POPULATION BASED PARTICLE FILTERING
Harish Bhaskar
1, Lyudmila Mihaylova
1and Simon Maskell
21
Dept. of Communication Systems, Lancaster University, UK
2QinetiQ, Malvern Technology Centre, Worcestershire, UK
{h.bhaskar,mila.mihaylova}@lancaster.ac.uk, [email protected]
Keywords: tracking, particle filter, population based methods, MCMC, SMC sampler
Abstract
This paper proposes a novel particle filtering strat-egy by combining population Monte Carlo Markov chain methods with sequential Monte Carlo chain particle which we call evolving population Monte Carlo Markov Chain (EP MCMC) filtering. Iter-ative convergence on groups of particles (popula-tions) is obtained using a specified kernel moving particles toward more likely regions. The proposed technique introduces variety in the particles both in the sampling procedure and in the resampling step. The proposed EP MCMC filter is compared with the generic particle filter [1], with a popula-tion MCMC [2] and a sequential Monte Carlo sam-pler [2]. Its effectiveness is illustrated over an ex-ample for object tracking in video sequences and over the bearing only tracking problem.
1
Introduction
Particle filtering [3, 4, 5] is a powerful technique that has proven its potential for estimation of non-linear systems, with non-Gaussian noises and multi-modal distributions. The posterior state probability density function is approximated by a set of samples (particles) and they are propagated over time with appropriate weighting coefficients. One of the main limitations of the particle filter-ing technique is degeneracy, the case when only one particle has a significant weight. The resam-pling step introduces variety in the particles and can help avoiding the degeneracy phenomenon by
eliminating particles with small weights and repli-cating particles with larger weights. The most fre-quently used algorithms for particle resampling are: residual resampling [6], systematic resampling [3], stratified resampling [4], and multinomial resam-pling [3]. The multinomial resamresam-pling procedure involves drawing uniform distributed samples and applying the inversion method [7]. Residual resam-pling, otherwise known as remainder resamresam-pling, is an efficient technique of decreasing the variance of the generated samples due to resampling [8]. Strati-fied resampling is based on ideas of survey sampling and involves pre-partitioning the uniform distribu-tion interval into disjoint sets and drawing indepen-dently and henceforth applying multinomial resam-pling approach [8]. Finally, systematic resamresam-pling improves the stratified approach further by deter-ministically linking all the variable in the subinter-vals. These techniques are sensitive to the order in which the particles are presented: a simple permu-tation can change the resampling process. Though these methods are widely used in the literature, they lack thorough theoretical analysis of their be-haviour, apart some theoretical works [8].
Population-based simulation methods [2, 9]
[10, 11, 12] can help resolving the particle filter-ing degeneracy problems. Population inference methods have been actively investigated during the recent years. The main idea behind these techniques can be summarised as a process of gen-erating a collection of random variables in parallel to approximate some target density π [2]. As a result, the population simulation methods generate a collection of samples in parallel as against a single independent or dependent sample. There are 1
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two main categories of population-based methods. The first one is the group of the Markov Chain Monte Carlo (MCMC) methods and the second is based on importance sampling and resampling ideas. The main difference between these groups of techniques is in the fact that the MCMC methods are directed by theoretically convergent iterations, whereas the sampling/ resampling techniques rely on handling a large number of samples in paral-lel. One of the first methods [13] referred to as population-based MCMC, attempts to build a new target density π∗ which admits π as a marginal.
A number of different categories of techniques have been proposed within the literature [6, 14, 2]. In [13] the new target density is generated by a population move. The exchange move procedure swaps information between chains. Similar types of moves such as those in the context of genetic algorithms have been specified by the crossover, mutation and snooker moves [15].
The main contribution of this paper is in the proposed novel evolving population MCMC (EP MCMC) filtering strategy. Iterative convergence on groups (populations) of particles is obtained by using a specified kernel. The effectiveness of the proposed technique is illustrated by a comparison with the generic particle filter (GPF) algorithm [1], with a population MCMC (PMCMC) [2] and a sequential Monte Carlo sampler (SMCS) [2]. The PMCMC technique investigated in [2], attempts to combine the strength of two or more competing samples within a population using cross-over and mutation operators in order to build a more reliable and robust population of samples. Whilst the PMCMC technique generates better samples within a population, the SMCS method [2] builds better blocks of populations consisting of these samples. This is done by initially generating a block of populations each consisting of several samples, applying crossover and mutation opera-tors over Monte Carlo cycles and finally generating a group of well behaving populations. In contrast to these techniques, the proposed EP MCMC filter aims to generate a block of faithful populations as in the SMCS and simultaneously combining samples within each of these populations in order to obtain the optimal solution. In other words, the operators of crossover, mutation and exchange are applied both locally between samples and globally
amongst populations. The remaining part of the paper is organised as follows. Section 2 describes briefly previous related techniques: the PMCMC algorithm and the SMCS [2]. Section 3 presents the novel EP MCMC filter. Section 4 presents results over over the range-bearing object tracking problem based on simulated data and over the object tracking problem in real video sequences. Finally, conclusions are given in Section 5.
2
Related Previous Works
The PMCMC algorithm [2] attempts to build a new target density π∗ which admits π as a
marginal. The technique is motivated by two main hypotheses. The sequence of M densities {πj}, j = 1, . . . , M will be selected such that
they remain related and in general easier to simulate than π. The algorithm comprises three main procedures: mutation, crossover and ex-change, and all these operations are performed sequentially in time. A population is defined
by X(i)k = [x(1:N )k ], i = 1, . . . , N where N is the
number of samples (particles) at discrete time k. The usage of a population of samples will allow global moves to be constructed resulting in faster mixing MCMC algorithms [16, 17, 18]. Table 1 presents the PMCMC Algorithm of Jasra et al. [2]. Table 1. A PMCMC Algorithm.
1. Initialisation
• Set discrete time instant k = 1
• For each sample i = 1, ..., N where N is the number of samples
– Draw X(i)k ∼ ηk, where X(i)k is a
population of i samples and ηk is a
known proposal distribution at time instant k.
2. For t = 1, ..., T , where T is the MCMC gener-ations, perform
• (MUTATION)
Perform Mutation as illustrated in Sec-tion 3.2
• Make a random choice between steps • (CROSSOVER)
Perform Crossover as illustrated in Sec-tion 3.1
• (EXCHANGE)
Perform Exchange as illustrated in Sec-tion 3.3
Experimental results have confirmed that the MCMC technique often produces satisfactory performances nevertheless it relies on clever pop-ulation moves to accomplish reliable performance in efficient computational time. The second approach, the sequential Monte Carlo sampler (SMCS) technique constructs samples from a sequence of related target distributions, using re-sampling techniques on the samples from previous target density. SMCS methods have been proposed as a framework for dealing with non-linear and non-Gaussian dynamic models. In principle, the method accomplishes a good representation of the posterior distribution by considering a number of hypotheses samples taken from an easy to sample distribution π and weighting them based on the likelihood of the observations and iterating over phases of selection of particles, resampling and reweighting [19, 20, 21]. Most of these population techniques share similar grounds and have common difficulties. As the size of the samples increase, the algorithm is able to better approximate the posterior distribution. However, degeneracy could occur, when all but one particles have weights turning negligible. In such cases resampling methods are then used for handling degeneracy. The resampling phase is usually performed when the measured Effective Sample Size (ESS) [6] is less than a predefined threshold Nthr. Generally,
resampling involves selecting new particle positions and weights such that the discrepancy between the resampled weights is reduced [4]. The SMCS proposed in [2] is summarised in Table 2.
Table 2. A SMCS Algorithm [2] 1. Initialisation
• Set discrete time instant k = 1
• For each sample i = 1, ..., N , where N is number of samples
– Draw the population X(i)k ∼ ηk
where X(i)k is a population of i sam-ples and ηk is a known proposal
dis-tribution at time instant k.
• Evaluate weights wk(X(i)k ) from the
like-lihood and normalise the weights to ob-tain Wk(i).
2. Iterate steps 3 and 4. 3. Resampling
• If ESS ≤ Nthr, where the ESS is
mea-sured as (PNi=1(Wk(i))2)−1 and N
thr is a threshold. – Resample – Set Wk(i)= 1 N 4. Sampling
• Increment the time instant k = k + 1, if k = p + 1, then stop.
• For each sample i = 1, ..., N
– Draw X(i)k ∼ Kk(X(i)k−1, .), where Kk
is a kernel function.
• Evaluate weights ewk(X(i)k−1:k) and
nor-malise the weights to obtain Wk(i)= W (i) k−1wke (X (i) k−1:k) PN j=1W (j) k−1wke (X (k) k−1:k) .
3
The Proposed EP MCMC
Filter
This Section presents an approach combining ideas from the PMCMC, SMC techniques [2], and genetic algorithms [22] for probabilistic infer-ence. The motivation to the proposed population particle filtering algorithm is drawn from the similarities between the structure of the PMCMC algorithms [2] and the genetic algorithms [22]. Genetic algorithms [22] are characterised with the candidates for optimal solution at each generation and usually comprise three steps: crossover, mutation and selection. The resampling process in particle filtering is analogous to the reproduction procedure of genetic algorithms. Both procedures are aimed at introducing variety in the population of samples. The main advantage of the proposed here EP MCMC technique compared with the SMC is that it provides a very systematic mecha-nism of drawing samples from a target density and is therefore easy to implement it. In Table 3 we
illustrate the EP MCMC filter algorithm. Table 3. The Proposed EP MCMC Filter
1. Initialisation
• Set discrete time instant k = 1 and num-ber of populations n = 1
• For each sample i = 1, ..., N
– Draw X(i)n,k ∼ ηk, where X(i)n,k is the
nth population containing i samples at kth time instant and ηkis a known
proposal distribution at time instant k.
• Evaluate weights wn,k(X(i)k ) from the
likelihood and normalise the weights to obtain Wn,k(i).
2. Iterate steps 3. and 4. 3. Resampling
• If ESS ≤ Nthr1, where the ESS is mea-sured as (PNi=1(Wn,k(i))2)−1 and N
thr1 is some threshold.
– For each population j = 1, ..., n Re-sample,
∗ Iterate over generations ∗ (MUTATION)
Perform Mutation as illustrated in section 3.2
∗ Make a random choice between steps
∗ (CROSSOVER)
Perform Crossover as illustrated in section 3.1
∗ (EXCHANGE)
Perform Exchange as illustrated in section 3.3
∗ Recompute the weights
wn,k(X(i)k ) from the
likeli-hood and normalise the weights to obtain Wn,k(i).
– End 4. Sampling
• Increment the time instant k = k + 1, if k = p + 1, then stop.
• If W (i)
n,k−1
Wn+1,k−1(i) ≤ Nthr2; where Nthr2 is some threshold,
• Independent Particle-to-Particle Sam-pling:
– Iterate over generations and For each sample i = 1, ..., N draw new popu-lations Xn,k(i) using,
∗ (MUTATION)
Perform Mutation as illustrated in section 3.2
∗ Make a random choice between steps
∗ (CROSSOVER)
Perform Crossover as illustrated in section 3.1
∗ (EXCHANGE)
Perform Exchange as illustrated in section 3.3
– End
• Evaluate weights ewk(X(i)k−1:k) from
cur-rent and previous states using the likeli-hood and normalise the weights to obtain Wk(i)
In the following subsections, the crossover, mu-tation and exchange operators that are effectively used in the PMCMC and the EP MCMC methods are illustrated.
3.1
CrossOver
In the proposed strategy the component vector is encoded as a string of binary bits. The crossover point lc is a random point on the string of bits of
length l. The crossover operator cannot be applied to all parts of the component vector. Some parts of the component vector may not undergo any change and thus for such components, the probability of crossover ρcis zero. This leaves the crossover being
operated only on components that are expected to undergo random changes. The crossover operator functions on two distinguished component vectors, for example, xs, xq. A detailed algorithm of the
crossover scheme used is as follows:
• Draw a uniform random number uc for every
component uc ≈ U (0, 1) and if uc ≤ ρc then
• Estimate a crossover point lc based on a
uni-form random integer between 1 and the length of the component l
• Generate two offsprings:
xs= (x1s, x2s, ..., x(lc−1)s, xlcq, ..., xdq) and
xq= (x1q, x2q, ..., x(lc−1)q, xlcs, ..., xds).
where ds and dq refers to the length of the samples xsand xq respectively.
3.2
Mutation
A probability of mutation is initially defined for each component. Such a probability is chosen in or-der to make sure that components need no stochas-tic fluctuations could be prohibited from undergo-ing mutation operation. For such components, the probability of mutation ρmis considered zero. The
components are assumed as a vector string of bi-nary units. According to the proposed mutation mechanism,
• Draw a uniform random number um for every
component um≈ U (0, 1) and if um≤ ρmthen
perform mutation
• Estimate a mutation point lm based on a
uni-form random integer between 1 and length of the component l
• Flip the lm bit
3.3
Exchange
Consider two independent chain of samples, for ex-ample, s and q. Swap the information between the two chains based on a probability min{1, A} as il-lustrated in Figure 1, where
A = πs(xq)πq(xs)
πn(xs)πq(xq). (1)
4
Results and Analysis
4.1
Bearings Only Tracking
Simulation results from the bearing only tracking problem (with the same parameters as in [23]) are presented in this subsection. The target motion is modeled using a constant velocity model
xk= 1 Tp 0 0 0 1 0 0 0 0 1 Tp 0 0 0 1 xk−1+ vk (2)
xt yt Øh Øa11 Øa12 Øa21 Øa22 Øl11 Øl12 Øl21 Øl22
0.6 0.6 0.4 0.5 0.7 0.5 0.6 0.5 0.2 0.2 0.4
x’t y’t Ø’h Ø’a11 Ø’a12 Ø’a21 Ø’a22 Ø’l11 Ø’l12 Ø’l21 Ø’l22
0.7 0.7 0.1 0.8 0.8 0.6 0.2 0.3 0.5 0.1 0.3
xt yt Ø’h Øa11 Øa12 Øa21 Ø’a22 Ø’l11 Øl12 Ø’l21 Ø’l22
Figure 1: Exchange Operation Illustrated on an ex-ample set of sex-amples (first row), their probabilities (second row) and the new hybrid offspring
where vk is a Gaussian process noise, vk ∼
N (0, Σ), the sampling period is Tp= 1 and,
Σ = 5 T3 p 3 T2 p 2 0 0 T2 p 2 Tp 0 0 0 0 Tp3 3 T2 p 2 0 0 Tp2 2 Tp . (3)
The state vector is xk=
¡ x1 k, x2k, x3k, x4k ¢0 , where x1 k and x3
k correspond to the horizontal and vertical
positions, while x2
k and x4kcorrespond to the
veloc-ities. The observation equation of the bearing only tracking problem is:
yk = tan−1 µ x3 k x1 k ¶ + wk, (4)
where, the measurement noise is wk ∼ N (0, R),
having a covariance R = 10−1. Experiments are
carried out comparing the proposed EP MCMC filter the PMCMC, SMCS [2] and GPF [1] (with recursively calculated weights). Their performance is evaluated using the combined (in x and y direction) root mean square error (RMSE) and the average number of unique particles. Figure 2 shows the root mean square error (RMSE) over 100 Monte Carlo runs. It is evident that the EP MCMC filtering mechanism produces smaller RMSE error than the PMCMC, SMCS and GPF across most intervals of time.
All filter are run with N = 1000 particles. The thresholds used are as folllows: Nthr = Nthr1 = Nthr2 = N/10. However, the algorithms maintain a different number of unique particles, as it is evident from Figure 3. Figure 3 shows the average number of unique particles that the EP MCMC filter uses in comparison with the GPF. The EP MCMC filter re-tains a higher number of unique particles compared
0 5 10 15 20 25 30 0 5 10 15 20 25 30 Cycles Combined RMSE
Root Mean Square Error Plot SMCS PMCMC GPF EP MCMC
Figure 2: Root mean square error (combined on x and y positions) of EP MCMC [2], SMCS [2], and the GPF [1] over 100 Monte Carlo runs.
with the GPF. This fact predetermines the superior performance of the EP MCMC algorithm compared with the other considered filters.The EP MCMC fil-ter introduces variety in the particles both in the sampling and resampling steps.
0 20 40 60 80 100 0 100 200 300 400 500 600 700 800 Number of Cycles
Average Unique Particles
Average Number of Unique Particles versus Cycles EPMCMC GPF SMCS PMCMC
Figure 3: Average number of unique particles (be-fore resampling) and for one run of the EP MCMC, PMCMC [2], SMCS [2] and GPF [1].
4.2
Video Body Parts Tracking
The performance of the proposed EP MCMC tech-nique is illustrated on the body parts tracking prob-lem [24]. The object detection is performed by means of the cluster background subtraction algo-rithm proposed in [25]. In the context of the hu-man body parts tracking the centre of the region surrounding each body part is evaluated. Video data sets [26] comprising different motions are used. Results are shown over a video sequence from the CMU data set [26] comprising a moving person and consisting of 153 colour image frames at a resolu-tion of 352 x 240 pixels. 0 10 20 30 40 0 100 200 300 400 500 0 50 100 150 200 250 300 Frame Number Trajectory Plot of Motion of the Object Center
x−cordinate Position y−cordinate Position EP MCMC GPF PMCMC SMCS Manual Ground Truth
Figure 4: Trajectory of the moving object esti-mated by the EP MCMC, PMCMC [2], SMCS [2], the generic PF [1] and ground truth data.
The location of the centre (tx, ty) of the torso can
be estimated and all other body parts (legs, arms and head) are located with respect to the torso, by taking into account certain geometrical/spatial constraints between the body parts. The centre of every other part within the body model is calcu-lated using the centre of the torso and accounting for their rotational degree of freedom. The over-all state vector of the population particle filter is assumed to contain the centre of the torso and the rotation angles of all the parts. A constant velocity model is assumed and the likelihood is a function of the colour distribution.
The experiments are conducted on a Intel Core-Duo processor with 1GB RAM. Figure 4, shows the actual trajectory of the moving object and the estimated trajectories obtained by the proposed EP MCMC filter, the PMCMC [2], SMCS [2] and GPF [1]. The actual trajectory of the torso is man-ually obtained and this is our ground truth data. It can be observed that the GPF looses track of the moving object at around the 27th time instant and this is a direct consequence of loss of diversity in the population of samples. The proposed EP MCMC algorithm is able to track the moving objects ac-curately in comparison with the manually labeled ground truth data.
Figure 5 presents the absolute mean error in the x and y co-ordinates for the EP MCMC, PM-CMC, SMCS, and GPF. It is evident that there is a big difference in error between the errors from the GPF compared with the other algorithms. The EP
0 10 20 30 40 0 100 200 300 0 20 40 60 80 Frame Number Mean Absolute Error in Spatial Location
Error in x−cordinates Error in y−cordinates EPMCMC GPF PMCMC SMCS
Figure 5: Absolute mean error in pixels in x and y co-ordinates for the proposed EP MCMC, PM-CMC [2], SMCS [2] and GPF [1] computed from the position estimates and the ground truth data. MCMC filter gives the least absolute mean error.
0 5 10 15 20 25 30 35 40 0 50 100 150 200 250 Frame Number Combined RMSE
Combined (in x and y directions) Root Mean Square Error EPMCMC
GPF PMCMC SMCS
Figure 6: Root mean square error (combined on x and y positions) for the proposed EP MCMC, PMCMC [2], SMCS [2] and GPF [1].
In Figure 6, the plot of the root mean square error combined in both x and y di-rections against time (frame number) is illus-trated. The proposed EP MCMC algorithm produces the least error in comparison to the other methods. This is further validated using the average root mean squared error over the considered time interval as: (5.4634, 5.8572) us-ing the EP MCMC approach, (34.0862, 17.9748) through PMCMC, (23.0862, 10.2334) using SMCS and (94.0862, 25.8453) for the GPF.
5
Conclusions
This paper proposes a novel evolving population MCMC technique for probabilistic inference. The proposed technique is compared with the population MCMC particle filtering algorithm [2], SMC sampler algorithm [2] and the generic PF. The results demonstrate that the main appealing property of the proposed EP MCMC filtering technique is in its ability to iterate through the processes to crossover, exchange and mutation for building a diverse population and simultaneously resample particle with a deterministic selection procedure.
Acknowledgement: The authors acknowledge the support of UK MOD Data and Information Fusion Defence Technology Centre under the Tracking Cluster project DIFDTC/CSIPC1/02.
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