2.1. Tracklet Representation
The compact HFSWR transmits a linear frequency-modulated interrupted continuous wave (FMICW) to illuminate the sea surface within the coverage area. The backscattered echoes are received by a linear array of antennas. The signal received by each antenna is digitally processed to attain the range and Doppler velocity information; thus, a range–Doppler map is obtained. Then, a constant false alarm rate (CFAR) algorithm is applied to the range–Doppler map to achieve target detection, and a direction of arrival (DOA) estimation method, such as multiple signal classification (MUSIC) or digital beamforming (DBF) is used to obtain the azimuth of the detected target. Therefore, a compact HFSWR locates a target in terms of range r and azimuth under the polar coordinate system with the radar site as its origin. Moreover, as a Doppler radar, it can measure the target velocity component along the radar radial direction (a.k.a. Doppler velocity). Thus, it represents a target with a state vector . HFSWR continuously observes the sea surface, at each sampling instant k, it acquires a frame of data containing plenty of “plots” from echoes of different targets and interferences. After several consecutive sampling periods, multiple frames of plot data can be collected. Then, a multi-target tracking algorithm can be applied to the obtained plot data sequence to produce target tracks.
In general, a multi-target tracking algorithm consists of three steps, i.e., track initiation, track maintenance, and track termination. Track maintenance includes state prediction, measurement-to-track association, and state estimation. The converted measurement Kalman filter (CMKF) and the minimal cost data association method were combined for target tracking in this paper and are described as follows.
(1) The dynamic and observation models.
The converted measurement Kalman filter operates with a dynamic model and an observation model. The dynamic model of moving vessels can be defined in a Cartesian coordinate system as
where
is the target’s true state vector at time
t,
and
are the target’s position components,
and
denote the target’s true velocity components along the
x and
y directions, respectively.
is the state transition matrix, defined as
and
T is the sampling time.
denotes the Gaussian process noise with a mean of zero and covariance matrix
.
The observation model is also defined in the Cartesian coordinate system as
where
is the target’s measured state vector at time
t,
and
represent the measured target’s position components,
and
denote the corresponding measured velocity components along
x and
y directions.
is the measurement matrix and
represents measurement noise following Gaussian distribution with a mean of zero and covariance matrix
.
(2) Track initiation.
Potential tracks are initiated using the logic method with the M-of-N rule [
22]. If there are more than M plots connected in the most recent N frames, the track is successfully initiated, and it will be added for track maintenance; otherwise, it will be discarded.
(3) Track maintenance.
A. State prediction. For each initiated or maintained track, denote as the estimated target state at time , the predicted state at time t can be obtained by In addition, the corresponding state prediction covariance matrix is calculated by
B. Coordinate conversion. For the subsequent measurement-to-track association procedure, the predicted state
is converted from the Cartesian coordinates to polar coordinates as
, in which
where
,
, and
denote the predicted range, azimuth, and radial velocity, respectively.
C. Measurement-to-track association. The minimal cost criterion is utilized to find the most likely measurements
for the current track at time
t within a predefined validation gate [
22]. If the measurement is associated with a track, go to step D; otherwise, go to step F.
D. Measurement conversion. The associated measurement
is converted from polar coordinates to Cartesian coordinates to obtain the measured target state
by
E. State estimation. The estimated target state
at time
t and the state estimation covariance matrix
are updated by
where
is the Kalman gain at time
t. Then the estimated target state
is used to update the current track.
F. Determine if the track termination conditions are satisfied. If the conditions are met, the track will be terminated; otherwise, t is increased by 1 and go to A.
(4) Track termination.
A maintained track will be terminated if one of the following conditions occurs:
A. There are no associated measurements in the past K frames out of the most recent L frames.
B. The estimated velocity reaches an unrealistic value .
(5) Track smoothing.
The obtained tracks by the above tracking procedure usually fluctuate significantly and deviate from their true positions due to a low positioning accuracy of the compact HFSWR. Denote the position data sequence in longitudes and latitudes of an estimated track with a length of
n as
, the track can be smoothed by a moving average filter with a window length of
m as
Then a smooth track can be generated. It should be noted that the main objective of this article is not target detection and tracking, but a target tracklet association method that is directly applied to the tracklets provided. Target detection and tracking are dependent on signal-to-noise ratio (SNR), the signal-to-clutter ratio (SCR), etc. The effects of these factors can be mitigated in target tracking, which involves state filtering and smoothing and is reflected in target parameter measurement errors. Therefore, investigating the influence of target parameter measurement errors on tracklet associations are more meaningful.
Due to the aforementioned shortcomings of target detection using compact HFSWR, the obtained tracks usually fluctuate and deviate from their true positions and are even fragmented into several short track segments. To improve the continuity of target tracking, several tracklets belonging to each (same) target should be associated and connected. Therefore, once a track is initiated, it is necessary to determine whether it comes from a new target or is a continuation of an existing target track. Two types of tracklets are defined as follows.
(1) Terminated tracklet. It represents a track that meets the termination condition [
23] and stops updating its state. The terminated tracklet set
is defined as
where
N denotes the number of terminated tracklets,
is the
jth terminated tracklet that contains
n plots. It should be noted that a terminated tracklet could be a fully completed track or a portion of a track that is interrupted.
(2) Initiated tracklet. It represents a new track that satisfies the track initiation condition [
22] and is defined as
where
M is the number of newly initiated tracklets,
denote the
ith initiated tracklet with a length of
l. It should be noted that an initiated tracklet could be an independent new track or a track portion that can be associated with an existing terminated tracklet.
2.2. Average Heading and Average Speed Calculation
Heading and speed are two important motion characteristics of a moving target. Compact HFSWR can only provide a coarse azimuth resolution; thus, the positions of the measured plots may deviate from their true values. Therefore, the instantaneous heading and speed cannot be accurately obtained using adjacent target positions. Fortunately, average heading and speed can be robustly estimated and reflect the overall motion characteristics of a moving target. An illustrative comparison between instantaneous and average headings is shown in
Figure 1.
In
Figure 1, the instantaneous and average headings are depicted in solid and dot dash lines, respectively, for a terminated tracklet
and its corresponding initiated tracklet
. It can be seen that the instantaneous headings at different sampling times change abruptly, while the average headings for
and
are almost the same. Therefore, the average heading is a more stable characteristic; the same for the average speed.
Taking the ith initiated tracklet with a length of l as an example, the average heading and speed can be calculated as follows.
(1) Average heading.
The instantaneous heading
of the tracklet
at time
k is defined as
where
represents the target position of the tracklet
at time
k in longitude and latitude, and it is determined by the measured range
, azimuth
at time
k as well as the radar site. Based on the above definition, the average heading of the tracklet
is calculated as
In order to further verify the feasibility of using the average heading as the track feature, two track segments were selected for validation, as shown in
Figure 2a. These two tracklets can be associated with the same automatic identification system (AIS) track using the track-to-track association method [
24], i.e., it is confirmed that they are derived from the same target. The instantaneous headings of the two tracklets were calculated separately using Equation (
17) and are shown in
Figure 2b, which illustrates that the instantaneous headings of the two tracklets fluctuate severely. In contrast, the average heading of tracklet 1 calculated by Equation (
19) is 121.69
and that of tracklet 2 is 120.98
, showing that the average headings have better consistencies.
(2) Average speed.
The instantaneous speed
of the tracklet
at time
k is defined as
where
is the geodesic distance between adjacent target positions
and
, and
T is the radar sampling interval. Based on the above definition, the average speed of the tracklet
is calculated as
A motion vector containing the average heading and average speed can be denoted as
Similarly, the motion vector of the terminated tracklet
can be represented as