Bernoulli track‐before‐detect filter for maritime radar

IET Radar, Sonar & NavigationVolume 14, Issue 3 p. 356-363 Research ArticleFree Access Bernoulli track-before-detect filter for maritime radar Branko Ristic, Corresponding Author Branko Ristic branko.ristic@rmit.edu.au orcid.org/0000-0001-8561-4412 RMIT University, School of Engineering, Melbourne, VIC, AustraliaSearch for more papers by this authorLuke Rosenberg, Luke Rosenberg Defence Science and Technology Group, Edinburgh, SA, AustraliaSearch for more papers by this authorDu Yong Kim, Du Yong Kim orcid.org/0000-0001-6882-2324 RMIT University, School of Engineering, Melbourne, VIC, AustraliaSearch for more papers by this authorXuezhi Wang, Xuezhi Wang RMIT University, School of Engineering, Melbourne, VIC, AustraliaSearch for more papers by this authorJason Williams, Jason Williams Data61/CSIRO, Robotics and Autonomous Systems Group, Pullenvale, QLD, AustraliaSearch for more papers by this author Branko Ristic, Corresponding Author Branko Ristic branko.ristic@rmit.edu.au orcid.org/0000-0001-8561-4412 RMIT University, School of Engineering, Melbourne, VIC, AustraliaSearch for more papers by this authorLuke Rosenberg, Luke Rosenberg Defence Science and Technology Group, Edinburgh, SA, AustraliaSearch for more papers by this authorDu Yong Kim, Du Yong Kim orcid.org/0000-0001-6882-2324 RMIT University, School of Engineering, Melbourne, VIC, AustraliaSearch for more papers by this authorXuezhi Wang, Xuezhi Wang RMIT University, School of Engineering, Melbourne, VIC, AustraliaSearch for more papers by this authorJason Williams, Jason Williams Data61/CSIRO, Robotics and Autonomous Systems Group, Pullenvale, QLD, AustraliaSearch for more papers by this author First published: 07 February 2020 https://doi.org/10.1049/iet-rsn.2019.0480Citations: 4AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract In this work, the authors study the problem of detecting and tracking small targets using high-resolution maritime radar, where sea clutter is correlated in range, and its amplitude fluctuations are characterised by occasional spikes. To tackle this problem, they develop a Bernoulli track-before-detect filter, as the optimal recursive Bayesian detector/estimator of target state and its presence in noise. A realistic clutter model, in the form of the K-distribution with unknown distribution parameters, is adopted. Target amplitude fluctuations are also included in the model. The detection and tracking improvement are demonstrated by simulations and compared against a conventional point target tracking algorithm. 1 Introduction Radar detection and tracking of small maritime targets is a challenging problem due to the dynamic motion of the sea and its non-Gaussian amplitude statistics. This is caused by different environmental conditions (waves, sea swell and wind speed) and radar characteristics (range resolution and grazing angle) [1]. For higher range resolutions and low grazing angles, the clutter amplitude distribution is characterised by longer tails and is described as being spiky. Also, the clutter amplitude is correlated both spatially and temporally, which further impacts the radar performance. The traditional approach to target tracking in the maritime domain is the first to detect the target using amplitude information, and then estimate the positional/kinematic characteristics (e.g. range–Doppler–azimuth). To deal with changes in the underlying mean of the clutter, constant false alarm rate (CFAR) processing is often employed, with many variations described in [1]. The point measurements (PMs) are subsequently inputted to a conventional target tracking system to initialise new and maintain existing tracks. A track typically consists of an estimate of the target state (e.g. position and kinematic characteristics) and is given a unique label. For targets with a low signal-to-interference (clutter plus noise) ratio (SIR), a common approach is to set the detection threshold very low and let the tracking algorithm deal with the high rate of false detections. However, this increases the difficulties in the track-to-measurement association, and the use of thresholding will cause some information in the processing chain to be irreversibly lost. A potentially more powerful approach to tracking low SIR targets is to use the raw output from the radar signal processing (e.g. a range–azimuth–Doppler map) as a measurement. By removing the detection threshold, the track-before-detect (TBD) approach means that potentially lower SIR targets can be detected and tracked. The TBD approach is computationally more intensive, yet can potentially achieve superior performance when compared with conventional trackers for low SIR targets [2, Ch. 17]. The concept was originally proposed for a staring satellite-based imaging sensor [3] with its first application to a scanning pulse–Doppler airborne radar discussed in [4]. Both [3, 4] use dynamic programming (DP) to solve the TBD problem. Subsequently, DP was the method of choice in numerous publications, e.g. [5-13]. Despite its popularity, DP is a fairly cumbersome framework for online implementation of TBD, mainly due to the impracticality of discretising the state space in realistic problems and its requirement for high-dimensional maximisation. The first recursive Bayesian estimator for a single-target TBD was reported in [14] and solved using the particle filter (PF) [15, Ch. 11]. Subsequently, a multi-target TBD PF was proposed in the context of radar processing [16] and image processing [17]. The recursive Bayesian framework for TBD soon became popular as the optimal Bayesian TBD for a single target as it could be formulated analytically as the Bernoulli TBD filter [18, 19]. Extensions to multi-target recursive Bayesian TBD were numerous, particularly using the random finite set framework [20-23]. Another noteworthy TBD method is based on the histogram probabilistic multi-hypothesis tracker (H-PMHT) [24]. In this approach, the raw output from radar signal processing (i.e. the range–azimuth map or the range–azimuth–Doppler map) is interpreted as a histogram of observations of an underlying random process. By a series of assumptions and approximations, the problem is simplified and solved in a computationally efficient manner using Kalman filters. Since the original H-PMHT is a track maintenance algorithm [25], a heuristic two-stage approach from [26] is adopted for track initiation and termination. A comparison between H-PMHT, DP and a recursive Bayesian TBD algorithm has been reported in [27] in the context of a Swerling 0 constant amplitude target in Rayleigh distributed clutter. The comparison favours the H-PMHT because it is fast and applicable to multiple targets. However, when using a high-resolution radar in the context of airborne maritime surveillance, the assumption of a constant amplitude target in Rayleigh distributed clutter is inappropriate. In reality, the target amplitude is fluctuating and is typically modelled by one of the Swerling models [28]. Furthermore, the probability distribution of the sea-clutter amplitude is characterised by long tails due to higher magnitude sea spikes. This led to a recent study [29], which considered a fluctuating amplitude target in K-distributed clutter and found that the H-PMHT-based TBD is difficult to calibrate and achieves worst performance than the Bernoulli-type recursive Bayesian TBD. In this paper, we focus on airborne radar maritime surveillance. The main contribution is a development and performance analysis of the Bernoulli TBD, as the optimal recursive Bayesian algorithm for detection and tracking of small targets in realistic heavy-tailed sea clutter. The target amplitude is fluctuating with a Swerling 1 model and the interference is formulated using the K-distributed compound Gaussian model [1], whose shape parameter can be related to variations in the data collection geometry and environment (sea-state and swell direction) [1]. Compared to the initial study in [29], we now show how the TBD scheme can work with spatially correlated clutter and unknown shape and scale distribution parameters. The detection and tracking performances of the proposed TBD algorithm are analysed extensively for different levels of sea-clutter spikiness and different target SIR values. Finally, a comparison is presented with a conventional target tracker, which processes PMs. 2 Mathematical models 2.1 Target dynamic models A discrete-time motion model is adopted with a constant sampling period T determined by the radar scan rate. The state of the target at time index k is fully characterised with the state vector , where are the Cartesian coordinates of the target position in the plane, is the target velocity vector and is the energy or mean power of the target signal, which could be spread across more than one measurement cell. The target state evolution satisfies the nearly constant velocity motion model [30] with the transition density given by (1) where denotes the Gaussian probability density function (PDF) evaluated at , with mean and covariance matrix . Matrices and represent the transition matrix and the process noise covariance matrix, respectively. They are given by [30] (2) (3) where and denote the process noise intensity in the spatial and amplitude domains, respectively. These standard expressions describe the discrete-time dynamics model resulting from the conversion of the continuous time model, where velocity and amplitude are random walks and position is the integral of velocity. To indicate the presence of a target in TBD, it is helpful to introduce a binary random variable , where by convention means that the target is present and inversely that the target is absent. The dynamics of are modelled by the first-order two-state Markov chain, whose transitional probability matrix is given by (4) where and are referred to as the probability of target birth and target survival, respectively. If the target appears at , its state vector can be seen as a random sample from the birth density . Birth density will be discussed in Section 3. 2.2 Measurement model The measurement at each time k is a range–azimuth map , obtained from an airborne scanning radar. For simplicity, however, we assume that the centre of each cell of the measured range–azimuth map is converted to the Cartesian coordinates in a tangential plane on the sea surface. The measured value in the cell of the map represents the magnitude of the complex backscatter signal, where and , with and being the number of range and azimuth cells. The radar measurement model is now presented for the two cases, with interference only and then for the target plus interference. 2.2.1 Interference only model If no target is present in a cell of the measurement map, the measured amplitude z in the cell is due to background clutter plus thermal noise. This interference is formulated using the compound Gaussian model [1], which comprises of a speckle component for modelling the small ripples on top of larger waves. These larger waves are then modelled by a slowly varying texture, which can be assumed constant over a typical radar dwell period. The PDF of the clutter component is modelled with the K-distribution [1] (5) where the speckle amplitude is Rayleigh distributed (6) with mean . The mean square is treated as a gamma-distributed random variable (7) where denotes the gamma function, is the shape parameter and is the scale. The final expression for the K-distribution, obtained from (5)–(7) is then [31] (8) where refers to the modified Bessel function of the second kind. The mean value is given by (9) and for , the distribution approaches the Rayleigh distribution. The PDF of clutter plus thermal noise amplitude can also be represented by a K-distribution, where the shape parameter is replaced by an effective shape . By matching the moments of the PDF with and without thermal noise [1], one can obtain the following relationship: (10) where C is the clutter-to-noise ratio (CNR). For the remainder of this paper, we will use as the shape parameter without explicitly defining the CNR. In summary, if no target is present in the cell of the measurement map at time index k, the PDF of its amplitude is modelled by (11) 2.2.2 Target plus interference model Suppose a Swerling type 1 target is present and contributing power to a cell of the measured map (note that a contribution to several neighbouring cells, in general, is possible, depending on the spatial extent of the radar signature). Then, the speckle amplitude in the cell of the map is Rayleigh distributed as in (6), but with the parameter (12) where is gamma distributed as in (7). The PDF of amplitude z in this case does not have an analytic expression and is given by [32] (13) The amplitude PDF of a Swerling 1 target in K-distributed clutter plus noise, given by (13), will be referred to as the L-distribution, and as the integral has no analytic solution, it will be computed numerically. If , the gamma distribution (7) approaches a delta function and the L-distribution becomes Rayleigh as in (12). In summary, if a target in the state is contributing power to the cell at time index k, the PDF of the measured amplitude in this cell is given by (14) 2.2.3 Spatially correlated clutter Owing to fluctuations in the swell, the mean clutter level fluctuates both spatially and temporally over a time scale of seconds [1]. These variations are typically modelled with a correlated texture . Fig. 1 shows a simulated range–azimuth map of K-distributed sea clutter with unity mean and shape parameter (i.e. ). Owing to the short time scale, the autocorrelation function is only represented in the range direction (x-axis in Fig. 1) with a negative exponential function , where r is the range bin and the correlation coefficient is . The correlated gamma-distributed samples have been generated using the memoryless non-linear transform method [1, 33]. Fig. 1Open in figure viewerPowerPoint Simulated range–azimuth map with unity mean and 3 Bernoulli TBD filter The Bernoulli filter is the optimal recursive Bayesian estimator for an appearing/disappearing target [18]. It propagates two quantities over time. The first is the posterior probability of target's presence , where denotes the measured amplitude map at time index n consisting of cells, discussed in Section 2.2, and is the sequence of measured maps from the initial index to the current time index k. The second quantity is the posterior spatial probability distribution , conditioned on target existence. If at time , both and are available, then the recursive computation of and can be carried out in two steps: prediction and update. 3.1 Equations The prediction equations of the Bernoulli filter are as follows [18]: (15) (16) where and are the probability of target survival and probability of target birth, respectively and , referred to as the target birth density, is the probability density over the state space, which models the state of the target if it has appeared at time index . A practical design of will be discussed in Section 3.2. The updated equations are given by (17) (18) where is the likelihood ratio defined as (19) with likelihood functions and introduced in (14) and (11), respectively. The likelihood ratio (19) plays a crucial role in the Bernoulli filter for TBD. As discussed in [34], when the clutter is spiky and has small , the likelihood ratio is conservative in the sense that it 'reacts' less dramatically to large amplitude values [32]. In practise, however, the parameters and b, which are necessary for the computation of the likelihood ratio, are unknown. Estimations of and b will be discussed in Section 4.1. The filter reports a detection only if , where is a parameter referred to as the reporting threshold. The choice of threshold will be discussed in Section 5.1. 3.2 Implementation notes The prediction and update of the probability of target presence are straightforward using the analytic expressions (15) and (17), respectively. The problem is how to represent the posterior spatial density and then predict and update this representation as new range–azimuth maps become available. For this purpose, we adopt the sequential Monte Carlo (SMC) method [15] and implement the Bernoulli TBD filter as a PF. Full details of this implementation can be found in [18, Section VIII.A]. Next, we explain how the particles representing the birth density are created. Let be approximated by weighted particles , where is the state of the ith random sample (particle) from and is its weight (the weights are non-negative and sum up to 1). Similar to [35], an adaptive measurement-driven birth density is implemented. Thus, in the subspace of the state space, is designed to cover the cells of the measurement map , whose amplitude is above a certain threshold and indicates the region, where targets can potentially appear. The position of particles , for , is hence a random draw, with equal probability, from this region. The threshold is adopted as the sample mean of , multiplied by the factor . In the subspace , and correspondingly , because does not contain information about target velocity, we adopt a uniform distribution from to , where is the maximum expected target speed. In the subspace of the mean target power , the birth density is a gamma distribution with parameters , chosen so that it covers the interval of SIR values of interest. The weights of birth particles are uniform, i.e. , for . For the remainder of this paper, the described PF implementation of the Bernoulli filter for TBD will be referred to as Bern-PF. 4 Pre-processing of range–azimuth maps 4.1 Parameter estimation The parameters of the K-distributed interference are unknown in practise and depend on the collection geometry and the evolving sea conditions. A number of methods for the estimation of the shape parameter have been proposed in the literature [36] such as the maximum likelihood estimate [37], the method based on moments [38, 39], expectation–maximisation [40], higher-order and fractional moments [41, 42] and the Bayesian approach [43]. While all these methods have some pros and cons, the fundamental limitation is that the theoretical Cramér–Rao lower bound for estimation of is proportional to [39], where L is the sample size. This implies that it is more difficult to estimate large values of (i.e. when the K-distribution approaches a Rayleigh) and for small sample sizes L. We adopt for simplicity the moment-based method [39]. Let be a sample from the K-distribution. Then, the first and second sample moments are estimated as (20) An approximate closed-form solution for is then [39] (21) with the scale estimate determined by . Next, we investigate numerically how many samples are required for a reliable estimation of using (21). For a given value of , we create range–azimuth maps, i.e. samples of size and from each sample compute the estimate , and its relative root-mean-square (RMS) estimation error (22) The range–azimuth maps are spatially correlated as described in Section 2.2.3, implying that the samples in range are only independent every fourth sample. Fig. 2 shows the result for , where one needs about samples for the RMS estimation error in to be < when or 3. For , this grows to ∼ samples, while for , L must be >12,000. To overcome this limitation, we can assume that the sea-clutter statistics are effectively constant over a defined spatial region and period. For example, consider a radar system with a scan rate of 24 rpm (2.5 rad/s), the pulse repetition frequency of 2 kHz and bandwidth of 100 MHz. If we assume the statistics are constant over a 1 km2 swath at a slant range of 20 km, then there will be 667 range bins and 39 pulses, where more than half of the azimuth beamwidth illuminates the swath. This gives 6503 independent samples per scan or samples if we consider that the statistics are unchanging over a 2 min interval, which is sufficient to estimate nearly any shape parameter. Fig. 2Open in figure viewerPowerPoint Relative RMS error in estimation of the shape parameter as a function of the sample size L 4.2 Spatial decorrelation Radars commonly employ a CFAR processor to deal with spatial fluctuations and achieve an approximately CFAR when scaled by an appropriate threshold multiplier [44]. To improve the TBD performance when the sea clutter is correlated, we will calculate the local averaging window as in the CFAR algorithm to form an estimate of the clutter map from . The clutter map represents an estimate of the spatially varying mean of . A spatially decorrelated range–azimuth map is then computed as (23) where is the sample mean of , introduced in (20). The idea behind (23) is to remove from the original map the spatially varying mean , and replace it with the (constant) sample mean . The input to the Bern-PF is then the decorrelated range–azimuth map , rather than the original . The clutter amplitude in the so-called cell under test (CUT) , , is estimated as the average value of the amplitudes in a block of cells (window) around the CUT (referred to as the training cells). The window for averaging is shown graphically in Fig. 3. The cells immediately adjacent to the CUT are referred to as the guard cells and not included because of the potential target signal leakage. As the correlation is in the range direction, the window size in azimuth direction is adopted as . Let us denote the range width of the outer and inner windows as and , respectively. For the example in Fig. 3, and . Fig. 3Open in figure viewerPowerPoint Averaging window indicated in ochre colour. The blue coloured cell is the CUT Note that after spatial decorrelation, some of the cells in will have negative amplitude. Since these cells in the original map have very low amplitude, they are unlikely to contain a target, and therefore their likelihood ratio (19) in the computation of the Bern-PF can be set to 1. Fig. 4 shows the decorrelated range–azimuth map , corresponding to the original map displayed in Fig. 1. Fig. 4Open in figure viewerPowerPoint Decorrelated range–azimuth map , corresponding to shown in Fig. 1 5 Numerical results Consider a generic scanning radar with range bins and azimuth cells. The clutter is modelled with unity; meanwhile, the shape parameter is varied. The following parameters, expressed in arbitrary units are used in simulations: sampling interval , process noise parameters , , correlation coefficient , decorrelation parameters , , Bern-PF parameters , and the number of particles used for the representation of spatial density is . The birth density parameters are , number of birth particles , , and . The parameters and are chosen to cover the target amplitude values () for a large span of SIR values and the size of each cell is units. Estimation of the parameters and b of K-distributed interference for time indices is based on the current and previous maps (i.e. and ). This results in correlated (i.e. 2048 independent) samples, which are intentionally set small to highlight any performance degradation in the proposed Bern-PF TBD algorithm. As discussed in Section 4.1, if we include more samples from previous scans, the accuracy of the parameter estimation can be improved. 5.1 Choosing the reporting threshold In this section, we investigate the probability of false-track declaration as a function of threshold . This is done in the absence of a target (interference only case) for a range of shape parameters , with the interference mean set to . Fig. 5 shows the estimated probabilities, obtained numerically by running the Bern-PF TBD algorithm over 3000 range–azimuth maps. Fig. 5Open in figure viewerPowerPoint Probability of false-track declaration as a function of for The following observations can be made from Fig. 5. First, the smaller the shape parameter , the higher the probability of false-track declaration. This is in line with our expectations: for smaller , the clutter is spikier (i.e. the K-distribution has a longer tail) and more frequently resembles a target return. Second, there appears to be a sudden change in behaviour between . Fig. 5 suggests that the developed TBD algorithm is applicable only to sea clutter characterised by . For lower shape values, we may adapt the parameters of the amplitude birth densities and to the estimated value of , but this is beyond the scope of the current work. For developed Bern-PF, we fix the reporting threshold to . Alternatively, could be selected so that the probability of declaring a false track is below a certain user-specified value. In this case, the choice of should be based on the estimated shape parameter . 5.2 Detection and tracking performance Joint detection and tracking performance is measured using the mean optimal sub-pattern assignment (OSPA) metric [45]. This metric penalises both the errors in the cardinality estimate (number of targets) and the error in the target position, but does not consider the target velocity and mean power in the performance assessment. It has two parameters: order p and the cut-off c. In simulations, we set , so the metric is an -norm measure. The parameter c, which expresses the penalty assigned to the cardinality error in terms of the number of cells, is set to . The mean OSPA metric is estimated by averaging over 100 Monte Carlo runs. A sequence of 40 range–azimuth maps is generated for every Monte Carlo run. The performance will be considered for two versions of the Bern-PF: (a) Bern-PF-1 'knows' exactly the true values of the K-distribution parameters and b; (b) Bern-PF-2 estimates and b from the data (as described in Section 4.1). For completeness, we also show the performance of a conventional tracking algorithm, which inputs the extracted PMs. For this purpose, we have developed a Bernoulli filter for PMs, which is implemented as a Gaussian sum filter with full details given in [18, Section VIII-B]. The PMs were extracted from range–azimuth maps using the cell averaging CFAR with an averaging window matching the one described in Section 4.2. The threshold multiplier is computed for each so that the probability of false alarm is ∼. The probability of detection in the filter is fixed to . Finally, the and parameters are equal to those used in the Bern-PF for TBD. This tracking algorithm is referred to as GM-Bern-PM in the text section. 5.2.1 Interference only case An assessment of the rate of false-track declarations by the TBD algorithm is first carried out in the absence of a target. The obtained average OSPA error curves for the three contesting algorithms are shown in Fig. 6 for shape parameter values . The low values in Fig. 6 indicate small errors in the cardinality estimate, and hence, a low false-track declaration rate. We conclude that all three algorithms perform similarly. Furthermore, in agreement with Fig. 5, the performance of TBD algorithms is slightly better for than for . Fig. 6Open in figure viewerPowerPoint Mean OSPA metric in the absence of a target (interference only) (a) , (b) 5.2.2 Target plus interference When a target is present, its signature may spread to more than one range or azimuth cell. The contributing power to the th cell at time k is related to the mean target power via the point spread function: (24) where and determine the spread in range and azimuth, respectively. In simulations, we set . Furthermore, we assume, during the observation period, a constant mean power . This is a reasonable assumption for a target in a range–azimuth sector at a large distance from the observing platform. The signal-to-interference ratio is defined (dB) as . The initial position of the Swerling type 1 target is randomly generated making sure the target remains in the surveillance volume. The initial velocity is 0.45 and 0.25 in the x and y directions, respectively, and the mean power is set to correspond to the following SIR values: 3, 5 and 8 dB. The resulting mean OSPA distance curves for the interference shape parameter values are shown in Figs. 7-9. Fig. 7Open in figure viewerPowerPoint Mean OSPA metric for the target plus interference case with an SIR of 3 dB (a) , (b) Fig. 8Open in figure viewerPowerPoint Mean OSPA metric for the target plus interference case with an SIR of 5 dB (a) , (b) Fig. 9Open in figure viewerPowerPoint Mean OSPA metric for the target plus interference case with an SIR of 8 dB (a) , (b) From Fig. 7, we can make the following observations about the performance of the two Bern-PFs. Initially, for small k, the mean OSPA error is large due to the cardinality error. This error reflects a delay in detecting and subsequently tracking the target. After about , both Bern-PFs reach the steady state with the mean OSPA error approximately at the level of 4. This elevated level of the mean OSPA is due to the occasional