IPDA Filters in the Sense of Gaussian Mixture PHD Algorithm

The Integrated Probabilistic Data Association (IPDA) type ﬁ lters provide estimates of the underlying target probability of existence as well as they track state maintenance. For each scan, IPDA recursive calculates the probability of target existence in order to resolve the uncertainty. Likewise, Random Finite Set (RFS) is a method for single target and multi-target tracking. It provides a Bayesian recursion of multi-target distribution through the Finite Set calculus. Practical implementation of multi-target posterior recursion is too difficult. It was analytically proved that IPDA algorithm can be derived from the RFS based filter recursion under the linear Gaussian assumptions. Probability hypothesis density (PHD) filter is an alternative to this problem where only the first order moment of the complete multi-target posterior is propagated in time. In this article, IPDA and Gausian Mixtures PHD (GM PHD) filters in a single target tracking scenario are derived and compared. Simulations have demonstrated the superiority of IPDA filters in heavy clutters.


Introduction
HEN tracking a single target in the presence of clutter, more than one measurement may be received at each scan after gating (measurement validation) process which eliminates measurements that fall outside a specified confidence region [1].In general, track maintenance using false measurements can lead to serious filter divergence problem.Therefore, a data association technique is required to differentiate target originated measurement from clutter.In a typical target tracking scenario, measurements originate from sources other than the desired target itself.The possible sources may be terrain, clouds or even thermal sources present in the sensor surveillance region.These unwanted measurements are generally termed as "clutter" tracking involving a data association technique [2].In essence, data association involves decision on which of the obtained measurements belong to target(s).It also hypothesizes the fact that no measurement may be target originated to clutter for the possibility of missed detection.The sequence of operations for a tracker in such case begins with the initiation phase where tracks are started based on the measurements of two successive scans.The IPDA, proposed by Mušicki, in [1,3], has two options on choice of Markov chain models of target existence propagation.Markov Chain One, the default one, recognizes two possibilities: the target either does not exist, or it exists and is visible with a probability of detection.Markov Chain Two, denoted with IPDA as IPDA-M2, also recognizes the possibility of target existing but not being visible.The IPDA filters provide estimates of the underlying target probability of existence as well as they track state maintenance.These quantities are conveniently used as track quality measures and can be used for the track confirmation and termination.
However, in [3], it was analytically proved that IPDA algorithm can be derived from RFS based filter recursion under linear Gaussian assumptions [4,5,6].A closed-form solution to the PHD recursion for linear Gaussian multi-target model was discovered [7,8].This result was reported in [9,10] together with the Gaussian mixture PHD filters for linear and mildly non-linear multi-target models.While more restrictive than SMC approaches, the Gaussian mixture implementations are much more efficient.Convergence results for the GM PHD filter were established in [11,12].In [13] the Gaussian mixture PHD filter is extended to linear Jump Markov multi-target model for tracking maneuvering targets, while in [14] it is extended to produce track-valued estimates.This study compared IPDA single target tracking filter in the sense of the Gaussian Mixture PHD algorithm effectiveness.
Paper is organized as follows: after introductory considerations, Section 2 presents the mathematical problem formulation.In Section 3 we derive IPDA and GM PHD in the sense of Gaussian filtering.The common algorithms comparisons and simulation results are presented in Section 4 followed by concluding remarks in Section 5.

Problem formulation
Any target tracking scenario is defined out of the probability of detection and clutter density.Again, the clutter density is depending on target dynamics and characteristics of sensor.Generally, clutter is defined by a number of selection measurement from the size of selection gate.At the beginning, the target state is to be considered.The scenario considers zero or at most one target.If the target exists, the target state x (which may consist of position, velocity or any other target dynamic parameter) follows the dynamic equation: w where F is the propagation matrix, and the process noise k ν is a zero mean and white Gaussian sequence with covariance k Q .Target measurement is modeled by the: where H is the measurement matrix and the measurement noise k w is a zero mean and white Gaussian sequence with the covariance R, independent of k ν .A measurement of target τ is present in each scan with a probability of detection P D .
The sensor is also assumed to detect the target with certain and known probability of detection.At every time instant, the sensor receives clutter measurements.The number of clutter measurements received at any particular time is random and assumed to be governed by a Poisson distribution with known average.The received clutter measurements are distributed uniformly in the surveillance region.Clutter measurements follow the non-uniform Poisson distribution by clutter measurement density y ρ .

Derivation of IPDA and GMPHD
Automatic track initiation in clutter will initialize true tracks which follow targets as well as false tracks which do not.We want to confirm true tracks and terminate false tracks.With on-line track quality measure, the IPDA type filters can be used for the track confirmation and termination, as well as the state estimation of tracks.

Derivation of IPDA
The IPDA proposed in [7] is derived based on PDAF [3] by introducing the concept of target existence.Two mutually exclusive and exhaustive events associated with the target existence were assumed, and modeled as a random variable E k .The occurrence of these two events is modeled as two states of the Markov Chain with transition probability matrix: 11 12 where is the transition probabilities for {i,j}th entries and P{.} denotes probability.A priori probabilities of the track existence (Markov Chain One): Applying definition of P W by the 1 ( ) , a priori PDF of a measurements from the target at scan k, given that it fell within the window is: -A priori PDF that no measurements originated from the target: -A priori PDF that the track exists and that no measurements originated from the target: -A priori PDF of measurements originated from the target, given that the track exists and that m k >0: The a priori probability density of measurements in the window, given that they are all false measurements: A priori probability density of measurements in the window, given that measurements i is a target and all others are false measurements is: Now, the track existence at scan k (event Using the theorem of total probability we have: And applying the Bayes rule we get: In case of no measurements in the window, m k =0 the probability of track existence can be calculated by the derivation: Data association probabilities are calculated as below: , The target state estimate conditioned on the target existence and its associated covariance is obtained as: where 0 is the corrected predicted error covariance, K is the Kalman gain, 0 q is known constant.

Derivation on GM PHD
Instead the IPDA, Gaussian model for individual targets includes certain assumptions on the birth, death and detection of targets.The GM PHD recursion has also been derived the state dependent target survival and detection probabilities.The GM PHD recursion propagates the intensity function that is approximated with a Gaussian mixture by analytically propagating the weights, means and covariances of the Gaussian mixture terms.The updated intensity function is also a Gaussian mixture.In case there is one sensor, this method can be employed.Hereafter, we outline the GMPHD filter with the assumption there are no spawning objects.In case there are spawning objects, the prediction equation is modified by adding Gaussian components representing for spawning objects [4].
Schematic diagram of GM PHD implementation is given in Fig. 1.Prediction step: Under the assumptions that target follow a linear Gaussian dynamical model, the survival and detection probabilities are constant, the intensities of the birth and spawned targets are equal to zero, and that the posterior intensity at time k-1 is a Gaussian mixture of the form Then the predicted intensity to time k is a Gaussian mixture, and is given: where 1 ( ) is the PHD of existing targets and determined from the linear Gaussian model by the: ( 1) where 1 1 ( 1) Update step: Under the above assumptions, and if the predicted intensity to time k is a Gaussian mixture: then the posterior intensity at time k is also a Gaussian mixture and is given by and the mean and covariance are updated with the Kalman filter update equations, , Given that the initial intensity function L 0 at time step k=0 is a known Gaussian mixture, the posterior intensity function at time step k>0 is also a Gaussian mixture from which the estimates of individual target states need to be extracted via peak extractions.

Results of Simulations
Consider a target motion scenario (Fig. 2) with nonmaneuvering constant velocity (CV) flight mode, similar to the previous work [15].Speed is constant with 311 m/s.The sampling period of radar sensor is T=1s.Duration of the scenario is 50 scans.Transition matrix and process noise matrix are given by: respectively, where q=0.0052 is a maneuver coefficient.The Integrated Track Splitting (ITS) simulation process is governed by a Markov Chain one.The calculations are based on Monte Carlo (MC) simulations using 100 MC N = realizations.During the one cycle of simulations, we have equal number of confirmed false tracks.This is achieved by adjusting initial probability of detection, while the confirmation threshold is the same and equal 0.95.In these circumstances, the number of false confirmed tracks is 20, overall the experiment for a period of 100 Monte Carlo simulations.
The measurement noise is also independent of the process noise.The probability of detection is assumed almost unity.Examined probability of detections is P D =0.7.Clutter is uniformly distributed over the observation space with an average rate of 50 points per scan.The results of experiments are given via three comparative parameters: number of confirmed true tracks, number of confirmed false tracks and root mean square error position (Fig. 3-5, respectively).Diagram of confirmed true tracks over all simulations (Fig. 3) shows better performance of IPDA instead of GM PHD filters, in heavy clutter environments.

Conclusions
The paper has compared the two GM-based target tracking algorithms: IPDA and GM PHD.The IPDA is the target existence and Bayes-based solution, newly represented here as a target density filter.The GM PHD is an approximation of the Bayes random set solution and the target density filter is its natural form.The modeling assumptions are translated into requirements for mutual exclusiveness of the same-track components and mutual exclusiveness of the feasible joint events.The GM PHD does not follow these requirements which indicate a future work to improve the GM PHD performance.The IPDA paradigm, on the other hand, may benefit from the elegant GM PHD birth and track splitting processes.Results of simulations with single target tracking scenario showed better track measure performance than GM PHD algorithms.In future, we should try multi target tracking experiments.Mots clés: détection de cible,suivi de cible,suivi par radar, processus de Gauss Markov, filtre IPDA, Filtre PHD, algorithme.

kE
-the target exists and is visible at time k, k E -the target does not exist at time k.

Figure 1 .
Figure 1.Schematic diagram of GM PHD implementation

Figure. 2
Figure.2 Simulations scenario: measurements and target The target moves in the region x=[0; 1000], y=[0;500] and can appear or disappear in the scene at any time.The target states consist of positions and velocities and move according to the linear and Gaussian target dynamics.The system input is modeled as follows: vector state ( ) [ ] T i x k x x y y = where , x y are the Cartesian coordinates of the target position, , x y are the appropriate velocities.Initial target state is given by 1 (0) [100; 16.0; 100; 7.0] T = x .

Figure 3 .
Figure 3. Diagram of confirmed true tracks over simulation

Figure 4 .
Figure 4. Diagram of confirmed false tracks over simulation