Kalman-Gain Aided Particle PHD Filter for Multitarget Tracking
Summary (3 min read)
Introduction
- 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.
B. Measurement Model
- 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.
- KALMAN-GAIN AIDED PARTICLE PHD FILTER FOR MULTITARGET TRACKING 2253, also known as DANIYAN ET AL.
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.
- 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 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.
- 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.
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.
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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Citations
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Cites methods from "Kalman-Gain Aided Particle PHD Filt..."
...[23] apply Kalman gain to minimise target error and Zhang, Ji...
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...Similar to the approach mentioned in [23] we use Kalman Gain to minimize the error between the estimated and actual measurements....
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...as mentioned in [23] is used to minimize the error between the estimated and actual values by the tracker module....
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References
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Additional excerpts
...Additionally, there are other works in the literature that combine the implementation of the GM and particle PHD filter into one filter such as the GM particle PHD filter in [27] and [29] and the Gaussian mixture SMC-PHD in [30]....
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...2) Extended Particle Filter (EPF): Given that the measurement model of (4) is nonlinear, but Gaussian, it is possible to use a proposal distribution that exploits a linear approximation to the posterior [19] in the same way as the extended Kalman filter (EKF) uses a local linearization about its estimates....
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...As an attempt to solve the importance sampling problem, Maskell and Julier [19] proposed an optimized proposal distribution for SMC filters with multiple modes in general....
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...These errors arise due to the stochastic nature of drawing samples from the proposal distribution and the stochasticity of the resampling process [19]....
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...Furthermore, in MTT which involves multiple modalities, if particles are in clusters representing the modes of the posterior, the iterative process of randomly drawing samples from proposal distributions results in random fluctuations in the total weight attributed to each mode [19]....
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...h(fk−1(xk−1)) is then evaluated at each sigma point and Hk computed from these samples [19]....
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Frequently Asked Questions (18)
Q2. What are the future works mentioned in the paper "Kalman-gain aided particle phd filter for multitarget tracking" ?
In their future work, the proposed filter will be extended to track maneuvering and closely spaced targets and applied to other target tracking applications including that in a multipleinput-multiple-output ( MIMO ) radar environment.
Q3. How many particles were used for the newborn track?
In the AP-PHD filter implementation, 1000 particles were used per existing track and 10005 particles were used for the newborn track.
Q4. What is the purpose of the Kalman gain as a correction technique?
the Kalman gain as a correction technique seeks to achieve minimal variance and thereby gives better accuracy (in approximating the posterior).
Q5. What is the auxiliary importance sampling process?
The auxiliary importance sampling [23] process starts with the selection of the measurements that are well described by the targets states extracted from the estimated PHD and this is achieved using the auction algorithm.
Q6. What is the proposed filter for MTT?
In their future work, the proposed filter will be extended to track maneuvering and closely spaced targets and applied to other target tracking applications including that in a multipleinput-multiple-output (MIMO) radar environment.
Q7. How many MC runs were used for the AP-PHD filter?
Tables V and VI show filter performance results averaged over 1000 MC runs with and without measurement set partition, respectively.
Q8. What is the t th target-originated nonlinear measurement model?
The t th target-originated nonlinear measurement model is given aszk = hk(xk, nk) (4) where hk(·) is a nonlinear function and nk is an i.i.d. process noise vector.
Q9. What are the assumptions that can be used to apply the measurement set partition approach?
the above measurement set partition approach can be applied under the following assumptions: that newborn targets exist for at least two consecutive time steps, the maneuvering of targets is not too abrupt, the sample period δt is not too large, measurement noise is not too large, clutter is not too dense, and clutter is not time correlated.
Q10. How many Guassian terms were set to the KGSMC-PHD filter?
The maximum number of Guassian terms was set to 100, with the merging (Tm) and pruning threshold (Tp) set at 10 m and 10−3, respectively.
Q11. How many particles per track is the proposed filter able to accurately track?
The plots indicate that the proposed filter with ρ = 500 particles per existing track is able to properly track all targets and in addition to being able to identify all target births and deaths while successfully accommodating nonlinearities under high clutter condition.
Q12. What is the main improvement in the tracking performance of the proposed filter?
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.
Q13. What is the reason why the proposed filter outperforms the SMC-PHD filter?
Fig. 7 clearly shows that the proposed filter outperforms the SMC-PHD filter as it maintains an average OSPA distance of less than 51 m up to clutter intensity of κk = 8 × 10−3 (radm)−1 due to the particle state correction technique in their approach while the SMC-PHD filter starts to exhibit breakdown from about κk = 6.4 × 10−3 (radm)−1 (i.e., λ = 40).
Q14. What is the reason why the KG-SMCPHD filter outperformed all other filters?
in Table V, it can be observed that with just 500 particles per existing track, the KG-SMCPHD filter outperformed all other filters by having lower OSPA distance.
Q15. What is the reason why the authors have discussed the filter limitations?
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.
Q16. How many particles are required to achieve an OSPA distance of less than 50 m?
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.
Q17. What is the PHD of the spawned targets?
respectively, asx̃lk|k−1 ≈ { qk(·|x̃lk−1, Żk), l = 1, . . . , Lk−1 pk(·|Żk), l = Lk−1 + 1, . . . ,Lk (30)with corresponding weightsw̃lk|k−1 =⎧ ⎪⎪⎪⎨⎪⎪⎪⎩φk|k−1(x̃lk, x̃ l k−1)qk(x̃k|k−1|x̃lk−1,Zk) wlk−1, l = 1, . . . , Lk−1γk(x̃lk) Jkpk(x̃k|k−1|Zk) , l=Lk−1 +1, . . . ,Lk(31) with the termφk|k−1(xk, xk−1) = pS(xk−1)fk|k−1(xk, xk−1) + bk|k−1(xk, xk−1)where Lk = Lk−1 + Jk , qk(·|·) and pk(·|·) denote the proposal distributions for persistent and newborn targets, respectively; γk(·) is the PHD of the spontaneous birth, pS(·) is the probability of 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 spawned targets;
Q18. Why is the EPF used as the importance sampling function for both filters?
This is because both EPF and UPF helps us to place generated samples “under” measurements as soon as measurements become available.