Historical progress of the PHD filter

Since its inception, the PHD filter has been receiving increasing interest from the research community. The PHD, being a density function in the space of

a single target, can naturally be implemented using a SMC approach. Early work on SMC based PHD filtering can be found in [161–164]; the SMC PHD filter described in [162] is later used in chapters 3 and 4 for comparison purpose. The auxiliary particle PHD filter implementation can be found in [165, 166]. The convergence result for the SMC implementation of the PHD filter is provided in [167, 168].

An important aspect of (SMC) PHD filtering is target state estimation.

One approach (that gained much popularity) to this problem, as proposed by Clark and Bell [169] in 2005, is clustering [58]. Zhao et al. [170] developed a partition method that divides the PHD function into sub-PHDs, each which is a single target PHD. The state estimates are then evaluated using these single target PHD functions. More recently Ristic et al. [171] incorporated the state estimation step into the update process of the SMC PHD filter as follows: if the predicted particle set {xⁱ_k, w_kⁱ}^I_i=1^k and the sensor observation RFS Σ_k = {z_k¹, ..., z_k^Mk} is available, then the weights for each element in Σ_k are updated and denoted by {w^i,m_k }_{i=1,...,Ik, m=1,...Mk+1}, i.e., the update process results in M_k+ 1 replicated weighted particle sets. The additional (M_k+ 1)th set corresponds to the case when no targets are present. Then the state estimate of the mth element in Σk is

ˆ x^m_k =

Ik

X

i=1

w^i,m_k xⁱ_k (2.99)

and ˆx^m_k is reported only if PI_k

i=1w_k^i,m is above a certain threshold.

Another important aspect of SMC PHD filtering is track labelling (or continuity), i.e., to be able to identify (or tag) the targets as time progresses.

Track labelling is difficult in PHD filtering because the amplitude of the peaks in the PHD function oscillate with time, i.e., some peaks may go flat at certain times. Lin et al. [172] proposed the SMC PHD resolution cell (RC) filter that successfully smoothed out the PHD peak oscillations by di-viding the state space into discrete cells and evaluating the PHD of each cell separately. Then the MHT filter (described in section 2.3.3) is applied to

associate peaks to targets. In [173], Panta et al. proposed to feed the SMC PHD filter state estimates as input measurements to the joint filter and used JPDA to associate targets to measurements, which in this context are PHD filter outputs. Clark and Bell [169] used k-means clustering¹³ [58, 59] of la-belled particles to achieve track labelling: the process can be summarised in three steps; a) particles are always clustered and so they have a label that identifies the cluster they belong, b) in the update step, particles are resam-pled along with their labels, and c) after resampling, particles are reclustered and if a majority of the particles in a cluster have the same label, then the clusters are associated. The authors later compared [174] the use of various clustering algorithms, for example, the expectation-maximisation (EM) al-gorithm [175], for track labelling and found that k-means is computationally more reliable than the others.

Hong et al. [176], were one of the first to focus on hardware implemen-tation of the PHD filter. The elements in the sensor observation RFS Σ_k are time varying and usually large in number. Hence a series hardware im-plementation of the PHD update step leads to prohibitive latency. Parallel hardware implementation of the PHD update is hard because the number of parallel units (equal to |Σk|) to be initialised every time sample is unknown.

The authors proposed a new update model for the SMC PHD filter by ar-ranging the time-varying elements of Σ_k in a series/parallel combination.

With a motivation to implement the SMC PHD filter in parallel hardware systems, Shi et al. [177] proposed the particle based observation selection (POSE) PHD filter that opportunistically selects only a fixed number of ele-ments in the observation RFS for PHD update processing. The POSE PHD filter was implementated on a Xilinx Vertex-11 Pro field programmable gate array (FPGA) platform.

In order to be able to detect new targets, the birth distribution in the

13This procedure will be used when implementing the SMC PHD filter for the compar-itive analysis of the contributions of this thesis.

SMC PHD filter has to span the entire single target space X, thus requir-ing an enormous number of particles. If fewer particles are used such that they do not densely populate the state space, then new-born targets could be undetected. A topic of great interest in this context is to develop efficient SIS procedures to initialise particles near regions of target activity, i.e., to initialise new particles based on observation driven SIS as b(x_k) → b(x_k|Z_k).

Yoon et al. [178] developed an improved SIS algorithm by using the un-scented transform technique. Other techniques to improve the SIS procedure in SMC PHD filtering can be found in [179, 180].

In 2005, Vo and Ma [181, 182] implementated the PHD filter using Gaus-sian mixtures (GMs) [183]. This GM PHD filter has the advantage of pro-viding a closed-form solution to the PHD prediction and update steps. In spite of being limited to linear Gaussian systems, the filter is computation-ally less expensive than a particle implementation. Moreover, track labelling and continuity can be performed very easily. The convergence result for the GM PHD filter can be found in [184]. The filter has been used for many ap-plications, for example, extended target tracking [185] and SONAR tracking [186]. A more general form of the PHD filter is the cardinalised PHD (CPHD) filter [187]. This filter propagates the PHD and the entire distribution on the target number — the cardinality. As a result, the state estimation can be improved. The CPHD filter can be implementated using GMs [188] and particles [32].

The PHD filter has been widely used in several phased array processing applications, a few examples of which include: autonomous navigation of underwater vehicles using SONAR [189, 190], tracking in far-field array pro-cessing [191], tracking moving vehicles using acoustic sensing [192], source separation for acoustic signal processing in reverberant environments [193], multiple input multiple output (MIMO) RADAR tracking [194], bistatic RADAR tracking [51], vehicular traffic monitoring [195], etc. However, since the observation model for the PHD filter is a RFS comprising of noisy point

detections, these applications first convert the phased array data into im-age models and then preprocess the imim-age to obtain the observation RFS.

One of the goals of this thesis is to perform Bayesian MTT without these preliminary data processing stages.