Variational Bayesian PHD filter with Deep LearningNetwork Updating for Multiple Human Tracking

Pengming Feng1, Wenwu Wang2, Syed Mohsen Naqvi1, Jonathon A. Chambers1

1. Communications, Sensors, Signal and Information Processing Group, Newcastle University, UKEmails: {p.feng2, s.m.r.naqvi, jonathon.chambers}@newcastle.ac.uk

2. Center for Vision Speech and Signal Processing, University of Surrey, UKEmail: w.wang@surrey.ac.uk

Abstract—We propose a robust particle probability hypothesisdensity (PHD) filter where the variational Bayesian method isapplied in joint recursive prediction of the state and the timevarying measurement noise parameters. The proposed particlePHD filter is based on forming variational approximation to thejoint distribution of states and noise parameters at each frameseparately; the state is estimated with a particle PHD filter andthe measurement noise variances used in the update step areestimated with a fixed point iteration approach. A deep beliefnetwork (DBN) is used in the update step to mitigate the effect ofmeasurement noise on the calculation of particle weights in eachframe. The deep learning network is trained based on both colourand oriented gradient histogram (HOG) features and then usedto mitigate the measurement noise from the particle selectionstep, thereby improving the tracking performance. Simulationresults using sequences from the CAVIAR dataset show theimprovements of the proposed DBN aided variational Bayesianparticle PHD filter over the traditional particle PHD filter.Index Terms—Multiple human tracking, PHD filter, deep

learning, variational Bayesian

I. INTRODUCTION

Unknown and varying number of targets cause the mainproblem in multiple target tracking (MTT); moreover the oc-clusion problem may occur which increases the challenge forreliable target tracking. A particular issue in MTT is that it isnot always possible to associate measurements with particulartargets and therefore false alarms and missed detections maybe generated particularly in the presence of clutter, occlusionand noise [1], which can introduce detection in the number oftargets and reduce the tracking accuracy in MTT.Both the Kalman filter and particle filter have been widely

used in tracking, however, in these approaches, the numberof targets is assumed to be known and fixed. For a variablenumber of targets, the random finite set (RFS) [2] basedprobability hypothesis density (PHD) filter has been recentlyproposed for the MTT problem. The advantage of the PHDfilter is that it can estimate both the number of targets andtheir locations, and thus avoids the need for data associationtechniques as part of the multiple target framework [3][4][5].Moreover, it mitigates the computational complexity issue thatoften occurs in other multiple target tracking approaches suchas the multiple hypothesis tracking (MHT) approach [5] byemploying the intensity instead of the posterior distribution.

However, the limitation of the PHD filter is that its perfor-mance can be easily affected by estimation errors caused bynoise.The particle PhD filter often assumes a priori knowledge of

the measurement and dynamic model parameters, includingthe noise statistics. However, such knowledge is not alwaysavailable in practical applications [6]. Variational Bayesian(VB) methods on the other hand have been used for a widerange of models to perform approximate posterior inferenceat low computational cost in comparison with the samplemethods [7], thereby establishing an analytically tractable formfor the joint posterior distribution of the state and measurementparameters. Moreover, in human tracking, the VB method canbe used with a factor-free form of distribution of the state andmeasurement noise model.In this paper, we therefore propose a novel robust PHD filter

for multiple human tracking where the VB method is used toapproximate the joint posterior distribution of the state andthe noise variance with a factorized free form distribution.The variance of the measurement noise is also updated bythis VB approach, thereby providing more stable updates forthe weights in the particle PHD filter. Since in an enclosedenvironment, only limited human features can be extractedwhen using a single camera and accurate measurement of thehumans can be difficult to obtain due to illumination and pos-ture changes, we employ a deep belief network (DBN) to aid incalculating the weights for the particle based PHD filter, whichutilizes the colour and oriented gradient histogram features.The DBN has the advantage that it is robust to backgroundnoise in the measurement due to the difference between humantarget and noises features. To evaluate the performance of ourproposed robust PHD filter, we employ sequences from theCAVIAR dataset [8] which include appearance, occlusion, anddisappearance of humans in the field of view of a camera. Nextwe introduce the proposed particle PHD filter.

II. PROPOSED PARTICLE PHD FILTER

A. Optimal Bayesian filteringWe choose the particle filter to form the fundamental

framework of the PHD filter so that the weight for each particlecan be calculated via a DBN to improve the accuracy in humantracking. We approximate the PHD filter with a set of weighted

978-1-4799-7444-3/15/$31.00 ©2015 IEEE

random samples using a sequential Monte Carlo method. Thecorresponding state and measurement model at time can bewritten respectively as

x = Fx−1 +w (1)

z = Hz−1 + v (2)

where x denotes the state vector at time including the 2-Dposition of the target, z is the measurement vector and w ∼(0P) is the Gaussian process noise, v ∼ (0R) isthe measurement noise with diagonal covariance matrix R.F and H are respectively the state and measurement transitionmatrix.We assume that the dynamic models of the states and thevariance parameters within (1) and (2) are independent [9],and thus can be described as:

(xR|x−1R−1) = (x|x−1)(R|R−1) (3)

The goal of Bayesian optimal filtering of the above modelis to compute the posterior distribution (xR|Z), whereZ = {z = 1 }. As described in [6], the filteringproblem consists of the following steps:1. Initialization: The recursion starts from the initial prior

distribution (x0R0).2. Prediction: The predictive distribution of the state x

and measurement noise covariance matrix R is given by theChapman-Kolmogorov equation:

(xR|Z−1) =Z

(x|x−1)(R|R−1)×(x−1R−1|Z−1)x−1R−1

(4)

where the integral has the dimension equal to the sum of(x) and (R).3. Update: Given the next measurement z, the predictive

distribution above is updated to a posterior distribution byBayes’ rule:

(xR|Z) ∝ (z|xR)(xR|Z−1) (5)

The integrations in the general solution are usually notanalytically tractable [6]; in the following, the recursion stepsare solved by using a variational Baysian approximation forthe posterior update.

B. Variational approximation

As described in Section II-A, the goal of VB is to computethe posterior distribution (xR|Z). Given that the inverse-Gamma distribution is the conjugate prior distribution forthe variance of a Gaussian distribution [10], a product ofinverse-Gamma distributions can be adopted to approximatethe posterior distribution of R; in this case, assuming theposterior distribution at time − 1 can be represented by

(x−1R−1|Z−1) =(x−1 −1P−1)Y=1

×

(2−1|−1 −1)(6)

where is the dimension of the measurement noise vectorand (x−1 −1P−1) denotes a Gaussian probabilitydensity function for the random variable x−1 with mean −1and covariance matrix P−1, and (2−1|−1 −1)denotes the inverse-Gamma distribution, which has the degreeof freedom parameter −1 and the scalar parameter −1.With the assumption that the dynamic models of the state andmeasurement noise variance are independent, the joint predic-tive distribution remains a factored form [9] of a Gaussiandistribution and an inverse Gamma distribution|−1(xR|Z) = |−1(x|Z−1)|−1(R|Z−1)

= (x|−1 |−1P|−1)Y=1

(2|−1||−1 |−1)(7)

where in the joint posterior distribution, the state and mea-surement noise variance will be coupled with the likelihoodfunction, which makes the exact posterior intractable.The next step is to derive an analytical expression for theposterior distribution within the update equation; in orderto make the computation tractable, an approximation to theposterior distribution is formed [7]. The standard VB approachis employed and a free form factored approximate distribution(xR|Z) can be described as

(xR|Z) ≈ x(x)R(R) (8)

where x(x) and R(R) are respectively a Gaussiandistribution and inverse Gamma distribution as follows:

x(x) = (x P) (9)

R(R) = (2| ) (10)

then the approximate posterior densities can be determinedby minimizing the Kullback-Leibler(KL) divergence betweenthe separable approximation and the true posterior densityexpressed as

{x(x)R(R)||(xR|Z)} =Zx(x)R(R) log

x(x)R(R)

(xR|Z) xR

(11)

In order to minimize the KL-divergence, methods fromcalculus of variations [6] are employed. Using the alternatingoptimisation, the probability densities x(x) and R(R)are calculated in turn, while keeping the other fixed, yielding:

x(x) ∝

½Zlog (zxR|Z1:−1)R(R)R

¾(12)

R(R) ∝

½Zlog (zxR|Z1:−1)x(x)x

¾(13)

Since the two equations are coupled, they cannot be solveddirectly, however, computing the expectation of the first equa-tion yields the following equationZ

log (zxR|Z1:−1)R(R)R =

− 05(z −Hx) hR−1 iR(z −Hx)

− 05(x −Fx−1) (P−1 )(x −Fx−1) +1

(14)

where h·iR =R(·)R(R)R denotes the expected value

with respect to the approximation distribution R(R) and1 denotes the terms independent of x.Similarly, the second expectation can be computed as followsZlog (zxR|Z−1)x(x)x = −

X=1

(3

2+ ) ln(

2)

−X=1

2

− 1

2

X=1

h(z −Hx)2 ix

2+ 2

(15)

where h·ix =R(·)x(x)x, and the parameters

P and are the solutions to the following coupledset of equations: = −1 +P|−1H

(H

P|−1H

+ R)

−1(z −H|−1)

P| = P|−1 −P|−1H (H

P|−1H

+ R)

−1HP|−1

= −1 +1

2

= −1 +1

2[(z −H)

2 + (HPH

)]

(16)

and the estimated covariance matrix R is

R =

½11

¾(17)

where ‘ · ’ denotes the estimate of the parameters. Following(16) and (17), the process of variational Bayes measurementparameter updating can be described as1. Prediction: Compute the parameters of the predicted distri-bution as follows:

−1 = −1

−1 = −1|−1 = F−1−1|−1P|−1 = F−1P−1|−1F

−1 +Q−1

(18)

where ∈ (0 1] is a scalar used for the extension of the noisefluctuations.2. Update: In the update step, a fixed point iteration method isemployed to achieve the best solution of the following equa-tions. First set 0 = |−1P0 = P|−1 0 =

12 + −1

and 0 = −1, for = 1 . Then use the fixed-pointiteration to achieve the solution of (16) and (17) for steps:then set = =

and = , after obtaining

the optimal solution for the equations, the parameters areupdated within the measurement model of the particle filterwhich thereby helps calculate the weights of each particle. Inthe next section, the VB aided particle filter is combined withthe DBN to update the weights of the particles.

C. Deep learning networks

Deep learning methods use a hierarchical learning methodwhen training the classifier. As described in [11], hierarchicallearning uses natural progression from low level to high levelstructure as seen in natural complexity, so it is easier tomonitor what is being learnt and to guide the machine evolvingto better subspaces. For example, when images are input into

the system, the first layer of the system represents the 0edges0

of the feature; the second layer represents the 0object parts0

of the feature and the third layer represents the objects inthe features [11]. As described in [12], wherein a restrictedBoltzmann machine (RBM) is employed, DBNs are graphicalmodels which learn to extract deep hierarchical representa-tionS of the training data. DBNs provide the advantage ofan intelligent method to use the limited information to traina classifier to calculate the weights of particles in humantracking, particularly with a single camera. The deep beliefnetwork we use is represented detailedly in [11], which iscomposed of one input layer, two RBMs based hidden layersand a one-class output layer. The process of training the RBMand DBN are described in [12] in detail. After training theDBNs following the process, the weight for each particle xcan be calculated as

(x) = (·1·W()) (19)

where 1 is a (1 × ) all-one vector, is a constant we setfor calculating the weights for the particles and W() is theweights from the last layer of DBN. In this way, the likelihoodfor each particle is obtained and these weights can then betaken as the input to the updating step of the PHD filter forMTT as discussed in the next section.

D. Particle PHD implementation

To formulate the PHD filter the RFS framework is em-ployed [13]. We denote D|(x) as the PHD filter at dis-crete time associated with the multi-target posterior density|(X|Z1:), whereX = {x = 1 } includes the2D position of all the human targets, x denotes the state ofthe target at time , is the number of targets and Z1:denotes the measurements up to time . The PHD predictionstep is defined as:

D|−1(x ) =

Z|−1(x

)D−1|−1()() +Υ (20)

where Υ is the intensity function of the new target birthRFS, |−1(x ) is the analogue of the state transitionprobability in the single target case which is calculated from

|−1(x ) = |−1()|−1(x

|) + |−1(x

|) (21)

in which |−1 is the multi-target transition density, |−1()is the probability that the target still exists at time and|−1(x |) is the intensity of the RFS that a target isspawned from the state . The PHD update step is definedas [14]:

D|(x ) =

⎡⎣ (x ) + X∈Z

(x )

+ hD|−1i

⎤⎦D|−1(x )

(22)

where is the missing detection probability, (x ) =(1− )(z|x ) is the single-target likelihood defining the

probability that a measurement z is generated by a target withstate x , is the clutter intensity.When the sequential Monte Carlo method is employed to

approximate the PHD filter, two fundamental steps in the parti-cle filter are sequential importance sampling and resampling.The basic principle of importance sampling is to representa PDF (X) by a set of random particles associated withthe weights, where X

= {x = 1 }, and is thenumber of particles we employed in the particle filter. Givena set of particles [15]

{−1x

−1}=1 (23)

which are independently drawn from importance samplingdensity (X) [15], the weight of each particle can be cal-culated as

= (x)(x

) (24)

thus (X ) can be approximated as

(X ) ≈

X=1

(X

− x) (25)

where (·) denotes the Dirac delta function.Assuming the particles for the PHD filter are indepen-

dently drawn from the PDF (X−1|Z1:−1), the particles

x = 1 are propagated and updated by the Gaussiandistribution, which are approximately distributed as (X

|Z)[16]. In this case, the proposed filter is an approximateimplementation of the relationship between the prediction andupdating step of the filter. The prediction and updating stepcan be described as follows.1. Prediction: Draw particle x−1 from X

−1 and feed it into

the prediction step to obtain particles at time . Thus theprediction model can be calculated as

(X |Z−1) =

Z(X

|X−1)(X

−1|Z−1)X

−1 (26)

2. Measurement update: Upon the receipt of the measurementZ, the likelihood of each prior sample x = 1 , can beevaluated and drawn independently from importance samplingdensity (X

|Z) [16]. The importance weight for each priorsample can be calculated as:

=

(Z|x)(X |Z−1)

(x|Z)(27)

Equations (26) and (27) above form the basis of the proposedparticle PHD filter. By introducing (26) into the PHD updat-ing formula (22), we can obtain the particle PHD updatingequation:

() =

⎡⎣(() ) + X∈Z

()

() + ()

⎤⎦()−1 (28)

where

() =

−1+X=1

(e() ) e()−1 (29)

The above work underpins the proposed particle PHD filterfor human tracking; in the next subsection, human features canbe extracted to train a deep belief network in the utilizationof particle selection and the updating part of the particle PHDfilter to mitigate the noise from the environment.When the variational Bayes method is applied within the

particle PHD filter, an iteration for updating the measurementnoise should be added before the updating step for the PHDfilter, in this case, the recursive particle PHD filter follows thefollowing process

Algorithm 1 Processing of the variational Bayes particle PHDfilterAt each time , sample particles for each new-born targetfrom background subtraction, after setting the weight to thenew-born target particles as 1, we can obtain the particleset {x }=1 where is the new-born target numberand is the number of particles at time .For each particle = 1 Initialize the state x and hyper parameters:

P

R

.

1 Predict the noise parameters using (18).2 Sample the particles.3 Update the measurement noise parameters using Varia-tional Bayesian approach with fixed point iteration with (16)to (18) described in Section II-B.4 Calculate the weights for each particle with DBN with(19) to aid the weights updating step.5 Evaluate the weights using (22).6 Resample the particles.END

From the above process, we can obtain the Variational Bayesparticle PHD filter; in the next section, simulation results andcomparisons will be given.

III. SIMULATION RESULTS

In order to evaluate the performance of the proposedrobust particle PHD filter for multiple target tracking, weemployed datasets from the EC Funded CAVIAR project[8], Video EnterExitCrossingPaths1cor and EnterExitCross-ingPaths1front. There are four human targets appearing, oc-cluding each other, and disappearing in the shopping mallenvironment. The number of particles is set to be 1000. Whencompared with the traditional PHD filter, our proposed PHDfilter shows its improvement in reducing the mean of the errorand standard deviation, where in scenario 1 which denotesthe video EnterExitCrossingPaths1cor, the mean of the erroris reduced from 13.45 to 11.89 with the standard deviation isreduced from 16.68 to 12.85; in scenario 2 which denotes thevideo EnterExitCrossingPaths1front, the mean of the error isreduced from 34.54 to 22.26 with the standard deviation isreduced from 19.87 to 11.85. To see this further the optimalsubpattern assignment (OSPA) [17] is employed. The OSPAcomparisons for both scenarios are shown in Figs. 1 and 2respectively.

0 50 100 150 200 250 300 3500

10

20

30

40

50

60

70

80

Frame number

OS

PA

val

ue

OSPA value comparison for traditional and proposed PHD filter

OSPA value for traditional PHD filterOSPA value for proposed PHD filter

Fig. 1: Comparison of target OSPA value for scenario 1 between theproposed and the traditional PHD filter, where the blue line denotesthe OSPA value for the traditional particle PHD filter with an averagevalue of 38.25 and the red line denotes the OSPA value for theproposed particle PHD filter with an average value of 11.79.

0 50 100 150 200 250 3000

20

40

60

80

100

120

140OSPA value comparison for traditional and proposed PHD filter

OSPA value for proposed PHD filterOSPA value for traditional PHD filter

Fig. 2: Comparison of target OSPA value for scenario 2 between theproposed and the traditional PHD filter, where the blue line denotesthe OSPA value for the traditional particle PHD filter with an averagevalue of 53.55 and the red line denotes the OSPA value for theproposed particle PHD filter with an average value of 20.62.

To verify the performance improvement, a new performancemetric for multiple target tracking based on optimal subpatternassignment, named OSPAMT [18] is employed, the OSPAMTincludes both cardinality and localization errors versus timefor the video 0EnterExitCrossingPaths1cor0 are shown as From

0 50 100 150 200 250 300 3500

2

4

OSPAMT cardinality and localization errors versus time

Frames number

OS

PA

MT

Err

or

0 50 100 150 200 250 3000

20

40

60

Frames number

OS

PA

MT

Err

or

OSPAMT cardinality erros for proposed PHD filter

OSPAMT localization erros for proposed PHD filter

Fig. 3: OSPAMT cardinality and localization errors versus time forvideo 0EnterExitCrossingPaths1cor0.the comparison of both OSPA and OSPAMT results, it can beobserved that our proposed PHD filter performs much betterthan the baseline PHD filter since in both scenarios the OSPAvalue and OSPAMT are much reduced, which means falsealarm and missing detection performance of our proposedPHD filter is likely to be much improved compared to thetraditional PHD filter. The error for the PHD filter is mostlyfrom the false alarm and miss detection caused by the noisefrom the background subtraction step, which can be mitigatedby the DBN classifier in our proposed PHD filter. Both themean of the error and the localization errors from OSPAMTare mitigated with the aid of our Variational Bayesian step forthe update of the measurement parameters. This is because theparameters are adaptive to the posterior of the joint distributionof the measurements parameters and the state model. From theresults, we can see that the proposed PHD filter improves thetracking accuracy over the traditional PHD filter.

IV. CONCLUSIONS AND FUTURE WORK

In this paper, we have presented a new DBN aided varia-tional Bayes particle PHD filter for human tracking, whichis based on recursively forming separable approximationsto the joint distribution of state and noise parameters byvariational Bayesian methods and the noise mitigating stepwith the aid of DBN. The performance of the algorithm hasbeen demonstrated by simulations. In our future work, moresequences will be used to evaluate the proposed PHD filterand an interaction model will be used to describe the statemodel of the targets.

V. ACKNOWLEDGMENT

This work was supported by the Engineering and Phys-ical Sciences Research Council (EPSRC) Grant numberEP/K014307 and the MOD University Defence ResearchCollaboration in Signal Processing.

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