CUSUM-Variance Ratio Based Markov Chain Monte Carlo Algorithm in Overlapped Vehicle

Tracking

W.Y. Kow W.L. Khong Y.K. Chin I. Saad K.T.K. Teo

Modelling, Simulation and Computing Laboratory School of Engineering and Information Technology

Universiti Malaysia Sabah Kota Kinabalu, Malaysia

msclab@ums.edu.my ktkteo@ieee.org

Abstract — Markov Chain Monte Carlo (MCMC) is one of the algorithms that have been widely implemented in tracking vehicle for traffic surveillance purposes. The sampling efficiency of the algorithm is essential to determine the vehicle position accurately. However, the sample size of the algorithm is still remaining an issue as non-optimal sample size will defect the tracking accuracy, especially when the moving vehicle is overlapped. Adaptive sample size of MCMC has been implemented using CUSUM Path Plot and Variance Ratio algorithms to perform vehicle tracking. CUSUM Path Plot determines the samples convergence rate by calculating the hairiness of the sample size whereas Variance Ratio method computes two sets of MCMC to determine the samples steady state. This paper proposes the fusion of CUSUM-Variance ratio algorithm to enhance the tracking efficiency. Experimental results shows that the CUSUM-Variance Ratio method have a better performance in tracking the overlapping vehicle with higher accuracy and more optimal sample size compared to the standalone CUSUM Path Plot and Variance Ratio approaches.

Keywords – Markov Chain Monte Carlo (MCMC); CUSUM Path Plot; Variance Ratio; Vehicle Tracking

I. INTRODUCTION Traffic surveillance using video sensor has gradually

become very important in the application of intelligent transportation system. This is because of various information can be provided by the video information such as vehicle speed, trajectories, collision and etc. [1]. However, tracking an overlapping vehicle has always been a n issue that remains to be solved due to the information lost when the vehicle is occluded by another vehicle. The overlapping situation happens especially when the vehicles encounter heavy traffic jam which causes the current tracking algorithm unable to keep track on the moving vehicle. Detection of the front view of the overlapping vehicle has been implemented in [2]. It detects the vehicles

using MCMC and shows high detection rate but the algorithm is implemented with static image where the vehicle is not in motion state. Even though it uses simulated annealing method to adaptively sample the MCMC but the algorithm still requires calibration for tracking the motion vehicle.

The implementation of MCMC in [3, 4, 5] has shown promising result in tracking multiple motion targets when the target objects are overlapped. These methods use fix sample size to track the target which causes drawback of the algorithm where lack of sample size will lead to insufficient tracking information, while exceeding sample size will lead to tracking error due to increasing of faulty samples. In order to overcome this problem, CUSUM Path Plot based MCMC in [6] has successfully track the overlapped vehicle with adaptive sample size but the method is still instable and leads to tracking error. The hairiness boundary of the CUSUM requires precise calculation which is difficult to be determined. Research study in [7, 8] shows that Variance Ratio method provides effective performance in calculating MCMC convergence rates. This algorithm computes multiple sets of MCMC and their corresponding variance are calculated to determine the steady state of the sample size. The implementation is capable to overcome the limitation on CUSUM Path Plot with the compensation of higher sample size using multiple MCMC sequences.

In this paper, adaptive MCMC using hybrid of CUSUM-Variance Ratio method will be developed and implemented to track the overlapped vehicle. The CUSUM-Variance Ratio will be used to obtain the third set of MCMC based on the two Variance Ratio sample sets by calculating their mean value. CUSUM Path Plot will then be implemented on the computed third chain and the MCMC is determined as converged when both methods’ stopping criteria has been met. The fusion of both methods is capable of overcoming their corresponding limitation by tracking the overlapped vehicle with higher accuracy and optimal MCMC sample size.

2011 International Conference on Computer Applications and Industrial Electronics (ICCAIE 2011)

978-1-4577-2059-8/11/$26.00 ©2011 IEEE 50

II. MARKOV CHAIN MONTE CARLO MCMC is a sampling based algorithm which depends on

the sampling efficiency. The algorithm initiates sample state, *θ to estimate the state of the target vehicle using proposal

distribution as shown in (1).

2

21*

2)(

21*

21)|( q

it

eQq

it

σθθ

πσθθ

−−−

− = (1)

The proposal distribution is a Gaussian distribution which is capable to provide sampling direction in the normal distribution since the vehicle might be moving in a random direction. The sample state *θ can be referred as the vehicle position coordinate },{ yx . Observation likelihood, )(θπ of the sample state is then calculated using Bhattacharyya colour likelihood in (2) and the edge distance likelihood in (3).

22

221)|( b

B

d

eC σ

πσθπ

−

= (2)

22

221)|( d

d

d

eE σ

πσθπ

−

= (3)

[ ] [ ])|()|()( θπγθπβθπ θθ EC ⋅= (4)

Variable B is the Bhattacharyya distance of the proposed sample state with the vehicle model and will approach zero if they are similar to each other. It is the same as edge distance likelihood in (3) as the larger d variable indicates the proposed sample state is not likely to the vehicle model. Equation (4) is the observation likelihood that fused up the colour likelihood and edge distance likelihood where θβ and

θγ are constant weight that used to emphasize the likelihood to have higher weightage on colour or edge distance. Hence, if the proposed sample state is approximately equal to the actual vehicle state then it will have higher likelihood value which increases the sample acceptance rate.

Prior probability distribution, )(θP of the sample state is calculated to ensure the proposed state is within the range that not too far from the vehicle’s previous state. The proposed sample state will be accepted as the new state in MCMC sampling by calculating the Metropolis-Hasting acceptance ratio,α as shown in (5).

⎟⎟⎠

⎞⎜⎜⎝

⎛= −−

−

−

)()|()()()|()(

,1min 11*1

**1*

it

it

it

it

QPQP

θπθθθθπθθθα (5)

Where i is the current MCMC sequence sample index and t is the current MCMC sequence index. The proposed sample state will be accepted into MCMC sampling set with probability α or it will be accepted with the probability of

α−1 . If the sample state did not reach to probabilityα , it will be rejected and the previous sample state, 1−i

tθ of the MCMC sample set is used as the new state in MCMC [9]. A new sample state will then be proposed for the next iteration and the procedure will be repeated until the stopping criteria have been reached. Thus, it is important for the MCMC to stop at the optimal sample size which provides accurate calculations for the target vehicle position.

III. CUSUM PATH PLOT CUSUM Path Plot determines the convergence of MCMC

by calculating the hairiness index of the output sample sets. Hairiness is the maximum or minimum point of the plotted sample set which shows the mixing rate of the accepted samples [10]. Sample set that are more hairy indicates that they are near to the convergence and suitable for evaluation of the target vehicle state. Euclidean distance of the MCMC samples will be implemented to calculate the state difference as shown in (6).

( )∑=

−=n

i

itiS

1μθ (6)

where μ is the mean of the samples. Hairiness index iD is then computed with increment of 1 when the state difference reaches maximum or minimum point. If the state difference is in a static state then 0.5 will be chosen for the hairiness index and no increment when undergoing a smooth plot. The index is defined as in (7).

⎪⎪⎩

⎪⎪⎨

⎧

====><<>

=+−

+−

+−

otherwiseSSandSSif

SSandSSorSSandSSif

Diiii

iiii

iiii

i

05.0

1

11

11

11

(7)

The hairiness of the sample sets will then be calculated using the hairiness index as shown in (8).

∑=

⎟⎠⎞

⎜⎝⎛=

n

iiD

nkH

1

1 (8)

k in (8) is a constant value that calibrates the hairiness to suit for variable tracking situations. The increasing of hairiness index indicates that the plot is more hairy and hence more possible to reach the convergence. If it is remains as constant or at a slow increasing rate, then more MCMC sample states are required to reach the convergence.

51

The MCMC sample sets are defined as converged when the hairiness has reached within the boundary in (9).

⎟⎠⎞

⎜⎝⎛+≤≤⎟

⎠⎞

⎜⎝⎛−

nkH

nk

4196.1

21

4196.1

21

(9)

When the hairiness reaches to this boundary, it indicates that 95% of the MCMC samples have reach steady state and the sampling iteration is ended for further evaluation.

IV. VARIANCE RATIO Variance Ratio method computes multiple sequences of

MCMC to calculate the steady state of the sample sets. The method performs simple analysis on variance between variable MCMC sequences and variance within a single MCMC. It provides basis estimation on how near is the sample set to the steady state as well as how long the algorithm needs to wait until convergence occurs [11]. Euclidean distance of the target vehicle state will be used as the parameter to calculate the chain variance. Hence, variance of multiple MCMC sequences is defined in (10).

( )∑=

−−

=m

jjmn

B1

2.1

1 θθ (10)

∑=

=n

i

ijj n 1

1 θθ (11)

∑=

=m

jj

m 1.

1 θθ (12)

jθ in (11) is the mean value of each MCMC sequence where

i is the MCMC sample state index and j is the index of

MCMC sequences. .θ in (12) is computed based on the mean value between multiple sets of MCMC sequence. The chain variance has smaller value when both MCMC sequences are near and will obtain larger difference when they are far from each other. The within chain variance difference, W will then be calculated as shown in (13).

( )∑=

=m

jjK

mW

1

1 (13)

( )∑=

−−

=n

ij

ijj n

K1

2

11 θθ (14)

Equation (14) determines the variance of accepted new state to the mean of its MCMC sequence. If the accepted sample state is far from the mean value of the sample set, the

value jK will be increased and the within chain variance

W will undergo increment as well. W is also the mean of multiple MCMC sequences, jK and it will not be affected much by small amount of defect samples in a single MCMC sequence if the other sequences have compensate it by accepting good sample state. However it will still encounter convergence error if majority of the MCMC sequences contain a lot of defected samples. This has caused the selection of proposal distribution to generate the sample state become important to improve the method’s accuracy. Therefore decrease value of W estimate that the MCMC sample sets are approaching to its steady state and can be halt for next iteration. The stopping criterion of the Variance Ratio method is determined by calculating the estimator R as shown in (15).

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ ++⎟

⎠⎞

⎜⎝⎛ −=

nWB

mnnDR 111

(15)

D in (15) is a constant value that enhances and calibrates the inherent approximation of the estimator R . As R approaching value of 1, the MCMC sample sets is diagnosed as converged and the generated MCMC sequences are expected to be near to the target distribution.

V. CUSUM-VARIANCE RATIO The CUSUM Path Plot and Variance Ratio methods

consist of sampling limitation which will lead to vehicle tracking error. CUSUM Path Plot is lack of consistency in accuracy whereas Variance Ratio provides stability with higher sample size by running multiple sequences of MCMC which is more computationally expensive. In order to overcome this, fusion of both CUSUM Path Plot and Variance Ratio method has been performed to compensate individual corresponding limitation.

In the proposed method, a third MCMC sequence will be generated using two MCMC sequences of Variance Ratio method as shown in Fig 1. Two sets of MCMC will be computed and each MCMC sequence will be initialized at different starting points which are near to the target vehicle distribution. The proposal distributions for both MCMC sequences will consist of different standard deviation value. This is to allow the algorithm to generate samples at different distribution areas and enhance its capability in tracking the target vehicle in a more robust way. Besides, the standard deviation has been made adaptive to further enhance the tracking accuracy as shown in (16).

( )ταλσσ −+=+ nn 1 (16)

where λ is a constant that calibrates the standard deviation preventing it from exceeding the target distribution and τ is the constant that decides the acceptance ratio quality. For instance, if the acceptance ratio is

52

Figure 1. Third MCMC generation

smaller than τ then it is considered as low acceptance probability and it will decrease scale parameter of 1+nσ to propose a new sample state. If the acceptance ratio is high, scale parameter will be increased for sample state proposal. When a new sample state is accepted or replicated from the previous sample state, the third MCMC sample is calculated using (17) and will be added as a new state in the third MCMC sequence.

∑=

=m

j

ij

ithird m 1

1 θθ (17)

Generation of the third MCMC sequence does not require high computational time and hence the proposed method will not burden the MCMC tracking algorithm. CUSUM Path Plot will be implemented to calculate its convergence rate and the Variance Ratio diagnostic is used to calculate the steady state of the two MCMC sequences. CUSUM Path Plot is having a drawback as it tends to determine the MCMC whether it has reached convergence at a lower sample size which will lead to insufficient information to track the vehicle. However the Variance Ratio method prevent the CUSUM Path Plot from stopping the sampling process and continues to generate samples until an optimal sample size has been achieved. In addition, the CUSUM Path Plot of the third MCMC sequence can enhance the Variance Ratio method. It will tend to compensate the Variance Ratio from faulty samples that are too far from the target distribution by continuing to generate more sample states. Since two MCMC sequences will be computed in this approach, one of the sequences might accept states that are distance away from the target distribution. The faulty samples will not be detected by the estimator R since the algorithm is based on the average value of the samples sets and will cause tracking error. The hairiness of CUSUM Path Plot is capable of detecting the faulty samples and prevents the Variance Ratio from stopping the sampling until better sample state has been accepted to compensate the defected samples. The CUSUM-Variance Ratio based MCMC tracking algorithm is shown in TABLE I.

VI. RESULTS AND DISCUSSIONS Tracking of overlapping vehicle has been implemented

using MCMC three different convergence methods. A single

white vehicle under overlapping situation is tracked via CUSUM Path Plot-Variance Ratio based MCMC and the tracking performance is shown in Fig. 2. The result in Fig. 2(a) shows the vehicle is tracked before overlapping occurs. Before fully occluded at frame 5, the vehicle begins to be overlapped by the black vehicle and the proposed method is still able to keep tracking on it. When most part of the target vehicle is overlapped in Fig. 2(c), slight tracking error has occurred as it tracks the target vehicle slightly higher than the actual position. This is because the value of observation likelihood has been decreased when the vehicle is almost fully overlapped. However, the algorithm has adaptively generated more sample size to compensate the information lost and tracked the vehicle with less error. Fig 2(d) shows the vehicle is partially overlapped and the algorithm begins to track the vehicle with better accuracy. When the overlapping is over at frame 13, it can be seen that the vehicle has been tracked accurately. Tracking has also been done by using CUSUM Path Plot and Variance Ratio method

TABLE I. CUSUM-VARIANCE RATIO MCMC ALGORITHM.

CUSUM-Variance Ratio MCMC Tracking Algorithm 1: for frame = 1 to t 2: Initialize two different sample 1θ at time t for two

sequence of MCMC 3: Loop 1 4: Loop 2 (for two MCMC) 5: Proposed two new states *θ with two different

standard deviations nσ of proposal distribution Q. 6: Compute prior probability and observation

likelihood for two proposed state. 7: Compute the acceptance ratio, α of both proposed

sample state 8: Accept proposed state with α or accept with α−1 . Else reject proposed sample and replicate previous

accepted state. 9: Compute and update 1+nσ with (16) end Loop 2 10: Compute third MCMC new sample state with (17).

11: ComputenB

, W and R with first and second

sequence of MCMC. 12: Compute hairiness H with third sequence of MCMC

13: if 1≈R and H lies within boundary 14: go to end Loop 1

15: else 16: go to Loop 1 17: end if 18: end Loop 1 19: Compute current vehicle position ( )∑

==

m

jjm

E1

.

1][ θθ

20: end for

53

(a) Frame 3

(b) Frame 5

(c) Frame 7

(d) Frame 11

(e) Frame 13

Figure 2. MCMC CUSUM-Variance Ratio Vehicle Tracking

and both their results are plotted are compared to the proposed approach as shown in Fig. 3 and Fig. 4.

In Fig. 3, CUSUM Path Plot has the highest Euclidean error in tracking the vehicle. This is due to the insufficient samples since CUSUM Path Plot tends to diagnose the MCMC sample set for reaching convergence at smaller sample size. The Variance Ratio method shows promising result as it achieved better accuracy and consistency on tracking the vehicle.

The CUSUM-Variance Ratio method has the similar Euclidean distance with the Variance ratio Method. This has showed the consistency of both methods in determining the convergence rate with less tracking error compared with CUSUM Path Plot. Other than this, the Euclidean errors of all the methods between frames 3 to 10 are higher compared with the other frame. This is because the vehicle is under overlapping situation between frame 3 till 10 and the observation likelihood will encounter information lost which will cause slight tracking error.

The plot in Fig. 4 is the MCMC sample size of the implemented methods. It shows CUSUM Path plot has most of the sample size at the range of 30 which is insufficient to track the vehicle accurately. However it still will adaptively generate more samples during frame 3 when the vehicle begins to overlap which is also the beginning of the observation likelihood information lost stage. The large sample size has compensated on the limitation and allows the

Figure 3. Euclidean Error of vehicle position

Figure 4. MCMC sample size

MCMC algorithm to keep track on the vehicle. The Variance Ratio method simulates two sets of MCMC which generates larger sample size to keep track on the vehicle. It is capable of adaptively sampling the MCMC when the vehicle is undergoing overlapping situation. The large sample size has led to more accurate tracking results and consistent Euclidean error which is essential for MCMC tracking algorithm. However the exceeding of sample size will have possibility in increasing the faulty sample sets which will reduce the tracking performance and it is also computationally more expensive.

The CUSUM-Variance Ratio methods have shown accurate tracking results with the most optimal sample size. It has the same Variance Ratio consistency and implements lesser sample size that is similar to CUSUM Path Plot. Hybrid of both methods when they are fused together to compensate each other limitation which Variance Ratio will give consistent tracking accuracy while CUSUM Path Plot will tend to compensate the Variance Ratio limitation. This is proven at the tracking frame of 13 in Fig. 4 where it

54

requires the higher sample size to track the vehicle even though the vehicle did not overlap. CUSUM Path Plot has detected the accepted defect sample states and disallowed the Variance Ratio method from converging and continued to generate more samples to compensate the defect samples. This can be seen in Fig. 3 where the CUSUM-Variance Ratio method has lower Euclidean error compared to the Variance Ratio method.

VII. CONCLUSION Adaptive MCMC tracking algorithm was implemented to

track the overlapped vehicle. CUSUM Path Plot, Variance Ratio and the proposed CUSUM-Variance Ratio adaptive sampling methods had been compared and analyzed. Results showed that CUSUM-Variance Ratio approach had given the best tracking performance with consistent accuracy at lower sample size. The proposed hybrid algorithm compensated the imperfection of both CUSUM Path Plot and Variance Ratio methods by providing an effective adaptive sampling of MCMC to track the overlapped vehicle more efficiently.

ACKNOWLEDGEMENT

The authors would like to acknowledge the funding financial assistance of the Ministry of Higher Education of Malaysia (MoHE) under Fundamental Research Grant Schemes (FRGS) No. FRG0220-TK-1/2010, and the University Postgraduate Research Scholarship Scheme (PGD) by Ministry of Science, Technology and Innovation of Malaysia (MOSTI).

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