The Probability Hypothesis Density Filter

The joint multi-target probability distribution (JMPD) filter is a theoretically optimal solution to the problem of multiple object tracking. This method is, however, computationally infeasible,

even for single object tracking [5, p. 1]. The JMPD filter, as the name implies, models all state distributions together in a single joint probability distribution [15, p. 2]. The JMPD does not solve the data association problem and is therefore known as an association-free method. Since this model is intractable for even relatively simple cases, it must generally be approximated by some means.

The probability hypothesis density filter (PHD) filter proposed by Mahler [5] is an approxi- mation to the JMPD filter that is analogous to the constant gain Kalman filter in that it only propagates the first order moment of the multi-target joint distribution [5]. This first order moment is called the probability hypothesis density or intensity function [16, p. 84]. The under- lying theory behind the PHD filter involves random finite sets (RFS) and Mahler’s multitarget calculus. Random finite sets are a generalization of random variables in the sense that they represent sets of random variables where the size of the set is itself random [17, p. 32]. A detailed discussion of random finite sets and the multitarget calculus is, however, beyond the scope of this work. Although the general PHD filter updates involve integrals that have no closed form solution [18, p. 1], it was shown by Vo and Ma [19] that a closed form solution can be obtained under linear Gaussian dynamics. This variant of the PHD filter is known as the Gaussian mixture PHD (GM-PHD) filter.

Furthermore, unlike the JMPD filter, the PHD filter operates on a single-target state space [16, p. 96], and the dimensionality of the state distributions therefore does not increase with the number of targets. The PHD filter essentially avoids the association problem by considering all association hypotheses and updating each state with all possible measurements (including no measurement) [20, p. 2]. If there were therefore n states and m measurements, the number of updated states would be n(m + 1). Avoiding the association problem allows the PHD filter to be less computationally expensive than association-based methods such as the MHT filter. This is, however, also the cause of the PHD filter’s biggest shortcoming: it cannot produce tracks with target identities. In addition to the process and measurement parameters required by most multiple object tracking methods, the PHD filter requires certain additional parameters to specified, namely the target detection probability, target survival probability, and clutter intensity. The PHD filter state and measurement update equations are given below.

The PHD Filter Equations [16, p. 96] State update: Vk|k−1(X) = pS Z fk|k−1(X|ζ)Vk−1(ζ))dζ + γk(X) where

pS= target survival probability,

γk(X) = PHD of the birth RFS Γk at time k,

Vk−1(ζ) = prior (previous posterior) state PHD,

fk|k−1(X|ζ) = single-target transition density at time k given previous state ζ.

Measurement update: Vk(X) =  (1 − pd) + X y∈Zk pdgk(y|X) κk(y) + pdR gk(y|X)Vk|k−1(X)dX  V_k|k−1(X) where

pd= target detection probability,

Vk(X) = predicted state PHD,

y = measurement,

gk(y|X) = likelihood of a measurement given a state X at time k,

κk(y) = PHD of the clutter RFS at time k.

From the equations above we can see that, if the PHD of the previous state Vk−1(ζ) is a

Gaussian mixture and the transition density1is a linear Gaussian, the predicted state,V_k|k−1(X) will also be a Gaussian mixture. Similarly, if the measurement likelihood is a linear Gaussian, the posterior intensity Vk(X) is also a Gaussian mixture. From the above equations we can

also see that with every prediction step, the existing state intensities (weighted by the survival probabilities), are propagated forward in time and the new birth intensities (γk(X)) are added.

These birth intensities correspond to the prior distributions over the states of possible new tar- gets. During the measurement update, each of the predicted intensities is updated with each of the measurements. The updated mixture associated with each measurement is then divided by the total mass of the entire updated mixture, plus the clutter probability of the measurement.

1

This is the conditional distribution of the next state (given the previous state) that encodes how the states change with time.

This will of course simply correspond to normalisation if the clutter probability is zero. The up- dated mixtures are also weighted by the detection probability. Furthermore, all the non-updated states will also be present in the posterior due to the 1 − pd,k constant term. The posterior inten-

sity is therefore a Gaussian mixture that contains weighted components of all the non-updated intensities and all possible permutations of the measurement-updated intensities. This will of course result in an unmanageable number of components, and methods to reduce the number of components are therefore typically applied. Component merging methods allow similar com- ponents to be merged together, while pruning methods are used to discard components with low weights. The detection and clutter weighting, as well as the weight adjustment inherent in Gaussian multiplication, cause the correct components2 to be more likely to have larger weights. On the other hand, the incorrect components will typically have very small weights and will therefore be removed by pruning operations.

The PHD filter also allows the number of targets in any specific region to be estimated by integrating the intensity function over the specific region [21]. The total number of targets at any given time can therefore be estimated simply by adding all the weights of the components of the Gaussian Mixture intensity3.

Several methods have been proposed to extract state estimates from the posterior intensity function. One such method is to first estimate n, the number of targets, and then select the n components with the largest weights [22, p. 684]. This method is of course rather sensitive to the accuracy of the target number estimate and is therefore more appropriate in PHD filter variations that have been designed for more accurate cardinality estimates, such as the cardinalised PHD (CPHD) filter. An alternative to the above is to simply choose a threshold (typically around 0.5) and, after each measurement update, select the means of the components with weights higher than this threshold as the estimates [7, p. 766].

This concludes our discussion on the PHD filter. We have given an overview of the operation of the PHD filter and discussed the update equations. In Section 10.2 we will compare the GM-PHD filter to the developed PGM model and discuss the differences between the two.