Improved Resampling Algorithms Based on Deterministic and Partial Resampling

2016 International Conference on Mathematical, Computational and Statistical Sciences and Engineering (MCSSE 2016) ISBN: 978-1-60595-396-0

Improved Resampling Algorithms Based

on Deterministic and Partial Resampling

Chao HE

, Wei-sheng XU

and Peng CHEN

1_{Department of Electrical & Information, CDHK, Tongji University, Shanghai, 200092, P.R. China} 2_{School of Electronics and Information Engineering, Tongji University, Shanghai, 201804, P.R.}

China

3

School of Mechanical Engineering, Southwest Jiaotong University, Chengdu, 610031, P.R. China *Corresponding author

Keywords: Particle filter, Resampling algorithm, Tracking.

Abstract. Particle Filter is widely used to track objects in autonomous driving. Resampling is one of the most important steps in the particle filter. In implementations, resampling is regarded as a bottleneck because of the increased complexity. To handle such kind of difficulties, an improved resampling algorithm is proposed. Firstly, the most frequently applied resampling algorithms for particle filters are introduced in this paper. A theoretical analysis is provided and the differences among these resampling algorithms are revealed. This facilitates a comparison of the algorithms about their resampling quality and computational complexity. Through the simulations, the results in the theoretical analysis are verified. It is found that optimized partial deterministic resampling is suitable for tracking in terms of a large amount of objects.

Introduction

Particle filters operate three basic processes sequentially: sampling step, importance step, and resampling step [1]. The resampling step is critical in every implementation of particle filtering, because without it the weights of most particles quickly become so small and will be given up, which leads to a bad result. The idea of resampling is to remove particles with small weights and replace them with particles with large weights [2]. The resampling step is an important and complex part in the particle filter. Therefore, a proper choice of resampling method is able to reduce the complexity or improves the quality of the filter. In the previous work, quite a few different resampling methods can be found. The most frequently applied algorithms are systematic resampling [3]. The algorithms are compared with respect to resampling quality and computational efficiency, both in theories and simulations.

In this paper, an improved resampling algorithm is proposed. In section 2 and 3, several algorithms are discussed. Section 5 is a discussion.

Basic Resampling Algorithms

As is known to all, after a few iterations of implementations, the weights of a lot of particles become small, only a few particles’ weights largely increased, thus the number of effective particles decreases immediately. On the other hand a large amount of computational calculation wastes in the estimation of probability distribution by using invalid sampling particles. In order to improve the performance of algorithms, resampling algorithms are reviewed at first. The resampling step modifies the probability density p_n to p_n by deleting particles having lower importance weights and multiplying particles having higher importance weights [4]. Formally:









1

|

N

i i

k k k k k

i

p  z    



















1 1

1 |

N N

i i i

k k k k k k

j i

n

p z

N N

      

 







(2)

In the literatures basic resampling algorithms can be found [5], the algorithm is as follows:

for m1:N

GenerateU u





0

c

s s  while su

1

c c

1

U  U M

end

ic

end

N is the number of input particles. M is the number of resampled particles.  is an array of weights for the importance step. The output i is an array of parameters, which shows how many times for each particle to be duplicated. It is shown that this method generates random numbers for every

particle. We found that basic resampling is implemented using two loops.

The first improving step of this method is to generate just one random number for all the particles. Then it is called the systematic resampling (SR). The SR algorithm performs the same way as the basic resampling algorithm. Instead of generating random number for every particle, SR generates only once for N particles in one literation.

Partial Resampling

The idea of partial resampling (PR) is to apply resampling algorithm only to particles with particular weights [5]. That is to say, particles with proper weights are not resampled. The advantage of this method is to apply resampling to smaller number of particles. In the previous work, there are two widely used partial resampling algorithms as following.

Partial Stratified Resampling

In partial stratified resampling (PSR), particles are grouped in three parts. The weight of particles is compared with two thresholds, higher one T_h and lower one T_l. Particles with weights between these thresholds are considered proper and are not resampled. The number of particles with weights larger than T_h is N_h, smaller than T_l is N_l. The sum of weights of particles that need to be resampled is

1

l

h

N j N j

hl j t N N t

S 





The PSR algorithm consists of two loops. The first one includes N iteration and classifies all the particles as proper or not. The second loop includes N_lN_h iterations. The algorithm is as follows:

1

,

,

j j

hl l h t h t l

j

t _j

t

S N N for T or T

otherwise

 



_

   

  

 (3)

l h

Partial Deterministic Resampling

In partial deterministic resampling (PDR) [5], T_h and T_l are still used as mentioned above. The difference of PDR from PSR is that, the weights of particles less than T_l are removed. Each particle of

h

N will be copied for n_i times. The weights of the particles and n_i are calculated as follows:

2, 0 < <

1,

l h t

i

l h t h

N N i N

n

N N N i N

            

 (4)

1

, 0 < <

,

i

t i h

i

t _i

t l h

n i N

N i N N

  _      

 , (5)

t l h h

N  _N N _N (6) Partial resampling was tested to find out how well the method works, by applying it to tracking problems in different conditions. The result shows that the tracking loss decreases when T_l decreases for unchanged T_h[1]. The reason for this phenomenon is inferred that particles with lower weight also carry out some information for the tracking cases.

Optimized Partial Deterministic Resampling

As is discussed in the third section, although partial resampling is really fast, particles with lower weights cannot be removed. The reason is that these particles also able to carry out some information to track the objects. In this paper, PDR is improved by adding the process to resample the lowest weights, called optimized partial deterministic resampling (OPDR). As mentioned before, the weight of particles is compared with two thresholds, higher one T_h and lower one T_l. Particles with weights between these thresholds are considered as proper and are not resampled. The algorithm is as follows:

1, 0 <

2, <

2, 0 <

1, <

l h t

l h t l

i

l h u

l h u h

N N i N

N N N i N

n

N N i N

N N N i N

_ _                  _{  }    (7) 1

, 0 <

, <

, <

i

t i l

i i

t t l h

i

t i h

n i N

N i N N

n N N i N

        _    _{ }  (8)

WhereNu  N Nl hNh,Nt  N Nl hNl.

Compared with PDR, OPDR works also fast because of only resampling part of the particles. Furthermore, it utilizes the potential information of particles with lower weights. Like the part with higher weight, particles with lower weights are also divided into 2 parts. Particles with higher weights are copied once more than those with lower weights.

Optimized Systematic Resampling and Discussion

the computational waste of SR is also very high. So in this section, one loop is added to SR. Particles with weights between thresholds T_h and T_l are not to be resampled. The other particles continue to be resampled as shown in the section 2. The pseudocode is as follows:

Systematic Resampling

j j

t h t

if T or T

do

   

(9)

For discussion, the complexity of resampling algorithms is calculated. The complexity of SR, PSR, OSR is 









 

Table 1. Comparison of the operations for resampling algorithms.

SR PSR OSR PDR OPDR

Comparisons cN

l h

NN N c(N_lN_h) 2N_h 2(N_l N_h)

The numbers of particles that should be updated are equal in PSR, OSR, and OPDR. The complexities of SR and OSR are relatively higher than the other algorithms. OPDR spends a little more computational wastes to conduct the resampling than PDR, because it is more complicated to take the particles with lower weights into account.

In Figure 1, it shows that the efficiency of the resampling algorithms of the different algorithms. Considering the following 3D tracking model of objects:

1

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

t t t

T

T

T

n

_ 

 

 

 

 

_{ } 

 

 

 

 

(10)

Where the state   p,py,p v v vz, x, y, zT contains position and velocity and nt N





independent Gaussian noise sequence. The simulation is performed on a PC. Figure 1 presents the result of speed to execute different resampling algorithms. The result shows that, the algorithms based on PR are the fastest, since the smallest number of particles is resampled. Although SR and OSR perform well, their computational wastes are very high.

Figure 1. Execution time of different algorithms.

Figure 2. Tracking efficiency of different algorithms.

Conclusions

In this paper the new resampling algorithm is presented. OPDR is based on PR and PDR, and it carry out more information than PDR. Although SR and OSR perform even better than OPDR, the new algorithm are suitable for the situation, where many objects need to be tracked. It can decrease the computational waste and present a better performance than PDR. By using this algorithm, higher speed can be achieved than the classic resampling, especially in parallel implementations. It also performs better than partial stratified PSR.

Acknowledgements

This work was supported by the National Natural Science Fund "Hybrid Model of Autonomous Driving Predict and Control Research Based on the Vehicle Social Behavior" (61473209).

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