular SAR [
7] may coherently process up to a complete cycle of target aspect rotation, with sophisticated and precise motion compensation in range. More notably, in the associated spatial frequency spaces, also known as
k-spaces [
2], traditionally intensive interpolation processing prior to image inversion processing may be necessary. Nevertheless, these forms of SAR and ISAR rely on the bandwidth resource to achieve high down-range resolution, and so can be considered as belonging to the category of ‘wideband radar tomography’.
Radar tomographic imaging with ultra-narrowband or single-frequency waveforms relies on spatial diversity as the only system resource for image formation [
8,
9,
10,
11]. Spatial diversity may be realized by: (i) having a radar with multiple receivers looking at the target from diverse angular locations, the received signals from which are processed coherently, or (ii) using a single receiver looking at a target undergoing relative rotational motion, i.e., changing target aspect. Both cases widen the angular extents of the measurement support of the received signal in the
k-spaces.
Previous work [
2] showed that narrowband radar tomography can be most effectively formulated in the
slow-time k-space in conjunction with the classical Doppler processing and Doppler radar tomography (DRT) [
12,
13]. The DRT algorithm applies the projection-slice theorem in which the inputs of the target’s cross-range projections are formed from Doppler profiles. The slow-time
k-space is not only convenient for describing the DRT algorithm, it is also a natural tool to formulate high-resolution DRT imaging with an
augmentation of its measurement support. Augmentation is the process of significantly enlarging the support of the slow-time
k-space by using longer coherent processing intervals (CPI) in the DRT algorithm and correcting for nonlinear phase effects due to strong rotational motion. This ‘augmentability’ is a unique characteristic of the slow-time
k-space.
The introduction of nonlinear phase terms in the
k-space augmentation causes a blurring effect in the resulting image. Spectral compression techniques for chirped signals can be used to address this problem using bilinear transforms such as the Wigner-Ville Distribution (WVD), the Cohen’s class and the time-frequency distribution series (TFDS) as discussed in [
14]. The problem with these techniques is the presence of undesirable cross terms when instantaneous component frequencies may overlap, which is the case for DRT imaging [
13]. The combination of the fractional Fourier transform and S-method was used to overcome the problem of cross terms in [
13], which demonstrated the slow-time
k-space augmentation with DRT. The current work is extended to a more novel technique based on the orthogonal matching pursuit (OMP) technique, inspired by related work in compressive sensing.
Radar imaging naturally is suited to compressive sensing techniques, given that real targets often resemble a sparse collection of discrete point scatterers [
5,
15]. OMP is fundamentally a technique for parameter estimation by matching a given signal to a dictionary of possible elemental functions spanning a finite parameter space. The dictionary is designed for the particular application and has been applied to the area of DRT imaging in varying contexts [
16,
17,
18]. The particular application in this work is to estimate the non-linear phase term in the radar signal to reduce the image blurring for improved resolution. The main contribution of the paper is two-fold: to highlight the augmentability of the slow-time
k-space as a fundamentally useful characteristic for narrowband radar imaging, and to present a novel application of the OMP technique to such augmentation processing.
The slow-time
k-space processing technique as presented in this paper provides a complimentary approach to traditional high-resolution ISAR imaging. The dependence of wide bandwidth signals for high resolution in ISAR imaging is not always readily achievable within the confines of the available spectrum and limitations at lower frequency bands [
9,
19]. The proposed high-resolution imaging scheme can lead to improved target recognition despite an absence of wide bandwidth signals, provided sufficient spatial diversity is available. This is an important capability of great interest to the radar research community [
20].
The rest of the paper is organized as follows. The next Section summarizes the fundamental theory: system geometry and signal model, cross-range bandwidth and resolution, and DRT.
Section 3 describes the slow-time
k-space and its augmentation with OMP processing.
Section 4 describes the experimental setup using simple point scatterers on a rotating turntable with imaging results for both standard and augmented DRT processing. The final Section presents some relevant discussion points and concluding remarks.
2. Background
This Section defines the signal model, the fundamental concept of cross-range bandwidth and resolution, and summarizes the known theory of Doppler radar tomography (DRT) in its standard version.
2.1. Signal Model
Consider a monostatic radar system geometry as illustrated in
Figure 1. Without loss of generality, an inertial local target reference frame, denoted as
with origin
O at the target’s nominal centre of rotation, is chosen to have the the
(‘down-range’) axis aligned with the radar line of sight (LOS), with
axis denoting cross-range. The plane
is often known as the image projection plane (IPP, or just ‘image plane’). The axis orthogonal to the IPP is denoted as the
-axis (sometimes referred to as ‘height’). The target’s
effective rotation vector
is defined as the projection of the target’s total rotational velocity vector
along the
-axis.
Using the definition above, the total rotational velocity vector can be written in
as
Physically, introduces cross-range dependent Doppler shifts in the radar backscatter and is the principal reason that motion-based target imaging is possible. In comparison, has minimal (sometimes deleterious) impact on radar imaging. For non-cooperative targets, neither nor , or the orientation of the IPP itself, are known a priori. In this paper, we further assume that , and is approximately constant during a coherent processing interval (CPI).
For this paper, we use an idealized point-scatterer model for the target: it is adequately modeled as an ensemble of
M point scatterers with reflectivity coefficients
, located in the far field of the radar. The approximate range to the
point-scatterer on the target with position vector
, defined relative to
O, can be defined as
in which
is the range vector to the
scatterer,
is the radar range to
O, the scatterer’s local down range is
and
is the unit vector along the radar LOS in the
frame.
Formulation in the frame is appropriate in traditional ISAR imaging where the change of aspect is small (a few degrees), or for signal analysis within a relatively short CPI. In contrast, radar tomography exploits spatial diversity through wide changes of target aspect. For this formulation, a second, dynamic local target frame denoted as , is needed. This frame rotates with the target and coincides with at a reference time, usually assumed to be . By the assumption, it follows also that and share the same -axis. The reason for choosing frame is that its axes are aligned with those of the underlying k-spaces and thus preserves angles across the frame and the k-spaces.
Let
denote the simple transmit continuous waveform at a single frequency
f, where only the slow time
is involved; there are no pulses and hence no ‘fast time’ spanning a pulse. The slow-time index is
, and we assume a total of
K time samples in a CPI. The received signal
is a delayed version of
, summed over all scatterers,
Here, we have also assumed that radar hardware perfectly removes the carrier frequency term
. The first factor in (
4) describes translational motion of the target as a whole; the second factor captures the target geometry and scattering reflectivities to be processed for imaging.
Furthermore, we shall assume a linear translational motion model for the target,
where
is the velocity, assumed known prior to DRT processing, and
is target range at a reference time
.
2.2. Cross-Range Bandwidth and Resolution
The position of each scatterer executing rotational motion with rotation vector
is described to a second order approximation by
where
for convenience. Relative to
, the local down range
in (
4) can be expressed as
where
are the initial (
) cross range and range, respectively, of the
m-th scatterer in
. The CPI duration is denoted by
. As has been thoroughly discussed in [
2], although
(dropping the subscript
m for brevity) cannot be directly estimated with a zero-bandwith signal, the
first-order term of (
6) suggests that a so-called
cross-range bandwidth,
can be used to estimate cross range
. In other words, the target’s rotation generates an effective bandwidth which allows for the resolving cross-range measurements, as long as the rotation angle through
,
is sufficiently small such that higher-order terms (quadratic and above) in (
6) can be ignored. In practice, the
is limited to a few degrees, which is consistent with wideband ISAR imaging. Note that the presence of the (unknown) zeroth-order term
means
cannot be directly estimated from the time-domain signal. Doppler tomography, as formulated below, overcomes such constraints to achieve target imaging.
Consider a segmented CPI of the received signal
as illustrated in
Figure 2. Taking a Fourier transform over
produces a Doppler profile
, with zero Doppler (
) corresponding to the centre of rotation at
O (ignoring any residual translational motion after preprocessing). For a segment of duration
, the achievable Doppler resolution is
Each Doppler profile contains contributions from all scatterers, with the down ranges coordinates
encoded as constant phase terms. Since the cross-range of a scatterer is directly proportional to its Doppler frequency
, namely
it follows that the
magnitude of the Doppler profile,
represents a cross-range projection of the target’s reflectivity function at angle
, the average aspect angle over the CPI. The achievable
cross-range resolution is
This expression is exactly analogous to the down-range resolution for wideband imaging with spectral bandwidth B.
2.3. Doppler Radar Tomography (DRT)
The Projection-Slice Theorem (PST) states that the Fourier transform of projection is a slice of the 2D FT of the target’s reflectivity function at aspect angle . This theorem can be used to invert the cross-range profiles accumulated from a range of aspect angles to recover the target reflectivity function, i.e., estimate the scatterer coordinates and in frame. For this to be effective, the target’s rotation must subtend a significant change in aspect angles; the 1D cross-range projections are computed in the frequency domain as discussed above, after which the target reflectivity function (image) can be reconstructed by a 2D inverse FT.
2.3.1. The Monostatic DRT Algorithm
To perform radar imaging using the DRT method, it is necessary to populate the slow-time k-space from the radar backscatter. The algorithm to generate the slow-time k-space samples consists of the following steps:
Data segmentation: Partition the N samples of the received signal into L overlapping CPIs of K samples, , ; . These are referred to as ‘segmented CPIs’ below. Denote the overlap factor with . At the midpoint of each segment, the target aspect angle (relative to ) is denoted as ;
Translational motion compensation (TMC): this step shifts the Doppler component induced by translational motion to zero Doppler frequency by modulating the segmented CPI by
, where
is the target’s translational velocity as noted in (
5). This quantity is assumed to be known or estimated by other methods. A discrete Fourier transform is then applied to the modulated segments to obtain the Doppler spectrum. The magnitude of the output,
is the cross-range (which is proportional to Doppler) profile for the target at an angle
from its original orientation. Accumulate all such cross-range profiles for all the corresponding aspect angles
, i.e., for all
L segmented CPIs.
Populating the k-space: The spatial Fourier transform of
at target aspect angle
are then used as the ‘measurement samples’ in the slow-time
k-space. As the target rotates, the measurements sweep out a region of support in slow-time
k-space as indicated in
Figure 2. Due to our choice of reference frames, the measurement population always starts close to the
-axis because
is the initial cross-range profile.
Image inversion: An inverse Fourier transform is applied to the populated support of the
k-space to yield the target image. Other works have either used filtered back projection, or interpolated the samples onto a rectangular grid to utilise a standard 2D inverse Fourier transform, for this task applied [
12,
13]. In this paper, we use the non-uniform Fast Fourier transform (NUFFT) [
21,
22,
23,
24].
It is worth noting that the image resolution is inversely proportional to the diameter of the span of the
k-space samples which is dependent on the cross range bandwidth
as defined in (
10). The resulting supportable size of the image is then determined from the image resolution cell multiplied by a factor of
K being the number of samples spanning the diameter of the
k-space. Although limited amounts of target rotation can reduce image resolution in the sparsely populated direction, here we focus on the case where a half cycle of the target scatterers is visible to the radar to completely populate the
k-space. Under this assumption, the angular sampling density of the
k-space samples drives the image contrast and is a trade off with computational cost [
25].
2.3.2. Standard DRT
By standard DRT, we refer to the case where the input cross-range profiles, as defined by (
11), are Doppler migration free (DMF), and the rotation angle corresponding to each profile formed under this condition is said to be within the linear limit (of phase variation). The DMF condition can be satisfied when the segmented CPI lengths are sufficiently short such that the nonlinear phase terms in (
6) are negligible and hence compensation is not necessary, or when
is small. The former case is particularly sensitive for scatterers at larger radial distances from the centre of rotation, while the latter case applies more to scatterers sufficiently close to the centre of rotation whose Doppler frequencies are small and Doppler migration effects (if any) are also small.
As derived in the
Appendix A, the standard DRT constraint on CPI rotation angle is
where
is an effective rotation angle required to induce a Doppler migration (DM) of one bin,
is the ‘linear limit’, while DRT image resolution, in both range and cross range, is
Here,
is the maximum radial dimension of the target. Note that
and
are independent of rotation rate and signal sampling rate, but only on radar wavelength and the dimension of the target (through maximum radial dimension
to any scatterer).
is roughly 10 degrees; Equations (
13) and (
14) can be used as a guide to predict the expected imaging performance or applicability of standard DRT for a specific radar wavelength and target size.
The limitations imposed by these nonlinear effects at wider rotation angles can be compensated by a processing technique described in the next Section. For differentiation from standard DRT, such cases are referred to as ‘Augmented DRT’.
5. Further Discussion
This paper is an expansion to the work reported earlier in [
2], demonstrating high-resolution DRT imaging with real experimental data. As this is not a real moving and rotating target in a typical operational scenario, a number of issues could be noted.
Firstly, the target’s translational velocity is exactly zero for the entire data collection. Nevertheless, this is not expected to be a sensitive factor. For most real moving targets, translational velocity can be readily compensated by shifting the ‘body Doppler’ line to zero Doppler. Sensitive propagation phases, as in the case of fast-time k-spaces, do not enter the slow-time k-spaces.
Secondly, the measured data were collected at precise angular sampling rates , which can only be estimated in typical operational scenarios. Errors in or would translate into errors of the locations of populated samples as well as image scaling factor. Hence both image focusing and image scaling could be affected. We have not fully addressed these issues in this work.
The experimental data does reveal interesting electromagnetic phenomenology, highlighting the limiting simplicity of the ideal point-scatterer assumption; creeping waves and nonlinear scattering effects do exist, which are not taken into account in the current DRT theory.
On application of the OMP algorithm, what this work has demonstrated its feasibility: techniques such as OMP can be used for slow-time
k-space augmentation. Other alternative sparse approximation techniques can possibly be used to yield higher performance. Numerous other aspects can also be considered, such as dictionary ‘learning’: how to select an optimum spatial scatterer grid for the best focusing performance while keeping computational cost at manageable levels? Or how to deal with the off-grid/mismatched scatterer problem [
30]. Many open questions remain, some of which will be addressed in future publications.
6. Concluding Remarks
We have demonstrated, with two datasets, the ability to improve image resolution using a rotating target with an ultra-narrowband radar. The enabling signal processing technique presented was a combination of Doppler radar tomography and a sparse reconstruction technique such as OMP, with a unifying mathematical framework based on the slow-time k-space. We have shown that closely spaced scatterers can be resolved by illustrating the creeping wave effect when the scatterer size is similar to the radar wavelength. The technique also performed well addressing the adverse effect of blurring in the image with scatterers at larger radial distances to the centre of rotation. By compensating for the blurred scatterer locations in the image, the ability to resolve closely spaced scatterers is improved providing finer details for target recognition.
Although the demonstration of this technique is effective, the application to a real complex target with many non-ideal scatterers may present additional challenges including discontinuous scattering effects, larger dictionaries affecting computational cost and inaccuracies due to signal mismatch with finite dictionary elements. In future work, we aim at investigating the use of multiple widely separated radar receivers to reduce the requirement on large target rotation angles for DRT imaging, where the direct application of OMP may not scale efficiently for large amounts of data. The increase in data may require a modified approach such as dictionary learning to help reduce the computational cost.