# Distributed Computation Particle PHD filter

###### Abstract

Particle probability hypothesis density filtering has become a promising means for multi-target tracking due to its capability of handling an unknown and time-varying number of targets in non-linear non-Gaussian system. However, its computational complexity grows linearly with the number of measurements and particles assigned to each target, and this can be very time consuming especially when numerous targets and clutter exist in the surveillance region. Addressing this issue, we present a distributed computation particle PHD filter for target tracking. Its framework consists of several local particle PHD filters at each processing element and a central unit. Each processing element takes responsibility for part particles but full measurements and provides local estimates; central unit controls particle exchange between processing elements and specifies a fusion rule to match and fuse the estimates from different local filters. The proposed framework is suitable for parallel implementation and maintains the tracking accuracy. Simulations verify the proposed method can provide comparative accuracy as well as a significant speedup with the standard particle PHD filter.

Distributed Computation Particle PHD filter

Wang Junjie |

Harbin Institute of Technology |

P.O. Box 319 |

No.92, West Da-zhi Street, Harbin, China |

jjwanghit@aliyun.com |

Zhao Lingling |

Harbin Institute of Technology |

P.O. Box 319 |

No.92, West Da-zhi Street, Harbin, China |

zhaolinglinghit@126.com |

Su Xiaohong |

Harbin Institute of Technology |

P.O. Box 319 |

No.92, West Da-zhi Street, Harbin, China |

larst@affiliation.org and |

Ma Peijun |

Harbin Institute of Technology |

P.O. Box 319 |

No.92, West Da-zhi Street, Harbin, China |

lleipuner@researchlabs.org |

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copyrightbox[b]

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PHD filter,SMC,distributed

Multi-target tracking(MTT) is to jointly estimate the number of targets and position from a set of uncertain observations. The classical approaches such as the nearest neighbor(NN)[?],Joint Probabilistic Data Association filter(JPDA)[?], and Multi-hypothesis tracking(MHT)[?] are based on the association algorithms.

Recently,much work has devoted to random finite set(RFS)-based approximations such as the probability hypothesis density(PHD)[?],[?],cardinalised PHD(CPHD)[?] [?] and multiple target multi-Bernoulli(MeMBer) filter[?].These methods avoid the data association problem. From implementation perspective, a particle PHD filter for nonlinear non-Gaussian MTT problems,was proposed in [?]. And Vo [?] proposed a close-form solution to PHD filter with assumptions on linear Gaussian system. It is called GM-PHD filter.

The demands of "real-time" MTT have been growing. Since the CPHD filter propagates both the intensity of the RFS and the posterior cardinality distribution,its real time characteristic is intrinsically not as good as the PHD filter.The MeMBer filter is more suitable for low clutter environments.And the GM-PHD is constrained to linear Gaussian system. The particle PHD filter is more suitable for nonlinear non-Gaussian MTT problems in dense clutter environment.However,the particle PHD filter computational complexity is very high,Therefore,we are interested in improving the real-time performance of the particle PHD filter.

Similar to the particle filter,the resampling is chief obstacles to parallel and distribute. [?] had proposed the distributed particle resampling algorithm and implement in many WSN applications.In this paper,we proposed a distributed particle PHD filter. In order to apply particle PHD filter theory to practice,distributed algorithm is better than centralized algorithm. Since the distributed algorithm can reduce the execution time by implementing the particle PHD filter using multiple processing elements. We distribute the particles to N PEs.Each PE can run particle PHD filter independently.All individual PEs compute local estimates based over observations in parallel,and transmit their estimate to a CU to obtain global estimate.

Moreover,if the PEs were let to run as independent particle PHD filter,each of them most likely have performance degradation caused by part of particles.The particles exchange will solve the problem.

The main contribution of this paper is summarized as follows. First,we proposed the distributed particle PHD filter architecture. Second,we use the stphd method to extract the state estimation and corresponding measurement label.

The real time performance is enhanced and the tracking performance is equal to traditional particle PHD filter.

The remainder of this paper is organized as follows, standard particle PHD filter is given in section 2.Section 3 ,we present our distributed particle PHD filter in detail.Simulation results are given in Section 4.Section 5 is devoted to the conclusions.

The PHD filter was developed in the framework of Finite Set Statistics(FISST) initially. The PHD function is the first order moment of the random finite set(RFS) and can be defined by

(1) where is the random density representation of . is the probability distribution of the RFS . The PHD has the properties that, the integral over a measurable subset is the expected number of target.In addition,the peaks of the PHD function give the estimates of the target states.

PHD filter consists of the prediction step and the update step.Assuming the RFS is Poisson, it has been shown that the recursion propagating the PHD of the multi-target posterior follows [?]

(2) where represents composition of functions, is the prediction operator and is the update operator,which are defined as follows:

(3) (4) for any function on ,where

As an approximate implementation of PHD filter,the particle PHD filter is composed of three steps:

At time ,let and denote the number of survival particles and birth particles at time k,repectively

1)Prediction step:For i=1,…,,sample and compute the predicted weights

(5) For i=,sample and compute the weights of new-born particles

(6) 2)Update step

For each compute

(7) For i=1,…, update weights

(8) 3)resampling step

Compute the total target number ,resample to get

Just like particle filter,the application of particle PHD filter is limited to its computational complexity which is mainly caused by the resampling and it also caused by the update which need all particles participate.

The method of DRNA(distributed resampling with non-proportional allocation) was initially proposed by Bolic().The idea of the DRNA PF is to divide the whole particles into several groups so that the resampling can be performed independently by group and thus be implemented in parallel.The general DRNA is outlined by:

1) Exchange particles among groups

2) For k=1,…K and i=1,…,N sample in parallel in each group

3) For k=1,…K and i=1,…,N compute the weights in each group in parallel

(9) 4) Normalize the weights of the particles with the sum of the weights in the group

5) Resample inside the groups

The probability hypothesis density(PHD) filter,which propagates only the first moment instead of the full multi-target posterior,still involves multiple integrals with no closed forms generally(by vol).So BA-NGU VO proposed the particle PHD filter.The particle PHD filter is suitable for problems that nonlinear non-Gaussian dynamics.However,it’s high computational is to hold back it’s application into real time system. To speed up the particle PHD filter, we propose a distributed approach that only uses a subset of particles to different computing cores. In other words,we use a subset of particles to different PE and acquire the same accuracy as all the particles were used together ,but avoid unnecessary communication among the PEs. We entitle this idea DCPFPHD in this paper,can be formalized as follows. The structure chart of DCPFPHD algorithm is shown in Fig.1

As the particle PHD filter is similar the particle filter,also involve three basic steps,we can use DRNA in particle PHD filter too.In this paper,we apply the DRNA scheme to particle-PHD filter.However,there is some differences in how calculate the weight and estimate target,the challenge is change the DRNA so we can use it in particle PHD filter(Fig 1).

The standard particle PHD filter requires all the particles be participated by a (single processor) in resampling step. In this paper,Assume we have K PEs and each can run a separate particle PHD with M particle.The total number of particles is N=MK. The distributed computation particle PHD filter can be outlines as follows,

Assume we have K groups and each can run a separate particle PHD with M particle.The total number of particles is N=MK. The distributed computation particle PHD filter can be outlines as follows,

In the following,we describe the algorithm steps in detail.

t=0:Generate M particles for each group.The number of all particles is N.All these particles are shared with same weights

t=t+1

Step 1:Prediction At time t-1,we assume the particle set . is available.For k-th group and m=1,…,sample from . For new-born particles we divide these particles to K group equal then join the groups.

Step 2:Update In update step,we can calculate the weights among groups.Let denote the measurements set.For each group,for each ,use the likelihood and compute and then update the weights.

Since we use the STPHD method to estimate state,the update step should make some change.zhao[?] has demonstrated that the can be calculated as

(10) where denotes the PHD of the measure undetected. For each observation ,,For each group j,j=1,…,K

(11) (12) then calculates the sub-weight of each particles for all observations

(13) And the particle sub-weight for the target without measurements obtained is

(14) Based on formula (Distributed Computation Particle PHD filter) ,the weights can compute through

(15) We use the STPHD method can extract the estimate targets and these targets’ sequence number. For each measurement ,,compute the sum of relevant to in group j.

(16) Compute the sum of sub-weight corresponding to targets without observations:

(17) Since the weight sum of all the particles equals to the target number,the local target number can be estimated by where is the nearest integer to

Find the largest sum weight and the index set relevant to . The local estimated target state can be calculated by where in index set ,

(18) The groups can send data to the CU. When the group get a local estimate like and transmit the pair to the CU.Then the CU can combine the local estimate state which depends on the measurement index from the group to construct a global estimate.

As the CU receives all the local information .Depend on the rule "at most one measurement per target"[?].If the local estimate state’s label is same that from different groups,then we use their mean value as the global estimate state. And the local estimate state may be from clutter,so we consider only the local estimate states’ number which have same label greater than half of groups’ number as valid estimate states.

The resampling can be carry out locally at the N groups.Normalize the weights of the particles with the sum of weights in the group.

The particles in the n-th group will degenerate when its aggregated weight becomes negligible relative to the aggregated weights of the other groups.Then the n-th group hardly contribute to the approximation of the posterior probability distribution.In order to keep the groups are valid,neighboring groups can exchange a portion of particles and weights.We select the L particles from k-th group() to replace the L particles from k-1 group in random.

for k=1,…K-1,i=1….,L do:

for k=K ,i=1….,L do:

The processors are connected using an interconnection network.There are many type of network can be used,we select a ring configuration in this paper.For a ring network ,the nth PE,n=1,…,N-1,can send data to the (n+1)th PE.The Nth PE transmits data to PE number 1.

The PEs can also send data to the CU. When the PE get a local estimate like and transmit the pair to the CU.Then the CU can combine the local estimate state which depends on the measurement index from the PE to construct a global estimate.

In the traditional particle PHD filter,all the particles have to be involved by serial. In our methods,since particles are divided into K groups and the group can run a particle PHD filter independently,thus it only utilise time than before in theory.

To evaluate the proposed distributed particle PHD filter,we consider a two-dimensional scenario with the target can disappear and appear at anytime.Each target moves according to the following model where is target state vector at time kT(k is the time index and T=1 is the sampling period).[,] is the position,while is the velocity. is the vector of independent zero-mean Gaussian white noise with standard deviations of There is just a signal sensor in the scenario and the target-originated measurement are given by

=g()+=The measurement variance . Clutter is modeled as a Poisson RFS with intensity The Target can disappear or appear in the scene at any time.The probability of target survival is and is detected with probability . Assume the target birth according to the Poisson distribution with the intensity where denotes a normal density with mean and covariance Q.

,

The surveillance region is rad.m. Assign 200 particles to an exist or new born target in each group.The number of group is 4.

We run 100 independent simulations of the BOT model given by….The overall number of particles was N = 2000 and ,for the DCPFPHD,we divide them into N=4 processors with M=500 particles each. The true trajectories of five tracks over 50 scans are plotted in Fig (a).Fig (a) also shows the positions of the estimated targets over 50 time steps. The individual x and y coordinates of the tracks and estimated targets for each time step are shown in Fig (b) and Fig (c),respectively.It can be seen that estimated position based on the traditional particle PHD filter which the number of particles is equals the DPHD filter’s particles and the DCPPHD are similar and they are all close to the true tracks.

Figure \thefigure: Target trajectories estimated by the DCPPHD filter and the particle PHD filter.(a)Estimated trajectories and true trajectories.(b)Estimated trajectories and true trajectories in x-axis direction.(c)Estimated trajectories and true trajectories in y-axis direction Another vital factor for the performance of the method is the estimated number of targets.The number of true targets and estimated target by our method ,standard particle PHD filter with same particles and only particles at each scan are given in Fig Distributed Computation Particle PHD filter.It is observed that,under the same simulation conditions and same particles ,the particle PHD filter and DCPPHD filter achieve

It was proposed in [?] to use the Optimal Sub-Pattern Assignment(OSPA) as a multi-target miss-distance metric,and the parameters in it are set as p = 1 and c = 100 inour evaluation. Fig Distributed Computation Particle PHD filter shows the OSPA distance of the DCPPHD filter and particle PHD filter.

The total number of particles for these methods is given in Tabel1.Implemented on a Dell computer using MATLAB,this approximate times used by these approachs are also given in Tabel1.And the Table1 shows the mean of OSPA as well.

method particle number times mean of OSPA std of OSPA r PHD 1 in 1,000 17.6540 3.0577 0.0548 0 DCPPHD 1 in 5 8.6331 3.0732 0.0830 0 partPHD 1 in 40,000 4.0935 3.1993 0.2590 0 PHD 1 in 1,000 72.2214 6.4432 19.0788 10 DCPPHD 1 in 5 27.1516 6.2220 5.4630 10 partPHD 1 in 40,000 17.2492 7.4752 54.4956 10 PHD 1 in 1,000 117.1504 8.0049 8.2928 20 DCPPHD 1 in 5 39.4615 8.0884 22.6434 20 partPHD 1 in 40,000 29.2382 9.7143 136.3478 20 Table \thetable: 12345 set simulation results 1.the target position

2.the ospa

3.the time

4.the numberand analyse the result.

In this paper,we proposed a DRNA particle PHD filter that want to improve the particle PHD filter runtime.The DRNA-PHD filter can speed up the particle PHD filter in theory.However, the feasibility of the proposed method needs to be tested in real applications.It note that divide the more groups will lead to the decrease of performance.

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