# Kalman-Gain Aided Particle PHD Filter for Multitarget Tracking

## Summary (4 min read)

### Introduction

- Kalman-Gain Aided Particle PHD Filter for Multitarget Tracking ABDULLAHI DANIYAN , Student Member, IEEE YU GONG, Member, IEEE SANGARAPILLAI LAMBOTHARAN, Senior Member, IEEE Loughborough University, Loughborough, U.K. PENGMING FENG, Member, IEEE JONATHON CHAMBERS, Fellow, IEEE Newcastle University, Newcastle upon Tyne, U.K. Besides the improved tracking accuracy, fewer particles are required in the proposed approach.
- Refereeing of this contribution was handled by B.-N. Vo.
- As a result, the measurements received at each time step are corrupted and consist of indistinguishable measurements that may be either target originated or due to clutter.
- In the literature, the PHD filter has been implemented in two distinct fashions, that is, as the Gaussian mixture PHD (GM-PHD) filter [16] and the sequential Monte Carlo PHD (SMC-PHD) filter [17].

### II. MTT PROBLEM FORMULATION

- The MTT problem relates to that of modeling a dynamical system.
- Two models are generally used, the state evolution model and the measurement model.

### A. State Model

- A nonlinear system governed by the state evolution model is considered xk = fk−1(xk−1, vk) (1) where xk denotes the t th target state at discrete time k, vk is an independent and identically distributed (i.i.d.) process noise vector, and fk−1(·) is the nonlinear system transition function.

### B. Measurement Model

- Under this assumption, (5) becomes g = ∫ n f (y)π(y)dy = ∫ n f (y) π(y) q(y) q(y)dy.
- To sum up, given a distribution π(y) that is difficult to sample from, importance sampling facilitates sampling from π(y) by sampling from an alternate distribution q(y) known as proposal distribution but weighting appropriately.
- This is the most popular choice of suboptimal proposal distribution for SMC-PHD filters and particle filters in general because its implementation is easy and straightforward [34].
- Hk by generating sigma points and applying a transform such that the new generated samples have fk−1(xk−1) as mean and Pk−1 as covariance.
- DANIYAN ET AL.: KALMAN-GAIN AIDED PARTICLE PHD FILTER FOR MULTITARGET TRACKING 2253.

### A. PHD Filter

- The PHD filter is a recursion of the PHD,Dk|k that is associated with the multitarget posterior density p(Xk|Zk), and p (Xk|Zk) ∝ p (Zk|Xk)p (Xk|Zk−1) (15) where p(Zk|Xk) and p(Xk|Zk−1) denote the multitarget likelihood and prior density, respectively.
- The prediction formula of the PHD,Dk|k is given as [8], [9] follows: Dk|k−1(xk|Zk−1) = γk(xk) + ∫ φk|k−1(xk, xk−1)Dk−1|k−1(xk−1|Zk−1)dxk−1 (16) with the factor φk|k−1(xk, xk−1) = pS(xk−1)fk|k−1(xk, xk−1) + bk|k−1(xk, xk−1) (17) where γk(·) is the PHD of the spontaneous birth, pS(·) is the probability of the target survival, fk|k−1(xk, xk−1) is the single target motion model, and bk|k−1(xk, xk−1) is the PHD of the spawned targets.
- 〈Dk|k−1, ψk,z〉 ⎤ ⎦Dk|k−1(xk|Zk−1) (18) with ν(xk) = 1 − pD(xk), ψk,z(xk) = pD(xk)g(z|xk), and κk(z) = λkck(z); where pD(xk) and ν(xk) denote the probability of target detection and nondetection for a given (xk), respectively, g(z|xk) is the measurement likelihood function for the single target, κk(z) is the clutter intensity, λk is the average number of Poisson clutter points per scan, and ck(z) is the probability density over the state-space of the clutter point; 〈·, ·〉 denotes inner product and is computed as [8], [9].

### B. Standard SMC-PHD Filter

- The PHD filter can be implemented either as in the SMC fashion (particle-PHD) or as the GM-PHD.
- The state 2254 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL.
- In B, as the latest measurement arrives, the particle weights are updated accordingly.
- Doing this will increase computational complexity.
- In the next section, the proposed SMC-PHD filter is presented.

### V. PROPOSED SMC-PHD FILTER

- The filter’s ability to estimate the posterior at a given time depends on how densely the state space is populated with samples and how well the estimated measurements match the actual measurements received in that time frame.
- The weights are then updated accordingly.
- The SMC-PHD filter does not provide for particle state correction to achieve particle improvement.
- The proposed method seeks to address this problem.
- The Kalman filter is a minimum MSE estimator, which in effect seeks to recursively minimize the MSE between the estimated measurements and actual measurements using the Kalman gain [36].

### A. Measurement Set Partition

- Given that Tk targets exist at time k, the measurements received at k may consist of target-originated measurements (i.e., measurements due to persistent target or newborn targets) and clutter.
- Therefore, a measurement set partition is needed to separate the measurement set into targetoriginated measurements and measurements due to clutter.
- The second step is to identify promising particles from the predicted target state using a validation threshold and improve their states using the Kalman gain while updating weights as measurement arrives.
- A predicted particle, x̃lk|k−1 is selected for correction if, for each clutter-free measurement żnk ∈.
- The selected particles from B are then corrected using (25) and (26) as shown in C.

### C. KG-SMC-PHD Implementation of the PHD Filter

- The authors now present the initialization, prediction, update, and resample steps of the KG-SMC-PHD filter.
- The authors draw Lk−1 and Jk particles from two proposal densities (chosen from the possibilities discussed in Section III, i.e., TP, EPF or UPF) to represent persistent and newborn 2256 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL.
- The updated PHD, Dk|k is then given as Dk|k(x̃k|Żk) = Lk∑ l=1 w̃lkδ(x − x̃lk).6 (35) 4) Resample: 1) The expected number of targets.
- This is to ensure that only weights belonging to corrected particles are chosen for resampling.
- DANIYAN ET AL.: KALMAN-GAIN AIDED PARTICLE PHD FILTER FOR MULTITARGET TRACKING 2257 Algorithm 2: KG-SMC-PHD.

### 1: Initialization

- Żk do 14: for l = 1 : Lk do 15: if g(żnk |x̃lk|k−1) ≥ τ then 16: x̃lk = x̃lk|k−1 + Kk(żnk − f (x̃lk|k−1)) 17: Compute (32) 18: else 19: x̃lk = x̃lk|k−1 20: Only compute (32) 21: end if 22: Compute (34) 23: end for 24: end for 25: Resample 26: Find all non contributing weights w̄k from w̃k such that w̄k ∈ w̃k and replace with where 0 < 1 ρ and resample as in Section V-C4.
- This is to ensure that only weights belonging to corrected particles are chosen for resampling.

### VI. SIMULATION RESULTS

- The nonlinear tracking performance of the proposed KG-SMC-PHD filter is demonstrated.

### A. Simulation Context and Filter Parameters

- The authors consider a two-dimensional (2-D) nonlinear range and bearing scenario with unknown and varying number of targets observed over a cluttered region.
- A plot of the ground truth (true trajectories) of the targets along with the start and end positions of each track is shown in Fig.
- Each persistent target has a probability of survival, pS(xk−1) = 0.9.
- The measurement sensor’s location, [xs, ys]T is at the origin.
- The OSPA distance metric enables us to compare multitarget filtering algorithms [37].

### B. Effect of Proposal Distributions

- Here, different importance sampling functions, TP, EPF, and UPF of Section III are applied to the SMC-PHD and KG-SMC-PHD filters to observe the effects of each choice on filter performance.
- The results obtained are shown in Tables I and II.
- This is primarily due to the generation of sigma points for each particle and the computation that follows during the unscented transform process.
- For this case, the EPF was chosen as the importance sampling density for both filters.
- It can be observed from both tables that the performance of the SMC-PHD filter appears to deteriorate further with more position and cardinality mismatch (high OSPA distance) as clutter density increases while the proposed filter is seen to maintain a consistent performance with improved accuracy in position and cardinality (low OSPA distance).

### D. Other Filters

- To further demonstrate the performance of the KGSMC-PHD filter, the proposed filter was evaluated along with the GM-PHD filter of [16], the GM-USMC-PHD filter of [21], and the AP-PHD filter in [23] in addition to the standard SMC-PHD filter.
- The GM-PHD filter was implemented with an EKF.
- The effect of the measurement partitioning process can be seen in Table V as the OSPA distance improved for the other filters.
- Here, 1000 particles were used per existing track for the KG-SMC-PHD filter while the parameters of the other four filters were maintained.
- As shown in Fig. 6, high values of OSPA distance occurs when new targets are born around time indices k = 10, 20, 40, and 60.

### E. Overall Evaluation

- The authors now discuss the filter limitations in terms of OSPA distance and number of clutter points, number of particles and CT as well as general filter performance.
- It is observed that the miss-distance increases for both filters as clutter intensity increases.
- In terms of miss-distance, Fig. 8 also suggests that the proposed filter is more efficient as only few a particles (less than 1000) are required to achieve an OSPA distance of less than 50 m while the SMC-PHD filter requires about 10 000 particles.
- A reduced computational burden is thus achieved as the unnecessary computation on measurements due to clutter is avoided during weight update.
- Second, the Kalman gain as a correction technique seeks to achieve minimal variance and thereby gives better accuracy (in approximating the posterior).

### VII. CONCLUSION

- The authors have proposed a new and efficient SMC-PHD filter for MTT, which seeks to minimize the MSE between received and estimated measurements at any given time.
- This was achieved by first partitioning the measurement set into target-originated measurements and clutter for weight computation and applying the Kalman gain to selected particles for state correction.
- The tracking performance was improved because only target-originated measurements were used for weight computation and the MSE at each time step was reduced resulting in fewer number of particles for state estimation.
- Simulation results demonstrate that their algorithm outperforms the standard SMC-PHD filter as well as other alternative implementations of the PHD filter.

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##### References

13,734 citations

### "Kalman-Gain Aided Particle PHD Filt..." refers methods in this paper

...The Kalman filter is a minimum MSE estimator, which in effect seeks to recursively minimize the MSE between the estimated measurements and actual measurements using the Kalman gain [36]....

[...]

4,591 citations

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### "Kalman-Gain Aided Particle PHD Filt..." refers background or methods in this paper

...given probability using the inverse cumulative χ2 function such that the Pr[d2 i,j,k ≤ d̃] falls within a given confidence region [3]....

[...]

...Multitarget tracking (MTT) is essential in many application areas, such as motion-based recognition, automated security, navigation and surveillance, medical imaging, traffic control, and human–computer interaction [1]–[3]....

[...]

...MTT belongs to a class of dynamic state estimation problems [3]–[5]....

[...]

...In [21], Yoon et al. proposed the Gaussian mixture unscented sequential Monte Carlo probability hypothesis density (GM-USMC-PHD) filter which uses the GM representation to approximate the importance sampling function and the predictive density functions via the unscented information filter (UIF)....

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...The newborn track initialization, resampling, and state extraction steps follow [21] and the mean and the covariance of Gaussian is computed using the UIF [3]....

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### "Kalman-Gain Aided Particle PHD Filt..." refers methods in this paper

...19: Resample Lk particles using resampling techniques such as in [34]....

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...Some common choices of importance density in SMC methods are given below: 1) The transitional prior (TP): This is the most popular choice of suboptimal proposal distribution for SMC-PHD filters and particle filters in general because it’s implementation is easy and straightforward [34]....

[...]

2,251 citations

### "Kalman-Gain Aided Particle PHD Filt..." refers background in this paper

...MTT belongs to a class of dynamic state estimation problems [3]–[5]....

[...]