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Combined penalized weights based GM-PHD for point target tracking in starry-sky background
Journal Pre-proof Combined penalized weights based GM-PHD for point target tracking in starry-sky background Qingqing Luo, Zhisheng Gao, Chunzhi Xie
PII:
S0030-4026(19)32044-3
DOI:
https://doi.org/10.1016/j.ijleo.2019.164145
Reference:
IJLEO 164145
To appear in:
Optik
Received Date:
28 November 2019
Accepted Date:
27 December 2019
Please cite this article as: Qingqing Luo, Zhisheng Gao, Chunzhi Xie, Combined penalized weights based GM-PHD for point target tracking in starry-sky background, (2020), doi: https://doi.org/10.1016/j.ijleo.2019.164145
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Combined penalized weights based GM-PHD for point target tracking in starry-sky background Qingqing Luoa , Zhisheng Gaoa,, Chunzhi Xiea of Computer and Software Engineering, Xihua University, Chengdu 610039, P. R. China.
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Abstract
It is well known that the point targets observed by the space-based platform are extremely difficult to track due to the interference of a large number of
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stellar targets and background noise. The traditional GM-PHD, due to its probability accumulation, is easy to mistakenly identify the star as a derived new target when the real target passes closely to the star. In order to achieve
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precise tracking of multiple moving point targets for space-based observations in a complex background environment such as a starry sky background, this paper
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proposes a simple and effective combined penalized weights based GM-PHD for filtering. First, a unique label is given for each target to determine which real target the estimate came from. Next, the moving region of target is discretized
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into 10 irregular intervals to construct ten different motion templates, numbered one by one, for discretization of the direction of the target motion. Next, the true target at time k-1 and all the estimated values having the same label at time k are used to calculate the direction penalty factor and the speed penalty factor to obtain the penalty strength matrix and penalized weight matrix, respectively. Finally, the penalized weight matrix is used to output the target with the highest
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weight in each label as the real target at time k. And a given penalty threshold is
used to remove some fuzzy weights with greater penalty in the penalty strength matrix, so as to selectively filter out some less relevant targets to further avoid the stars participating in the next iteration. In order to verify the effectiveness ∗ Corresponding
author Email address: [email protected] (Zhisheng Gao)
Preprint submitted to Journal of LATEX Templates
November 28, 2019
of the proposed algorithm, we used the Tycho-2 catalog to simulate four kinds of starry-sky data sets with different complexity and conducted comparative experiments on each data set. The experimental results show that the proposed algorithm can effectively track multiple space-based infrared weak point targets in complex starry-sky scenarios. Keywords: combined penalize, starry-sky background, space-based
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observation, point target tracking.
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1. Introduction
In traditional tracking scenarios, multi-target tracking refers to estimating
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the state and quantity of a target from a series of measurements that are interfered by noise and clutter [1]. New targets, derived targets, clutter, and noise, 5
etc., will make the number and state of the target change over time. Therefore,
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how to overcome the influence of these factors to maintain the correct tracking of the target has always been the focus and difficulty of multi-target tracking.
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Ideally, there is an assumption that in each time step, one target has only one observation, and one observation can only originate from one target (one10
to-one)[2]. However, due to the influence of noise and clutter, a target often has
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multiple observations, and an observation often comes from multiple targets (many-to-many). And in the starry-sky background, the stars always exist and are easier to be detected, which leads to a greater probability of a many-tomany situation with real targets. However, we find that unlike many-to-many
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cases that occur in traditional scenarios, the predicted values of the target in the starry-sky scenario and the observed values of the star are often calculated to
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be greater than the given pruning threshold. Also, as the number of iterations increases, the weight will gradually increase by more than 0.5 and even exceed the true target. Finally, the star is incorrectly identified as a derived new target
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of the real target and is incorrectly tracked. So how to deal with the falsederived new targets interfered by the stable strong noise (stars) is the problem to be solved in this paper.
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In recent years, based on the random finite set theory [3] and the point process theory [4], the researchers proposed a method of using the probability 25
hypothesis filtering (PHD) filtering to track time-varying targets [5]. The PHD filtering uses the first-order statistic of the multi-target posterior probability distribution to replace the posterior probability distribution of the target, avoiding the traditional data association and effectively reducing the computational cost
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[6]. The Gaussian mixture PHD (GM-PHD) filtering [7] and the Sequential Monte Carlo PHD (SMC-PHD) filtering [8] are two important variants of the
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PHD algorithm. Moreover, the former is suitable for nonlinear non-Gaussian tracking scenarios, and the latter is suitable for linear Gaussian tracking scenarios.
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In the tracking environment, new targets may appear anywhere in the observation area, which allows traditional algorithms to extend the scope of the
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emergence of new targets to the entire observation area. Wang [9] proposed a trajectory initialization to generate the trajectory of a new target at an unknown location. Ristic [10] adopted a measurement driver to adaptively calculate the
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birth intensity of the new target in each time step from the received observation set, which improves the ability of PHD and CPHD filtering to distinguish between the new target and the survival target. In reference [11], based on the
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fact that at least one Gaussian component of each measured value is retained after the conventional pruning and merging, a solution for simply and effectively processing the intensity of the new target was proposed.
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For multi-target tracking problems, many researchers have proposed their
own solutions because traditional filtering cannot identify individual targets. Panta [12] provided a unique label for each target and obtained a set of state
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trajectories for a single target by correlating the state with the target in each time step. Some novel label-based trajectory extraction schemes have also been
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proposed in [13–15], but the problem of identification of trajectories of nearby targets is not well solved. A distribution scheme based on detection of dynamic weights was used to reduce the weight error between neighboring targets and to use labels to form and maintain the track of a single target [16], effectively 3
alleviating the problem of identification of trajectories of nearby targets. More55
over, reference [17] proposed a track extraction method based on target topology information to achieve target recognition and trajectory management. Inspired by the idea that the historical measurement information of the target can increase the tracking accuracy of the algorithm to the target, in 2010, Mahler [18, 19] proposed a Gaussian mixture closed solution for forwardbackward smoothing filtering based on the smoothing scheme. Moreover, a
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simple and effective N-scan algorithm was proposed in [20], which used the set
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of historical weights of the target to overcome the shortcoming of GM-PHD filter that some real targets might be trimmed for the reduction of weights. These
algorithms have improved the utilization of measurement information, but when the weight of the star is greater than the weight of the target, the star would
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replace the real target for output.
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In recent years, some researchers have proposed the concept of weight penalty and weight redistribution for target-intensive space. Yazdian-dehkordi proposed CGM-PHD [21] and PGM-PHD [22], which provided a strategy for weight penalty in the update step and redefined the weight distribution of the target.
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In reference [2], Wang proposed a filter that jointly penalizes the Gaussian mixture probability. The algorithm provides a single target’s track management,
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and also uses the unique label of each target to jointly penalize weights of the dense target, which improves the tracking ability of GM-PHD filtering in a close
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motion space. A new weight update strategy is proposed in [23] and provides a label for each target. The irregular sliding window detection algorithm was proposed in reference [24]. The algorithm uses a new weight redistribution scheme for the weights of the same label in the weight matrix, making the algorithm
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more consistent with the one-to-one assumption.
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For the extended target used in the tracking system, reference [25] proposed
to construct a joint penalty coefficient by using the Euclidean distance and the color similarity between the predicted value and the observed value to realize the penalty for the fuzzy weight. Moreover, the paper [26] has reported that in a human sperm tracking system, in order to improve the ability of the algorithm to 4
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track sperm in dense space, the GM-PHD filtering pruning and merging process has been introduced and corrected and has achieved a good effect. The above algorithms have a certain effect on improving the correct rate of GM-PHD filtering in dense space targets. However, the abnormal weights brought by the stars are not caused by the dense space between the target-
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s. Therefore, the existing weight penalty and weight redistribution algorithms
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cannot fully achieve the correct tracking of the targets.
This paper proposes a combined penalized weights strategy for multi-target
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tracking in the background of starry-sky for space-based observations. First,
each target is given a unique label that identifies the updated estimate of the 95
corresponding target and all observations. Through a large number of investi-
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gations, we found that the movement direction and velocity of the target in the background of the space-based and starry-sky are Gaussian. That is, when the
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target is in motion, the gradient of the change in direction and speed does not rise or fall abruptly, and the change is mainly distributed around the center. 100
So next, the concept of the target motion template is introduced. The moving
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direction of the target can be discretized by different motion templates, and then be calculated according to the number of the template at time k-1 and all template numbers with the same label at time k, so that the trend of each
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updated value at time k can be obtained. For the change of the speed of the target, we use the speed at k-1 and the estimated speed of the corresponding label at time k to calculate the Euclidean distance between the speeds. Then, a penalty factor is obtained by combing the direction and the speed and is used to penalize the weighted matrix after the pruning and obtain the penalty strength matrix. Finally, the final tracking target is extracted from the penalty weight matrix, and the penalty weight matrix is used to filter out the heavily penalized
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Gaussian components to prevent strong interference information such as stars from entering the next iteration. The main structure of this paper is as follows. The second part is a de-
scription of some basic knowledge and problems under the new scenario. The 115
third part is the combined penalty algorithm proposed in this paper. The fourth 5
part is the experimental simulation results and analysis and the fifth part is a summary of this work.
2. Background knowledge 2.1. Random finite set In the target tracking, the state and quantity of the target will be dynamical-
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ly changed by factors such as clutter and noise. Mahler proposed a theoretical
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framework for multi-sensor multi-objective data fusion based on finite set statistical theory [27, 28]. In this framework, the state set and the observation set of the target at time k can be defined as a random finite set, as follows:
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A random finite set of states of the target:
Xk = { xk,1 , xk,2 ....xk,M (k) } ∈ F (ℵ),
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(1)
A random finite set of measurements of the target: Zk = {zk,1 , zk,2 ...., zk,N (k) } ∈ F (Z),
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(2)
where ℵ ∈ Rnx and Z ∈ Rnx are the state space and the observation space of the target, respectively. F (ℵ) and F (Z) are the respective collections of all finite
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subsets of ℵ and Z. M (k) and N (k) are the number of targets and the number of measurements at time k, respectively. Combined with the multi-target tracking,
the state set Xk and the measure set Zk of the target at time k are expressed
as follows:
Xk =
[
Sk—k - 1 (x)
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[
Bk—k - 1 (x)
[
Γk ,
(3)
x∈xk−1
x∈xk−1
" Zk = Kk
[
[
# [
Θk(x) ,
(4)
x∈Xk
where Sk—k - 1 is the target random finite set that survives at k-1 and at k. Bk—k - 1 (x) is a random finite set of derived new targets generated at time k-1.
Γk is a random finite set of new targets for k. Kk is a random finite set of 6
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measurements interfered by noise at time k.
Θk(x) is a random finite set
x∈Xk
of measurements of the target at time k. There are stellar measurement information at every moment in the starry-sky scenario, but these information are different from the factors such as false alarms and clutter. So, we extend on equation (4) to make it more fully representative of the target observations at k: Zk = Kk
[
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[
Θk(x) ,
(5)
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x∈{Xk ,Gk }
where Gk is a random finite set of measurements of stars at k.
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2.2. The GM-PHD filter
Mahler proposed PHD filtering based on a random finite set theory. PHD filter uses the first-order statistic of the multi-target posterior probability dis-
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tribution to replace the posterior probability distribution of the target [5]. Although the PHD filter can greatly reduce the computational complexity of the
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multi-target Bayesian filter, it needs to be approximated by some numericalbased methods. Recently, Vo and Ma provided a closed-form solution for PHD filter by performing a linear Gaussian assumption on a multi-objective model
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and summing the mixed weights of the Gaussian components [29]. Assume that N (c; m, P ) is a Gaussian density function with a mean of m and
a variance of P . In GM-PHD filtering, the transition model and the observation
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model can be modeled as a linear Gaussian, as follows: f (xk |xk−1 ) = N (xk ; Fk−1 , Qk−1 ),
(6)
g(zk |xk ) = N (zk ; Hk xk , Rk ),
(7)
where Fk , Hk , Qk and Rk are the transfer matrix, the observation matrix, the process noise covariance, and the observed noise covariance, respectively. Based on some of the assumptions in [29], the posterior strength at time k-1 can be
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modeled as a Gaussian mixture using Jk−1 components, as follows: Jk−1
vk−1 (x) =
(i)
X
(i)
(i)
wk−1 N (x; mk−1 ; pk−1 ),
(8)
i=1 (i)
where wk−1 is the weight of the ith Gaussian component.
The prediction
strength at time k can also be modeled as a Gaussian mixture using Jk|k−1
Jk|k−1
vk—k - 1 (x) =
X
(i)
(i)
(i)
wk|k−1 N (x; mk|k−1 ; pk|k−1 ).
(9)
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i=1
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components, as follows:
Then, the posterior intensity at time k can also be expressed as a Gaussian
vk (x) =
X Jk|k−1 X z∈Z
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mixture model, as follows:
j wzj (z)N (x; mjk|k (z), Sk|k )
j=1
(10)
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− (1 − PD,k )vk|k−1 (x)
where PD,k is the probability of detection of the target. More details on GM-
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PHD can be found in [25].
2.3. Problems with existing filtering methods in starry-sky scenario
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(1) False derived targets interfered by the stellar. Ideally, the traditional multi-target tracking algorithm satisfies the one-to-one
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principle. However, in a real tracking scenario, one target is updated with multiple observations to obtain multiple prediction targets with the same label. However, in a starry-sky scenario, since the stars are easy to detect and exist near the same location at all times, the weight of the target updated with the stellar
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information tends to be greater than the clipping threshold. Especially when the
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target passes closely to the star, its weight tends to be gradually greater than 0.5 in the process of multiple iterations and is mistakenly recognized as the derived targets of the real target. Therefore, if the algorithm can selectively filter out targets that are most likely to be stars before entering the next iteration, then the generation of derived new targets can be avoided to some extent. 8
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(2) The new target is updated with the measurement information of the star. Another shortcoming of the GM-PHD algorithm is that in the tracking scenario when a new target has not yet appeared and there is a star or false alarm around its possible location, the traditional GM-PHD filter will directly update with these measurement information and the new target to get the wrong target number and state estimation. Irregular sliding windows can be used to improve
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the above situation [24]. The essence of the algorithm is to take advantage of the irregularities and low frequencies that occur in false alarms. However, in the
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starry-sky scenario, the star is always there. So, when there are stars around a new target, the existing algorithms can not completely avoid the problem. (3) The weight of the star is greater than the weight of the real target.
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The algorithm of weight penalty or weight redistribution mainly penalizes and
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normalizes the weights of the same label in the weight matrix, so that the fuzzy weight satisfies the one-to-one principle as much as possible. In the starry-sky
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scenario, the false-derived new target generated by the target through the star has the same label as the target, so the weight of the same label in the weight matrix may be more than 0.5. Thus, when the weight of the star is greater than the weight of the real target, it is impossible to increase the weight of the
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real target and reduce the weight of the star only by weight penalty or weight redistribution. As a result, the algorithm will lose the position information of
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the real target from this moment.
3. Combined penalized weights based GM-PHD
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Through observation, we found that when the target is close to the star, the
traditional algorithm can mistakenly identify the star as a new target of the real target easily. This is because the traditional algorithm obtains a weight
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that is easily greater than 0.5 after updating the predicted value of the target and the observed value of the star. Considering that the moving direction and speed of the target in the infrared weak point target motion will not jump, that 9
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Figure 1: Ten motion templates and their corresponding ranges.
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Table 1: The specific angular ranges of the ten motion templates and their corresponding values. Template number angle
1 [90 ◦ ,56 ◦ )
2 [56 ◦ ,36 ◦ )
3 [36 ◦ ,20 ◦ )
4 [20 ◦ ,9 ◦ )
5 [9 ◦ ,0 ◦ ]
[0,0.5592)
[0.5592,0.809)
[0.809,0.9396)
[0.9396,0.9876)
[0.9876,1]
6
7
8
9
angle
(0 ◦ ,-9 ◦ )
[-9 ◦ ,-20 ◦ )
[-20 ◦ ,-36 ◦ )
[-36◦ ,-56 ◦ )
10 [-56 ◦ ,-90 ◦ )
range
(-1,-0.9876)
[-0.9876,-0.9363)
[-0.9396,-0.809)
[-0.809,-0.5592)
[-0.5592,0)
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Cosine value Template number
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is, the gradient of the moving direction and speed of the target will not rise or fall too much, and the change is in accordance with the Gaussian distribution. 185
In order to avoid the drawbacks of the traditional algorithm completely relying
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on the weight value and improve the anti-interference ability of the algorithm to the star, the moving direction and the moving speed of the target with the same label are used to design the combined penalized weights and used in the GM-PHD algorithm to penalize the estimated weights of targets.
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3.1. Discrete non-uniform weights of moving
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First, the moving region of targets is discretized into ten motion intervals
with inconsistent ranges, and numbered one by one as shown in Figure 1. The motion template consists of a half-sided unit circle with a quantized degree range of [-90◦ , 90◦ ]. The closer the interval difference is to 0◦ , the smaller the
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discretization range is. Table 1 shows the specific angular ranges of the ten motion templates and their corresponding values.
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The moving direction of all estimated targets labeled t at the time of k can be obtained by: →
cos <
i,j i Vk−1,t , Vk—k−1,t
→
i,j i Vk−1,t • Vk—k−1,t >= → → × V i,j V i k−1,t k|k−1,t
(11)
+ τ,
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i,j i is the where Vk−1,t is speed of the ith real target labeled t at time k -1, Vk|k−1,t
estimated speed obtained after updating the ith target and the j th observation
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i,j i at time k. cos < Vk−1,t , Vk—k−1,t > is the cosine value between the Vki - 1 and i,j the Vk—k−1 , τ is the bias of the random noise in the simulated observations.
Since noise is already included in the measured value, the value of τ may not
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i,j i be additionally specified when calculating cos < Vk−1,t , Vk—k−1,t >.
The cosine value between the speeds can be obtained by the formula (11),
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and then the motion template number of all the estimated targets whose labels are t at the time of k can be obtained by Table 1. Finally, by formula (12), the difference between the template number of the real target labeled t at time
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k-1 and the template number of all estimated targets having the same label at time k can be obtained. This difference is the discrete non-uniform weights of moving proposed in this paper.
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i,j i T rendi,j k,t = N umk−1,t − N umk|k−1,t ,
(12)
where N umik−1,t is the motion template number of the ith target labeled as t at
th time k-1, T rendi,j real target and the k,t is the number difference between the i
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j th observation labeled t at k time. T rendi,j k,t physically expresses the degree
of change in the moving direction of the target at the time of k compared to
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the moving direction of the real target at the previous moment. T rend•k,t will be used as an item in the joint penalty algorithm to penalize the weights of all estimates with the label t.
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3.2. Weights consist of the distances between speeds When the multiple estimates with the label t have the same motion template
number, it is not enough to use T rendi,j k,t to achieve the penalty for the estimated 11
target. Therefore, the Euclidean distance between the real target ith labeled t at the time of k-1 and all estimated targets with the same label at time k is used to further penalize the same motion template: i,j i,j − Vki - 1,t |, P Vk,t = | Vk|k−1,t
(13)
i,j i,j where P Vk,t is the Euclidean distance between Vk|k−1,t and Vki - 1,t . The physical
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i,j meaning of P Vk,t is the magnitude of the speed change of all the estimated
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targets of t at the time of k with respect to the unique real target at time k-1. 3.3. Combined penalized weights
i,j P Vk,t and T rendi,j k,t are combined to penalize the weights of the estimates
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in the traditional algorithm. The combined approach and the penalized the weights are as follows:
(14)
i,j i,j pwk,t = wk|k−1,t − pi,j k,t ,
(15)
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i,j i,j pi,j k,t = λT rendk,t + βP Vk,t ,
where λ and β are the penalty coefficients, pi,j k,t is the penalty strength obtained i,j by the combined penalty formula, wk|k−1,t is the weight of the estimated value
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i,j obtained by the ith target at time k-1 and the j th observation at time k, pwk,t
is the weighted value after the penalty. From equations (11)-(15), p•k and pwk•
of all estimates at time k can be obtained. Using the weights of the estimated
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values at time k, p•k and pwk• , three weighting matrices are constructed as shown in Figure (2), Figure (3) and Figure(4) respectively. Where Jk|k−1 is the number
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of targets that exist in the k-1 time and participate in the iteration of k, and Nm|k is the number of all observations at time k. In the weight matrix of the estimated value, the ith row is the weight values obtained by updating
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the ith target with all the observations, and the j th column is weight values obtained by updating each target and the j th observations. The elements in the
penalty strength matrix and the penalized weight matrix respectively represent
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Figure 2: Weight matrix of estimated values.
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Figure 3: Matrix of penalty strength.
Figure 4: Weight matrix composed of penalized estimates.
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the penalty strength for the position weight and the weight value after the penalty in the estimated weight matrix. 230
When the element in the penalty strength matrix is greater than the given threshold ξ, the value of the corresponding position in the estimated weight
i,j pwk|k−1,t
=
k|k−1,t
, pi,j k,t < ξ
0
, others
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pwi,j
(16)
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matrix and the penalty weighted matrix is set to 0, as follows: wi,j , pi,j k,t < ξ k|k−1,t i,j = wk|k−1,t 0 , others
(17)
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Finally, we extract the target with the highest weight value from the estimated weight matrix after the penalty as the real target extracted at time k,
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that is, extract only one target for each type of label. At the same time, all the targets corresponding to the weights of 0 in the weight matrix of the estimated
the next iteration. 240
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value will be deleted, and the remaining targets will be used as the input for
The proposed combined penalized weights based GM-PHD for point target tracking is shown in Algorithm 1.
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3.4. Iterative steps of the proposed algorithm After the update step of the GM-PHD filter, the penalty weight matrix
and the penalized weight matrix are obtained by penalizing the weights of all
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the estimated values using the combined penalized weights algorithm proposed in this paper. Finally, the penalty strength matrix is used to filter out some
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targets with excessive penalty, and the target with the largest weight value in the penalty weight matrix is output as the final tracking result, so as to improve the anti-interference ability of GM-PHD filtering on the star-like noise in the
250
sky background. The iterative steps of the improved GM-PHD filtering are as follows: Step 1: Initializing 14
Algorithm 1 Combined penalized weights based GM-PHD for tracking Require: i,j Xk|k−1,t
k|k−1
i , Xk−1,t (i ∈ Jk|k−1 and j ∈ Nm|k ). k−1
Ensure: xk,t 1: tk|k−1 ← tk−1 2: Get the number of
i,j Xk|k−1,t
k|k−1
i and Xk−1,t assigned to nowLength and k−1
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lastLength respectively. 3: if nowLength 6= 0 then
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if lastLength 6= 0 then Initialize the template number N umik−1,tk|k−1 as 5.
5: 6:
end if
7:
for i ← 1 to lastLength do for j ← 1 to nowLength do
8:
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4:
i,j i,j Get pwk|k−1,t , wk|k−1,t through formula (11)-(17).
9:
i,j Put pwk|k−1,t in set P Wk|k−1,t
11:
i,j Put wk|k−1,t in set Wk|k−1,t
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end for
12: 13:
end for
14:
Output the index of the largest weight in the set P Wk|k−1,t and assign it to m and n.
i,j Extract the final result xk,t according to index m and n from Xk|k−1,t
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15:
k|k−1
.
16: end if
v0 (x) =
N0 X
w0i N (x; mi0 , P0i ),
(18)
i=0
At time step k=0, the intensity function of the state set is initialized by N
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mixed Gaussian components, where w0i , mi0 and P0i are the weight, mean and
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covariance of each Gaussian component in the initial model, respectively [7]. The number of the motion template for each target is initialized to N umi0 =5. Step 2: Forecasting
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Z vk|k−1 (x) =
ps,k (ς)fk|k−1 (x|ς)vk−1 (ς)d(ς) (19)
Z +
βk|k−1 (x|ς)vk−1 (ς)d(ς) + γk(x), tik|k−1 = tik−1 = ti ,
(20)
where ps,k (·) is the probability of survival of the target at time k, fk|k−1 (·|·) is
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the intensity function of the surviving target, βk|k−1 (·|·) is the intensity function of the derived target, and γk(·) is the intensity function of the random finite
set Γk of the new target at time k. ti is the ith target label, and each surviving
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target’s label is derived from its original label, and each new target and derived target will be given a new label respectively.
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Step 3: Updating
In the update step, the predicted value is updated not only by the measurement
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information of the target, clutter, etc., but also by the measurement information of all the stars in the observation area. The proposed method updates the target
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state at time k in a conventional update manner.
vk (x) = 1 − pD ,k (x) vk—k - 1 (x)+ X PD,k (x)gk (z|x)vk|k−1 (x) R , kk(z) + PD,k (ξ)gk (z|ξ)vk|k−1 (ξ)d(ξ) z∈Z
(21)
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i ti,j k = tk—k - 1 ,
(22)
where PD ,k (·) is the probability of detection of the target at k, gk (·|·) is the th observed intensity function of the target, and ti,j Gaussian k is the label of the j
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component of the ith target.
Step 4: Pruning and merging To prevent the Gaussian component from growing endlessly with iteration, GM-
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PHD filter uses the given pruning threshold τ and merging threshold U to pruning and merging Gaussian components, respectively. The specific way is as follows: 16
Pruning is performed when the following conditions are not met: Ik = {i = 1, ..., Jk |wki > τ },
(23)
j = arg max(wkr ) ,
(24)
M = {i ∈ Ik |(mjk − mik )(Pkj )−1 (mjk − mik )T ≤ U },
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Merging is performed when the following conditions are met:
r∈Ik
(25)
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where Jk is the total number of predictions at time k, and Ik is the number of predictions that satisfy the pruning threshold. M is a set of targets that satisfy the merge threshold.
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Step 5: Penalizing weights
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4. Experiments and analyses
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This step is described in Algorithm 1.
In order to verify the effectiveness of the proposed algorithm, we selected three representative multi-target tracking algorithms and conducted compara-
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tive experiments on four carefully designed data sets. These three methods are traditional GM-PHD [7], N-SCAN algorithm [20] and GM-PHD-IW [24]. In the experiments, the observation range of each scenario is [-400,400]×[-400, 400]. A linear Gaussian transfer model and an observation model at time k are defined
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as follows:
xk|k−1
t2 0 0 2 t 0 1 0 0 q , xk−1 + 2 k 0 t2 0 1 t 0 t 0 0 1 1 0 0 0 xk + rk , zk = 0 1 0 0
1 0 = 0 0
t
0
17
(26)
(27)
where t is the time step, qk and rk are process noise and measurement noise, 280
respectively, and are set as Qk = diag([0.5, 0.1]) and Rk = diag([0.25, 0.25]). The target detection probability is set to 0.9, the target survival probability is set to 0.99, the pruning threshold is set to τ = 0.01 , the merge threshold is set to U =4, the birth intensity for a new target is Bk = diag([1, 2, 1, 2]), the maximum Gaussian distribution is set to Jmax = 200, and the coefficients for the combined penalized weights algorithm are λ = 0.1 and β = 0.5.
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4.1. Datasets
We used the Tycho-2 catalog to simulate four starry data sets with different complexity. Considering that the moving curve of the target in the scenario
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may be linear or non-linear, a mixture with linear and nonlinear forms is used to simulate the target motion. In each data set, the number of initialization
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targets is 4, and two new targets appear after running 20 time steps. The data sets with the four different complexity we simulated are as follows:
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(1) Data set A
Dataset A simulates a scenario with target linear motion, as shown in Figure 295
5. The target was interfered by many stars during the movement, especially
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when there are two targets with trajectories intersecting and there is a slowly moving star around them. In general, the mode of target motion in the data set A is single, the noise interference is not serious and the difficulty of tracking is generally easy.
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(2) Data set B
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Data set B simulates a scenario in which the target moves in a linear or nonlinear manner. In Figure 6, except for the intermediate target passing closely to the star during the motion, almost all of the targets are rarely interfered by the star during the movement. Therefore, in this scenario, the difficulty of tracking
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is generally moderate. (3) Data set C 18
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Figure 5: Data set A (containing 5 targets, moving 50 time steps).
Figure 6: Data set B (containing 5 targets, moving 50 time steps).
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Figure 7: Data set C (containing 5 targets, moving 50 time steps).
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In dataset C, we simulated a more complex motion scenario based on dataset B, as shown in Figure 7. It can be seen in Figure 7 that the target on the right
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is interfered by a star at the beginning of the birth position; the target in the middle position changes greatly when moving through the star; the target on the lower left is interfered by multiple stars. Therefore, the simulation data in this scenario is relatively diverse and the tracking scenario is relatively complicated,
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and the difficulty of tracking is more high. (4) Data set D
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Based on the data set C, we further increase the complexity of the scenario to form the data set D. In this scenario, the number of stars increases, and the
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distance between the star and the target is more compact. The three targets in the middle of Figure 8 are interfered by at least two stars in motion. The target in the left part of Figure 8 has been interfered by the two closely spaced stars
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from the birth position. 4.2. Estimation of the number of tracking targets (1) On data set A 20
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Figure 8: Data set D (containing 5 targets, moving 50 time steps).
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The estimation of the target number and target trajectory of four algorithms on data set A are as shown in Figure 9. It can be seen that the tracking error rate of the traditional GM-PHD algorithm is the highest. When the target
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did not pass closely to the star (the two targets on the left side of Figure 9(a)), the GM-PHD algorithm could effectively keep the correct tracking of
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the target. While when the target passed closely to a star during motion (the three targets on the right side of Figure 9(a)), it generated a false-derived new
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target, which is consistent with the problem (1) mentioned in section 2.3. The N-SCAN and the GM-PHD-IW had stronger anti-interference ability against the stellar than the GM-PHD in this scenario. However, when there were two targets with trajectories intersecting, both algorithms had a very obvious missed
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detection. Compared with the other three methods, the proposed algorithm had
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very good tracking results and number estimation, which indicates that the joint penalty strategy proposed in this paper is effective for linear tracking of multiple targets in starry-sky scenarios, and can effectively resist the stable interferences of stellars. (2) On data set B 21
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The results of various algorithms on data set B for tracking are shown in Figure 10. As can be seen from the figure, when the middle target passed the star, the GM-PHD algorithm produced two new targets. The N-SCAN algorithm incorrectly treated the star as a real target for continuous tracking while losing the real target. Therefore, the result of GM-PHD is that the estimated number
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of targets is erroneously increased, and the result of N-Scan is that although
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the estimated number of targets remains unchanged, the real target has been
lost. which is consistent with the problem (3) mentioned in section 2.3. So, for
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the tracking scenario simulated by data B, the GM-PHD-IW algorithm and the
algorithm proposed in this paper have obtained good tracking results, and there are almost no target loss and tracking errors.
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(3) On data set C
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The scenario simulated by data set C is more difficult than A and B, and the experimental results are shown in Figure 11. In this tracking scenario, GMPHD, N-SCAN and GM-PHD-IW had the situation described in problem (2) of section 2.3 due to the presence of a star at the birth position of the target on the
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right side of the graph. The result is that these algorithms directly misidentify stars as real targets and track them. Moreover, the target in the lower left
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corner of the figure was interfered by five stars. In contrast, although GM-PHD produced 5 derived new targets, N-SCAN and GM-PHD-IW had very serious
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missing detections. However, the algorithm proposed in this paper still showed very good tracking ability. (4) On data set D
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In data set D, the scenario is more complex, the target signal-to-noise ratio is lower, and tracking is extremely difficult. The experimental results of all the
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algorithms are shown in Figure 12. By analyzing the experimental results, we found that in such a data set that is more in line with the real tracking scenario, the GM-PHD algorithm output all the stars that the target passed during the motion as derived new targets. The N-SCAN and the GM-PHD-IW had showed
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very serious missing detections when the target met the star. However, the 370
algorithm proposed in this paper dealt with the problems described in Section 2.3 very robustly, and could stably and reliably track multiple targets. 4.3. OSPA distance This paper also introduced the Optimal Sub-Pattern Assignment (OSPA)
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[30] to evaluate the overall performance of the algorithm. The OSPA distance is an evaluation method for the overall performance of the target tracking result,
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which can be used to measure the error between the real track and the estimated track. The OSPA distance separates the error between the real track and the estimated track into two parts: the distance error and the associated error. The
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OSPA distance also uses two adjustable parameters, the distance sensitivity parameter p and the associated sensitivity parameter c. In this experiment
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c=70, p=2. The OSPA distance is defined as shown in equation (28):
|Xk | h 1 X ˆ π(i) ))p (dc (xi , X min ˆ π∈π ˆ | |X |Xk | k i=1 i1/p ˆ k | − |Xk |) + cp × (|X ,
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ˆk ) = ospac,p (Xk , X
(28)
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Figure 13 shows the OSAC distance curves for each algorithm on four different complexity data sets. Through observation and analysis, the OSPA curve of the GM-PHD algorithm results had a large value on the four data
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sets. This is because that GM-PHD mistakenly identified the stars as derived new targets. On dataset A and dataset B, the OSPA curves of the N-SCAN and GM-PHD-IW results were almost no different from the proposed algorith-
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m. This is because the tracking scenarios simulated by dataset A and dataset B are relatively simple, and existing algorithms have certain feasibility in this
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scenario. In addition, the N-SCAN algorithm was better at processing data set A, while GM-PHD-IW was better at processing data set B. On dataset C and dataset D, the GM-PHD, N-SCAN, and GM-PHD-IW algorithms did not solve the three problems described in Section 2.3. This is because the scenarios
23
simulated by dataset C and dataset D are more complex, and the targets are 395
interfered by factors such as stars and clutter in their motion, which is more in line with the actual tracking scenario. The algorithm proposed in this paper has obtained more robust and reliable tracking results by using the idea of combined penalized weights.
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5. Conclusions
In traditional tracking scenarios, the focus and difficulty of correct tracking of
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the target are how to estimate from a series of measurements that are interfered by clutter and noise to get the state and number of targets. In the starry-
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sky scenario, the focus and difficulty of correctly tracking the target are how to deal with the anomaly of the weight caused by the star. In order to solve 405
this problem, the existing algorithms start from the aspects of penalty weights
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and weight redistribution and propose various algorithms for solving weight anomalies, such as PGM-PHD, CGM-PHD, CPGM-PHD, etc. However, these
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algorithms are good at dealing with weight anomalies caused by tight spaces, and can not solve the problem of derived new targets caused by stars in the starry410
sky scenario. This paper proposes a simple and effective combined penalized weights based on the idea of penalizing weights, which makes the proposed
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algorithm maintain a very good tracking effect in different complexity tracking environments.
We constructed four typical scenarios with different background complexity,
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different target signal-to-noise ratios, and different tracking difficulty. Detailed comparative experimental analysis was carried out with three typical methods,
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GM-PHD, N-SCAN, and GM-PHD-IW. The experimental results show that the GM-PHD algorithm does not have any discriminating power for stars. When there are stars around the target, the algorithm will indiscriminately identify
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the star as the derived new target of the real target, so its tracking ability is the worst of the four algorithms. The N-SCAN algorithm and the GM-PHD-IW algorithm have better tracking capabilities in a simple tracking scenario but are
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not very stable. And in complex tracking scenarios, the tracking capabilities of these two algorithms may drop dramatically. The algorithm proposed in this 425
paper shows strong stability and accuracy in both simple and complex tracking scenarios.
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Acknowledgment This work has been partially supported by the Ministry of education Chunhui project(Grant No: Z2016149), the Key scientific research fund of Xihua U-
niversity (Grant No: Z17134), Xihua University Key Laboratory Development
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Program (Grant No: szjj2017-065), Xihua University Graduate Innovation Fund
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Research Project (Grant No: ycjj2018044) and Sichuan science and technology program (Grant No: 2019YFG0108).
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References
[1] Y. Bar-Shalom, P. K. Willett, X. Tian, Tracking and data fusion, YBS
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publishing Storrs, CT, USA:, 2011.
[2] Y. Wang, H. Meng, Y. Liu, X. Wang, Collaborative penalized gaussian mixture phd tracker for close target tracking, signal processing 102 (2014)
440
ur na
1–15.
[3] R. Mahler, D. Hall, J. Llinas, Random set theory for target tracking and identification (2001).
[4] D. J. Daley, D. Vere-Jones, An introduction to the theory of point process-
Jo
es: volume II: general theory and structure, Springer Science & Business Media, 2007.
445
[5] R. P. Mahler, Multitarget bayes filtering via first-order multitarget moments, IEEE Transactions on Aerospace and Electronic systems 39 (4) (2003) 1152–1178.
25
[6] R. Mahler, Phd filters of higher order in target number, IEEE Transactions on Aerospace and Electronic systems 43 (4) (2007) 1523–1543. 450
[7] B.-N. Vo, W.-K. Ma, The gaussian mixture probability hypothesis density filter, IEEE Transactions on signal processing 54 (11) (2006) 4091–4104. [8] T. Zajic, R. P. Mahler, Particle-systems implementation of the phd mul-
of
titarget tracking filter, in: Signal Processing, Sensor Fusion, and Target
Recognition XII, Vol. 5096, International Society for Optics and Photonics, 2003, pp. 291–299.
ro
455
[9] Y. Wang, Z. Jing, S. Hu, J. Wu, Detection-guided multi-target bayesian
-p
filter, Signal Processing 92 (2) (2012) 564–574.
[10] B. Ristic, D. Clark, B.-N. Vo, B.-T. Vo, Adaptive target birth intensity
460
re
for phd and cphd filters, IEEE Transactions on Aerospace and Electronic Systems 48 (2) (2012) 1656–1668.
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[11] H. Zhang, J. Wang, B. Ye, Y. Zhang, A gm-phd filter for new appearing targets tracking, in: 2013 6th International Congress on Image and Signal Processing (CISP), Vol. 2, IEEE, 2013, pp. 1153–1159.
465
ur na
[12] K. Panta, D. E. Clark, B.-N. Vo, Data association and track management for the gaussian mixture probability hypothesis density filter, IEEE Transactions on Aerospace and Electronic Systems 45 (3) (2009) 1003–1016.
[13] S. Reuter, B.-T. Vo, B.-N. Vo, K. Dietmayer, The labeled multi-bernoulli filter, IEEE Transactions on Signal Processing 62 (12) (2014) 3246–3260.
Jo
[14] B.-T. Vo, B.-N. Vo, Labeled random finite sets and multi-object conjugate
470
priors, IEEE Transactions on Signal Processing 61 (13) (2013) 3460–3475.
[15] B.-N. Vo, B.-T. Vo, D. Phung, Labeled random finite sets and the bayes multi-target tracking filter, IEEE Transactions on Signal Processing 62 (24) (2014) 6554–6567.
26
[16] H. Zhang, J. Yang, H. Ge, L. Yang, An improved gm-phd tracker with track 475
management for multiple target tracking, in: 2015 International Conference on Control, Automation and Information Sciences (ICCAIS), IEEE, 2015, pp. 185–190. [17] F. Yang, Y. Su, W. Zhang, X. Yao, A track extraction method based
480
of
on target topology for phd filter, in: 2016 31st Youth Academic Annual Conference of Chinese Association of Automation (YAC), IEEE, 2016, pp.
ro
449–453.
[18] B.-N. Vo, B.-T. Vo, R. P. Mahler, A closed form solution to the probability hypothesis density smoother, in: 2010 13th International Conference on
485
-p
Information Fusion, IEEE, 2010, pp. 1–8.
[19] B.-N. Vo, B.-T. Vo, R. P. Mahler, Closed-form solutions to forward–
re
backward smoothing, IEEE Transactions on Signal Processing 60 (1) (2011) 2–17.
lP
[20] M. Yazdian-Dehkordi, Z. Azimifar, Novel n-scan gm-phd-based approach for multi-target tracking, IET Signal Processing 10 (5) (2016) 493–503. 490
[21] M. Yazdian-Dehkordi, Z. Azimifar, M. Masnadi-Shirazi, Competitive gaus-
ur na
sian mixture probability hypothesis density filter for multiple target tracking in the presence of ambiguity and occlusion, IET Radar, Sonar & Navigation 6 (4) (2012) 251–262.
[22] M. Yazdian-Dehkordi, Z. Azimifar, M. A. Masnadi-Shirazi, Penalized gaus-
495
sian mixture probability hypothesis density filter for multiple target track-
Jo
ing, Signal Processing 92 (5) (2012) 1230–1242.
[23] H. Zhang, H. Ge, J. Yang, Improved gaussian mixture phd for close multitarget tracking, in: 2014 IEEE 7th Joint International Information Technology and Artificial Intelligence Conference, IEEE, 2014, pp. 311–315.
27
500
[24] H. Zhang, H. Ge, J. Yang, Y. Yuan, A gm-phd algorithm for multiple target tracking based on false alarm detection with irregular window, Signal Processing 120 (2016) 537–552. [25] X. Zhou, Y. Tang, J. Yang, Z. Xie, S. Chen, Penalized gaussian mixture probability hypothesis density tracker with multi-feature fusion, in: 2014 IEEE International Conference on Robotics and Biomimetics (RO-
of
505
BIO 2014), IEEE, 2014, pp. 1415–1420.
ro
[26] H. D. Hesar, H. A. Moghaddam, A. Safari, P. Eftekhari-Yazdi, Multiple sperm tracking in microscopic videos using modified gm-phd filter, Machine
510
-p
Vision and Applications 29 (3) (2018) 433–451.
[27] R. P. Mahler, ” statistics 101” for multisensor, multitarget data fusion,
re
IEEE Aerospace and Electronic Systems Magazine 19 (1) (2004) 53–64. [28] R. Mahler, An introduction to multisource-multitarget statistics and ap-
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plications, Lockheed Martin, 2000.
[29] B.-N. Vo, W.-K. Ma, A closed-form solution for the probability hypothesis 515
density filter, in: 2005 7th International Conference on Information Fusion,
ur na
Vol. 2, IEEE, 2005, pp. 8–pp.
[30] D. Schuhmacher, B.-T. Vo, B.-N. Vo, A consistent metric for performance evaluation of multi-object filters, IEEE transactions on signal processing
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56 (8) (2008) 3447–3457.
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(a) Tracking trajectory of the GM-PHD
(b) Number estimation of the GM-
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(d) Number estimation of the N-SCAN
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(c) Tracking trajectory of the N-SCAN
(e) Tracking trajectory of the GM-PHD-IW (f) Number estimation of the GM-
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PHD-IW
29 (g) Tracking trajectory of our algorith- (h) Number estimation of our algorithm
m Figure 9: Tracking trajectory and number estimation on data set A.
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(b) Number estimation of the GM-PHD
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(a) Tracking trajectory of the GM-PHD
(d) Number estimation of the N-SCAN
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(c) Tracking trajectory of the N-SCAN
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(e) Tracking trajectory of the GM-PHD-IW (f) Number estimation of the GM-PHD-IW
(g) Tracking trajectory of our algorithm
(h) Number estimation of our algorithm
30 Figure 10: Tracking trajectory and number estimation on data set B.
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(b) Number estimation of the GM-PHD
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(a) Tracking trajectory of the GM-PHD
(d) Number estimation of the N-SCAN
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(c) Tracking trajectory of the N-SCAN
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(e) Tracking trajectory of the GM-PHD-IW (f) Number estimation of the GM-PHD-IW
(g) Tracking trajectory of our algorithm
(h) Number estimation of our algorithm
31 Figure 11: Tracking trajectory and number estimation on data set C.
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(a) Tracking trajectory of the GM-PHD
(d) Number estimation of the N-SCAN
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(c) Tracking trajectory of the N-SCAN
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(e) Tracking trajectory of the GM-PHD-IW (f) Number estimation of the GM-PHD-IW
(g) Tracking trajectory of our algorithm
(h) Number estimation of our algorithm
32 Figure 12: Tracking trajectory and number estimation on data set D.
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(b) OSAC curve on data set B
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(a) OSAC curve on data set A
(c) OSAC curve on data set C
(d) OSAC curve on data set D
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Figure 13: The OSA distance of the compared algorithms on the four data sets.
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PII:
S0030-4026(19)32044-3
DOI:
https://doi.org/10.1016/j.ijleo.2019.164145
Reference:
IJLEO 164145
To appear in:
Optik
Received Date:
28 November 2019
Accepted Date:
27 December 2019
Please cite this article as: Qingqing Luo, Zhisheng Gao, Chunzhi Xie, Combined penalized weights based GM-PHD for point target tracking in starry-sky background, (2020), doi: https://doi.org/10.1016/j.ijleo.2019.164145
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier.
Combined penalized weights based GM-PHD for point target tracking in starry-sky background Qingqing Luoa , Zhisheng Gaoa,, Chunzhi Xiea of Computer and Software Engineering, Xihua University, Chengdu 610039, P. R. China.
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Abstract
It is well known that the point targets observed by the space-based platform are extremely difficult to track due to the interference of a large number of
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stellar targets and background noise. The traditional GM-PHD, due to its probability accumulation, is easy to mistakenly identify the star as a derived new target when the real target passes closely to the star. In order to achieve
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precise tracking of multiple moving point targets for space-based observations in a complex background environment such as a starry sky background, this paper
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proposes a simple and effective combined penalized weights based GM-PHD for filtering. First, a unique label is given for each target to determine which real target the estimate came from. Next, the moving region of target is discretized
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into 10 irregular intervals to construct ten different motion templates, numbered one by one, for discretization of the direction of the target motion. Next, the true target at time k-1 and all the estimated values having the same label at time k are used to calculate the direction penalty factor and the speed penalty factor to obtain the penalty strength matrix and penalized weight matrix, respectively. Finally, the penalized weight matrix is used to output the target with the highest
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weight in each label as the real target at time k. And a given penalty threshold is
used to remove some fuzzy weights with greater penalty in the penalty strength matrix, so as to selectively filter out some less relevant targets to further avoid the stars participating in the next iteration. In order to verify the effectiveness ∗ Corresponding
author Email address: [email protected] (Zhisheng Gao)
Preprint submitted to Journal of LATEX Templates
November 28, 2019
of the proposed algorithm, we used the Tycho-2 catalog to simulate four kinds of starry-sky data sets with different complexity and conducted comparative experiments on each data set. The experimental results show that the proposed algorithm can effectively track multiple space-based infrared weak point targets in complex starry-sky scenarios. Keywords: combined penalize, starry-sky background, space-based
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observation, point target tracking.
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1. Introduction
In traditional tracking scenarios, multi-target tracking refers to estimating
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the state and quantity of a target from a series of measurements that are interfered by noise and clutter [1]. New targets, derived targets, clutter, and noise, 5
etc., will make the number and state of the target change over time. Therefore,
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how to overcome the influence of these factors to maintain the correct tracking of the target has always been the focus and difficulty of multi-target tracking.
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Ideally, there is an assumption that in each time step, one target has only one observation, and one observation can only originate from one target (one10
to-one)[2]. However, due to the influence of noise and clutter, a target often has
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multiple observations, and an observation often comes from multiple targets (many-to-many). And in the starry-sky background, the stars always exist and are easier to be detected, which leads to a greater probability of a many-tomany situation with real targets. However, we find that unlike many-to-many
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cases that occur in traditional scenarios, the predicted values of the target in the starry-sky scenario and the observed values of the star are often calculated to
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be greater than the given pruning threshold. Also, as the number of iterations increases, the weight will gradually increase by more than 0.5 and even exceed the true target. Finally, the star is incorrectly identified as a derived new target
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of the real target and is incorrectly tracked. So how to deal with the falsederived new targets interfered by the stable strong noise (stars) is the problem to be solved in this paper.
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In recent years, based on the random finite set theory [3] and the point process theory [4], the researchers proposed a method of using the probability 25
hypothesis filtering (PHD) filtering to track time-varying targets [5]. The PHD filtering uses the first-order statistic of the multi-target posterior probability distribution to replace the posterior probability distribution of the target, avoiding the traditional data association and effectively reducing the computational cost
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[6]. The Gaussian mixture PHD (GM-PHD) filtering [7] and the Sequential Monte Carlo PHD (SMC-PHD) filtering [8] are two important variants of the
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PHD algorithm. Moreover, the former is suitable for nonlinear non-Gaussian tracking scenarios, and the latter is suitable for linear Gaussian tracking scenarios.
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In the tracking environment, new targets may appear anywhere in the observation area, which allows traditional algorithms to extend the scope of the
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emergence of new targets to the entire observation area. Wang [9] proposed a trajectory initialization to generate the trajectory of a new target at an unknown location. Ristic [10] adopted a measurement driver to adaptively calculate the
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birth intensity of the new target in each time step from the received observation set, which improves the ability of PHD and CPHD filtering to distinguish between the new target and the survival target. In reference [11], based on the
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fact that at least one Gaussian component of each measured value is retained after the conventional pruning and merging, a solution for simply and effectively processing the intensity of the new target was proposed.
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For multi-target tracking problems, many researchers have proposed their
own solutions because traditional filtering cannot identify individual targets. Panta [12] provided a unique label for each target and obtained a set of state
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trajectories for a single target by correlating the state with the target in each time step. Some novel label-based trajectory extraction schemes have also been
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proposed in [13–15], but the problem of identification of trajectories of nearby targets is not well solved. A distribution scheme based on detection of dynamic weights was used to reduce the weight error between neighboring targets and to use labels to form and maintain the track of a single target [16], effectively 3
alleviating the problem of identification of trajectories of nearby targets. More55
over, reference [17] proposed a track extraction method based on target topology information to achieve target recognition and trajectory management. Inspired by the idea that the historical measurement information of the target can increase the tracking accuracy of the algorithm to the target, in 2010, Mahler [18, 19] proposed a Gaussian mixture closed solution for forwardbackward smoothing filtering based on the smoothing scheme. Moreover, a
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simple and effective N-scan algorithm was proposed in [20], which used the set
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of historical weights of the target to overcome the shortcoming of GM-PHD filter that some real targets might be trimmed for the reduction of weights. These
algorithms have improved the utilization of measurement information, but when the weight of the star is greater than the weight of the target, the star would
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replace the real target for output.
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In recent years, some researchers have proposed the concept of weight penalty and weight redistribution for target-intensive space. Yazdian-dehkordi proposed CGM-PHD [21] and PGM-PHD [22], which provided a strategy for weight penalty in the update step and redefined the weight distribution of the target.
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In reference [2], Wang proposed a filter that jointly penalizes the Gaussian mixture probability. The algorithm provides a single target’s track management,
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and also uses the unique label of each target to jointly penalize weights of the dense target, which improves the tracking ability of GM-PHD filtering in a close
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motion space. A new weight update strategy is proposed in [23] and provides a label for each target. The irregular sliding window detection algorithm was proposed in reference [24]. The algorithm uses a new weight redistribution scheme for the weights of the same label in the weight matrix, making the algorithm
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more consistent with the one-to-one assumption.
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For the extended target used in the tracking system, reference [25] proposed
to construct a joint penalty coefficient by using the Euclidean distance and the color similarity between the predicted value and the observed value to realize the penalty for the fuzzy weight. Moreover, the paper [26] has reported that in a human sperm tracking system, in order to improve the ability of the algorithm to 4
85
track sperm in dense space, the GM-PHD filtering pruning and merging process has been introduced and corrected and has achieved a good effect. The above algorithms have a certain effect on improving the correct rate of GM-PHD filtering in dense space targets. However, the abnormal weights brought by the stars are not caused by the dense space between the target-
90
s. Therefore, the existing weight penalty and weight redistribution algorithms
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cannot fully achieve the correct tracking of the targets.
This paper proposes a combined penalized weights strategy for multi-target
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tracking in the background of starry-sky for space-based observations. First,
each target is given a unique label that identifies the updated estimate of the 95
corresponding target and all observations. Through a large number of investi-
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gations, we found that the movement direction and velocity of the target in the background of the space-based and starry-sky are Gaussian. That is, when the
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target is in motion, the gradient of the change in direction and speed does not rise or fall abruptly, and the change is mainly distributed around the center. 100
So next, the concept of the target motion template is introduced. The moving
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direction of the target can be discretized by different motion templates, and then be calculated according to the number of the template at time k-1 and all template numbers with the same label at time k, so that the trend of each
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updated value at time k can be obtained. For the change of the speed of the target, we use the speed at k-1 and the estimated speed of the corresponding label at time k to calculate the Euclidean distance between the speeds. Then, a penalty factor is obtained by combing the direction and the speed and is used to penalize the weighted matrix after the pruning and obtain the penalty strength matrix. Finally, the final tracking target is extracted from the penalty weight matrix, and the penalty weight matrix is used to filter out the heavily penalized
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Gaussian components to prevent strong interference information such as stars from entering the next iteration. The main structure of this paper is as follows. The second part is a de-
scription of some basic knowledge and problems under the new scenario. The 115
third part is the combined penalty algorithm proposed in this paper. The fourth 5
part is the experimental simulation results and analysis and the fifth part is a summary of this work.
2. Background knowledge 2.1. Random finite set In the target tracking, the state and quantity of the target will be dynamical-
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ly changed by factors such as clutter and noise. Mahler proposed a theoretical
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framework for multi-sensor multi-objective data fusion based on finite set statistical theory [27, 28]. In this framework, the state set and the observation set of the target at time k can be defined as a random finite set, as follows:
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A random finite set of states of the target:
Xk = { xk,1 , xk,2 ....xk,M (k) } ∈ F (ℵ),
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(1)
A random finite set of measurements of the target: Zk = {zk,1 , zk,2 ...., zk,N (k) } ∈ F (Z),
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(2)
where ℵ ∈ Rnx and Z ∈ Rnx are the state space and the observation space of the target, respectively. F (ℵ) and F (Z) are the respective collections of all finite
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subsets of ℵ and Z. M (k) and N (k) are the number of targets and the number of measurements at time k, respectively. Combined with the multi-target tracking,
the state set Xk and the measure set Zk of the target at time k are expressed
as follows:
Xk =
[
Sk—k - 1 (x)
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[
Bk—k - 1 (x)
[
Γk ,
(3)
x∈xk−1
x∈xk−1
" Zk = Kk
[
[
# [
Θk(x) ,
(4)
x∈Xk
where Sk—k - 1 is the target random finite set that survives at k-1 and at k. Bk—k - 1 (x) is a random finite set of derived new targets generated at time k-1.
Γk is a random finite set of new targets for k. Kk is a random finite set of 6
S
measurements interfered by noise at time k.
Θk(x) is a random finite set
x∈Xk
of measurements of the target at time k. There are stellar measurement information at every moment in the starry-sky scenario, but these information are different from the factors such as false alarms and clutter. So, we extend on equation (4) to make it more fully representative of the target observations at k: Zk = Kk
[
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[
Θk(x) ,
(5)
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x∈{Xk ,Gk }
where Gk is a random finite set of measurements of stars at k.
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2.2. The GM-PHD filter
Mahler proposed PHD filtering based on a random finite set theory. PHD filter uses the first-order statistic of the multi-target posterior probability dis-
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tribution to replace the posterior probability distribution of the target [5]. Although the PHD filter can greatly reduce the computational complexity of the
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multi-target Bayesian filter, it needs to be approximated by some numericalbased methods. Recently, Vo and Ma provided a closed-form solution for PHD filter by performing a linear Gaussian assumption on a multi-objective model
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and summing the mixed weights of the Gaussian components [29]. Assume that N (c; m, P ) is a Gaussian density function with a mean of m and
a variance of P . In GM-PHD filtering, the transition model and the observation
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model can be modeled as a linear Gaussian, as follows: f (xk |xk−1 ) = N (xk ; Fk−1 , Qk−1 ),
(6)
g(zk |xk ) = N (zk ; Hk xk , Rk ),
(7)
where Fk , Hk , Qk and Rk are the transfer matrix, the observation matrix, the process noise covariance, and the observed noise covariance, respectively. Based on some of the assumptions in [29], the posterior strength at time k-1 can be
7
modeled as a Gaussian mixture using Jk−1 components, as follows: Jk−1
vk−1 (x) =
(i)
X
(i)
(i)
wk−1 N (x; mk−1 ; pk−1 ),
(8)
i=1 (i)
where wk−1 is the weight of the ith Gaussian component.
The prediction
strength at time k can also be modeled as a Gaussian mixture using Jk|k−1
Jk|k−1
vk—k - 1 (x) =
X
(i)
(i)
(i)
wk|k−1 N (x; mk|k−1 ; pk|k−1 ).
(9)
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i=1
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components, as follows:
Then, the posterior intensity at time k can also be expressed as a Gaussian
vk (x) =
X Jk|k−1 X z∈Z
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mixture model, as follows:
j wzj (z)N (x; mjk|k (z), Sk|k )
j=1
(10)
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− (1 − PD,k )vk|k−1 (x)
where PD,k is the probability of detection of the target. More details on GM-
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PHD can be found in [25].
2.3. Problems with existing filtering methods in starry-sky scenario
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(1) False derived targets interfered by the stellar. Ideally, the traditional multi-target tracking algorithm satisfies the one-to-one
145
principle. However, in a real tracking scenario, one target is updated with multiple observations to obtain multiple prediction targets with the same label. However, in a starry-sky scenario, since the stars are easy to detect and exist near the same location at all times, the weight of the target updated with the stellar
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information tends to be greater than the clipping threshold. Especially when the
150
target passes closely to the star, its weight tends to be gradually greater than 0.5 in the process of multiple iterations and is mistakenly recognized as the derived targets of the real target. Therefore, if the algorithm can selectively filter out targets that are most likely to be stars before entering the next iteration, then the generation of derived new targets can be avoided to some extent. 8
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(2) The new target is updated with the measurement information of the star. Another shortcoming of the GM-PHD algorithm is that in the tracking scenario when a new target has not yet appeared and there is a star or false alarm around its possible location, the traditional GM-PHD filter will directly update with these measurement information and the new target to get the wrong target number and state estimation. Irregular sliding windows can be used to improve
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the above situation [24]. The essence of the algorithm is to take advantage of the irregularities and low frequencies that occur in false alarms. However, in the
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starry-sky scenario, the star is always there. So, when there are stars around a new target, the existing algorithms can not completely avoid the problem. (3) The weight of the star is greater than the weight of the real target.
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The algorithm of weight penalty or weight redistribution mainly penalizes and
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normalizes the weights of the same label in the weight matrix, so that the fuzzy weight satisfies the one-to-one principle as much as possible. In the starry-sky
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scenario, the false-derived new target generated by the target through the star has the same label as the target, so the weight of the same label in the weight matrix may be more than 0.5. Thus, when the weight of the star is greater than the weight of the real target, it is impossible to increase the weight of the
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real target and reduce the weight of the star only by weight penalty or weight redistribution. As a result, the algorithm will lose the position information of
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the real target from this moment.
3. Combined penalized weights based GM-PHD
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Through observation, we found that when the target is close to the star, the
traditional algorithm can mistakenly identify the star as a new target of the real target easily. This is because the traditional algorithm obtains a weight
180
that is easily greater than 0.5 after updating the predicted value of the target and the observed value of the star. Considering that the moving direction and speed of the target in the infrared weak point target motion will not jump, that 9
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Figure 1: Ten motion templates and their corresponding ranges.
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Table 1: The specific angular ranges of the ten motion templates and their corresponding values. Template number angle
1 [90 ◦ ,56 ◦ )
2 [56 ◦ ,36 ◦ )
3 [36 ◦ ,20 ◦ )
4 [20 ◦ ,9 ◦ )
5 [9 ◦ ,0 ◦ ]
[0,0.5592)
[0.5592,0.809)
[0.809,0.9396)
[0.9396,0.9876)
[0.9876,1]
6
7
8
9
angle
(0 ◦ ,-9 ◦ )
[-9 ◦ ,-20 ◦ )
[-20 ◦ ,-36 ◦ )
[-36◦ ,-56 ◦ )
10 [-56 ◦ ,-90 ◦ )
range
(-1,-0.9876)
[-0.9876,-0.9363)
[-0.9396,-0.809)
[-0.809,-0.5592)
[-0.5592,0)
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Cosine value Template number
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is, the gradient of the moving direction and speed of the target will not rise or fall too much, and the change is in accordance with the Gaussian distribution. 185
In order to avoid the drawbacks of the traditional algorithm completely relying
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on the weight value and improve the anti-interference ability of the algorithm to the star, the moving direction and the moving speed of the target with the same label are used to design the combined penalized weights and used in the GM-PHD algorithm to penalize the estimated weights of targets.
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3.1. Discrete non-uniform weights of moving
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First, the moving region of targets is discretized into ten motion intervals
with inconsistent ranges, and numbered one by one as shown in Figure 1. The motion template consists of a half-sided unit circle with a quantized degree range of [-90◦ , 90◦ ]. The closer the interval difference is to 0◦ , the smaller the
195
discretization range is. Table 1 shows the specific angular ranges of the ten motion templates and their corresponding values.
10
The moving direction of all estimated targets labeled t at the time of k can be obtained by: →
cos <
i,j i Vk−1,t , Vk—k−1,t
→
i,j i Vk−1,t • Vk—k−1,t >= → → × V i,j V i k−1,t k|k−1,t
(11)
+ τ,
of
i,j i is the where Vk−1,t is speed of the ith real target labeled t at time k -1, Vk|k−1,t
estimated speed obtained after updating the ith target and the j th observation
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i,j i at time k. cos < Vk−1,t , Vk—k−1,t > is the cosine value between the Vki - 1 and i,j the Vk—k−1 , τ is the bias of the random noise in the simulated observations.
Since noise is already included in the measured value, the value of τ may not
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i,j i be additionally specified when calculating cos < Vk−1,t , Vk—k−1,t >.
The cosine value between the speeds can be obtained by the formula (11),
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and then the motion template number of all the estimated targets whose labels are t at the time of k can be obtained by Table 1. Finally, by formula (12), the difference between the template number of the real target labeled t at time
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k-1 and the template number of all estimated targets having the same label at time k can be obtained. This difference is the discrete non-uniform weights of moving proposed in this paper.
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i,j i T rendi,j k,t = N umk−1,t − N umk|k−1,t ,
(12)
where N umik−1,t is the motion template number of the ith target labeled as t at
th time k-1, T rendi,j real target and the k,t is the number difference between the i
205
j th observation labeled t at k time. T rendi,j k,t physically expresses the degree
of change in the moving direction of the target at the time of k compared to
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the moving direction of the real target at the previous moment. T rend•k,t will be used as an item in the joint penalty algorithm to penalize the weights of all estimates with the label t.
210
3.2. Weights consist of the distances between speeds When the multiple estimates with the label t have the same motion template
number, it is not enough to use T rendi,j k,t to achieve the penalty for the estimated 11
target. Therefore, the Euclidean distance between the real target ith labeled t at the time of k-1 and all estimated targets with the same label at time k is used to further penalize the same motion template: i,j i,j − Vki - 1,t |, P Vk,t = | Vk|k−1,t
(13)
i,j i,j where P Vk,t is the Euclidean distance between Vk|k−1,t and Vki - 1,t . The physical
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i,j meaning of P Vk,t is the magnitude of the speed change of all the estimated
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targets of t at the time of k with respect to the unique real target at time k-1. 3.3. Combined penalized weights
i,j P Vk,t and T rendi,j k,t are combined to penalize the weights of the estimates
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in the traditional algorithm. The combined approach and the penalized the weights are as follows:
(14)
i,j i,j pwk,t = wk|k−1,t − pi,j k,t ,
(15)
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i,j i,j pi,j k,t = λT rendk,t + βP Vk,t ,
where λ and β are the penalty coefficients, pi,j k,t is the penalty strength obtained i,j by the combined penalty formula, wk|k−1,t is the weight of the estimated value
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i,j obtained by the ith target at time k-1 and the j th observation at time k, pwk,t
is the weighted value after the penalty. From equations (11)-(15), p•k and pwk•
of all estimates at time k can be obtained. Using the weights of the estimated
220
values at time k, p•k and pwk• , three weighting matrices are constructed as shown in Figure (2), Figure (3) and Figure(4) respectively. Where Jk|k−1 is the number
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of targets that exist in the k-1 time and participate in the iteration of k, and Nm|k is the number of all observations at time k. In the weight matrix of the estimated value, the ith row is the weight values obtained by updating
225
the ith target with all the observations, and the j th column is weight values obtained by updating each target and the j th observations. The elements in the
penalty strength matrix and the penalized weight matrix respectively represent
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Figure 2: Weight matrix of estimated values.
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Figure 3: Matrix of penalty strength.
Figure 4: Weight matrix composed of penalized estimates.
13
the penalty strength for the position weight and the weight value after the penalty in the estimated weight matrix. 230
When the element in the penalty strength matrix is greater than the given threshold ξ, the value of the corresponding position in the estimated weight
i,j pwk|k−1,t
=
k|k−1,t
, pi,j k,t < ξ
0
, others
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pwi,j
(16)
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matrix and the penalty weighted matrix is set to 0, as follows: wi,j , pi,j k,t < ξ k|k−1,t i,j = wk|k−1,t 0 , others
(17)
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Finally, we extract the target with the highest weight value from the estimated weight matrix after the penalty as the real target extracted at time k,
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that is, extract only one target for each type of label. At the same time, all the targets corresponding to the weights of 0 in the weight matrix of the estimated
the next iteration. 240
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value will be deleted, and the remaining targets will be used as the input for
The proposed combined penalized weights based GM-PHD for point target tracking is shown in Algorithm 1.
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3.4. Iterative steps of the proposed algorithm After the update step of the GM-PHD filter, the penalty weight matrix
and the penalized weight matrix are obtained by penalizing the weights of all
245
the estimated values using the combined penalized weights algorithm proposed in this paper. Finally, the penalty strength matrix is used to filter out some
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targets with excessive penalty, and the target with the largest weight value in the penalty weight matrix is output as the final tracking result, so as to improve the anti-interference ability of GM-PHD filtering on the star-like noise in the
250
sky background. The iterative steps of the improved GM-PHD filtering are as follows: Step 1: Initializing 14
Algorithm 1 Combined penalized weights based GM-PHD for tracking Require: i,j Xk|k−1,t
k|k−1
i , Xk−1,t (i ∈ Jk|k−1 and j ∈ Nm|k ). k−1
Ensure: xk,t 1: tk|k−1 ← tk−1 2: Get the number of
i,j Xk|k−1,t
k|k−1
i and Xk−1,t assigned to nowLength and k−1
of
lastLength respectively. 3: if nowLength 6= 0 then
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if lastLength 6= 0 then Initialize the template number N umik−1,tk|k−1 as 5.
5: 6:
end if
7:
for i ← 1 to lastLength do for j ← 1 to nowLength do
8:
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4:
i,j i,j Get pwk|k−1,t , wk|k−1,t through formula (11)-(17).
9:
i,j Put pwk|k−1,t in set P Wk|k−1,t
11:
i,j Put wk|k−1,t in set Wk|k−1,t
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10:
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end for
12: 13:
end for
14:
Output the index of the largest weight in the set P Wk|k−1,t and assign it to m and n.
i,j Extract the final result xk,t according to index m and n from Xk|k−1,t
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15:
k|k−1
.
16: end if
v0 (x) =
N0 X
w0i N (x; mi0 , P0i ),
(18)
i=0
At time step k=0, the intensity function of the state set is initialized by N
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mixed Gaussian components, where w0i , mi0 and P0i are the weight, mean and
255
covariance of each Gaussian component in the initial model, respectively [7]. The number of the motion template for each target is initialized to N umi0 =5. Step 2: Forecasting
15
Z vk|k−1 (x) =
ps,k (ς)fk|k−1 (x|ς)vk−1 (ς)d(ς) (19)
Z +
βk|k−1 (x|ς)vk−1 (ς)d(ς) + γk(x), tik|k−1 = tik−1 = ti ,
(20)
where ps,k (·) is the probability of survival of the target at time k, fk|k−1 (·|·) is
260
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the intensity function of the surviving target, βk|k−1 (·|·) is the intensity function of the derived target, and γk(·) is the intensity function of the random finite
set Γk of the new target at time k. ti is the ith target label, and each surviving
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target’s label is derived from its original label, and each new target and derived target will be given a new label respectively.
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Step 3: Updating
In the update step, the predicted value is updated not only by the measurement
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information of the target, clutter, etc., but also by the measurement information of all the stars in the observation area. The proposed method updates the target
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state at time k in a conventional update manner.
vk (x) = 1 − pD ,k (x) vk—k - 1 (x)+ X PD,k (x)gk (z|x)vk|k−1 (x) R , kk(z) + PD,k (ξ)gk (z|ξ)vk|k−1 (ξ)d(ξ) z∈Z
(21)
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k
i ti,j k = tk—k - 1 ,
(22)
where PD ,k (·) is the probability of detection of the target at k, gk (·|·) is the th observed intensity function of the target, and ti,j Gaussian k is the label of the j
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component of the ith target.
Step 4: Pruning and merging To prevent the Gaussian component from growing endlessly with iteration, GM-
270
PHD filter uses the given pruning threshold τ and merging threshold U to pruning and merging Gaussian components, respectively. The specific way is as follows: 16
Pruning is performed when the following conditions are not met: Ik = {i = 1, ..., Jk |wki > τ },
(23)
j = arg max(wkr ) ,
(24)
M = {i ∈ Ik |(mjk − mik )(Pkj )−1 (mjk − mik )T ≤ U },
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Merging is performed when the following conditions are met:
r∈Ik
(25)
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where Jk is the total number of predictions at time k, and Ik is the number of predictions that satisfy the pruning threshold. M is a set of targets that satisfy the merge threshold.
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275
Step 5: Penalizing weights
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4. Experiments and analyses
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This step is described in Algorithm 1.
In order to verify the effectiveness of the proposed algorithm, we selected three representative multi-target tracking algorithms and conducted compara-
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tive experiments on four carefully designed data sets. These three methods are traditional GM-PHD [7], N-SCAN algorithm [20] and GM-PHD-IW [24]. In the experiments, the observation range of each scenario is [-400,400]×[-400, 400]. A linear Gaussian transfer model and an observation model at time k are defined
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as follows:
xk|k−1
t2 0 0 2 t 0 1 0 0 q , xk−1 + 2 k 0 t2 0 1 t 0 t 0 0 1 1 0 0 0 xk + rk , zk = 0 1 0 0
1 0 = 0 0
t
0
17
(26)
(27)
where t is the time step, qk and rk are process noise and measurement noise, 280
respectively, and are set as Qk = diag([0.5, 0.1]) and Rk = diag([0.25, 0.25]). The target detection probability is set to 0.9, the target survival probability is set to 0.99, the pruning threshold is set to τ = 0.01 , the merge threshold is set to U =4, the birth intensity for a new target is Bk = diag([1, 2, 1, 2]), the maximum Gaussian distribution is set to Jmax = 200, and the coefficients for the combined penalized weights algorithm are λ = 0.1 and β = 0.5.
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4.1. Datasets
We used the Tycho-2 catalog to simulate four starry data sets with different complexity. Considering that the moving curve of the target in the scenario
290
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may be linear or non-linear, a mixture with linear and nonlinear forms is used to simulate the target motion. In each data set, the number of initialization
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targets is 4, and two new targets appear after running 20 time steps. The data sets with the four different complexity we simulated are as follows:
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(1) Data set A
Dataset A simulates a scenario with target linear motion, as shown in Figure 295
5. The target was interfered by many stars during the movement, especially
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when there are two targets with trajectories intersecting and there is a slowly moving star around them. In general, the mode of target motion in the data set A is single, the noise interference is not serious and the difficulty of tracking is generally easy.
300
(2) Data set B
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Data set B simulates a scenario in which the target moves in a linear or nonlinear manner. In Figure 6, except for the intermediate target passing closely to the star during the motion, almost all of the targets are rarely interfered by the star during the movement. Therefore, in this scenario, the difficulty of tracking
305
is generally moderate. (3) Data set C 18
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Figure 5: Data set A (containing 5 targets, moving 50 time steps).
Figure 6: Data set B (containing 5 targets, moving 50 time steps).
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Figure 7: Data set C (containing 5 targets, moving 50 time steps).
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In dataset C, we simulated a more complex motion scenario based on dataset B, as shown in Figure 7. It can be seen in Figure 7 that the target on the right
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is interfered by a star at the beginning of the birth position; the target in the middle position changes greatly when moving through the star; the target on the lower left is interfered by multiple stars. Therefore, the simulation data in this scenario is relatively diverse and the tracking scenario is relatively complicated,
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and the difficulty of tracking is more high. (4) Data set D
315
Based on the data set C, we further increase the complexity of the scenario to form the data set D. In this scenario, the number of stars increases, and the
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distance between the star and the target is more compact. The three targets in the middle of Figure 8 are interfered by at least two stars in motion. The target in the left part of Figure 8 has been interfered by the two closely spaced stars
320
from the birth position. 4.2. Estimation of the number of tracking targets (1) On data set A 20
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Figure 8: Data set D (containing 5 targets, moving 50 time steps).
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The estimation of the target number and target trajectory of four algorithms on data set A are as shown in Figure 9. It can be seen that the tracking error rate of the traditional GM-PHD algorithm is the highest. When the target
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325
did not pass closely to the star (the two targets on the left side of Figure 9(a)), the GM-PHD algorithm could effectively keep the correct tracking of
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the target. While when the target passed closely to a star during motion (the three targets on the right side of Figure 9(a)), it generated a false-derived new
330
target, which is consistent with the problem (1) mentioned in section 2.3. The N-SCAN and the GM-PHD-IW had stronger anti-interference ability against the stellar than the GM-PHD in this scenario. However, when there were two targets with trajectories intersecting, both algorithms had a very obvious missed
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detection. Compared with the other three methods, the proposed algorithm had
335
very good tracking results and number estimation, which indicates that the joint penalty strategy proposed in this paper is effective for linear tracking of multiple targets in starry-sky scenarios, and can effectively resist the stable interferences of stellars. (2) On data set B 21
340
The results of various algorithms on data set B for tracking are shown in Figure 10. As can be seen from the figure, when the middle target passed the star, the GM-PHD algorithm produced two new targets. The N-SCAN algorithm incorrectly treated the star as a real target for continuous tracking while losing the real target. Therefore, the result of GM-PHD is that the estimated number
345
of targets is erroneously increased, and the result of N-Scan is that although
of
the estimated number of targets remains unchanged, the real target has been
lost. which is consistent with the problem (3) mentioned in section 2.3. So, for
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the tracking scenario simulated by data B, the GM-PHD-IW algorithm and the
algorithm proposed in this paper have obtained good tracking results, and there are almost no target loss and tracking errors.
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(3) On data set C
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The scenario simulated by data set C is more difficult than A and B, and the experimental results are shown in Figure 11. In this tracking scenario, GMPHD, N-SCAN and GM-PHD-IW had the situation described in problem (2) of section 2.3 due to the presence of a star at the birth position of the target on the
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355
right side of the graph. The result is that these algorithms directly misidentify stars as real targets and track them. Moreover, the target in the lower left
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corner of the figure was interfered by five stars. In contrast, although GM-PHD produced 5 derived new targets, N-SCAN and GM-PHD-IW had very serious
360
missing detections. However, the algorithm proposed in this paper still showed very good tracking ability. (4) On data set D
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In data set D, the scenario is more complex, the target signal-to-noise ratio is lower, and tracking is extremely difficult. The experimental results of all the
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algorithms are shown in Figure 12. By analyzing the experimental results, we found that in such a data set that is more in line with the real tracking scenario, the GM-PHD algorithm output all the stars that the target passed during the motion as derived new targets. The N-SCAN and the GM-PHD-IW had showed
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very serious missing detections when the target met the star. However, the 370
algorithm proposed in this paper dealt with the problems described in Section 2.3 very robustly, and could stably and reliably track multiple targets. 4.3. OSPA distance This paper also introduced the Optimal Sub-Pattern Assignment (OSPA)
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[30] to evaluate the overall performance of the algorithm. The OSPA distance is an evaluation method for the overall performance of the target tracking result,
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which can be used to measure the error between the real track and the estimated track. The OSPA distance separates the error between the real track and the estimated track into two parts: the distance error and the associated error. The
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OSPA distance also uses two adjustable parameters, the distance sensitivity parameter p and the associated sensitivity parameter c. In this experiment
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c=70, p=2. The OSPA distance is defined as shown in equation (28):
|Xk | h 1 X ˆ π(i) ))p (dc (xi , X min ˆ π∈π ˆ | |X |Xk | k i=1 i1/p ˆ k | − |Xk |) + cp × (|X ,
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ˆk ) = ospac,p (Xk , X
(28)
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Figure 13 shows the OSAC distance curves for each algorithm on four different complexity data sets. Through observation and analysis, the OSPA curve of the GM-PHD algorithm results had a large value on the four data
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sets. This is because that GM-PHD mistakenly identified the stars as derived new targets. On dataset A and dataset B, the OSPA curves of the N-SCAN and GM-PHD-IW results were almost no different from the proposed algorith-
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m. This is because the tracking scenarios simulated by dataset A and dataset B are relatively simple, and existing algorithms have certain feasibility in this
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scenario. In addition, the N-SCAN algorithm was better at processing data set A, while GM-PHD-IW was better at processing data set B. On dataset C and dataset D, the GM-PHD, N-SCAN, and GM-PHD-IW algorithms did not solve the three problems described in Section 2.3. This is because the scenarios
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simulated by dataset C and dataset D are more complex, and the targets are 395
interfered by factors such as stars and clutter in their motion, which is more in line with the actual tracking scenario. The algorithm proposed in this paper has obtained more robust and reliable tracking results by using the idea of combined penalized weights.
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5. Conclusions
In traditional tracking scenarios, the focus and difficulty of correct tracking of
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the target are how to estimate from a series of measurements that are interfered by clutter and noise to get the state and number of targets. In the starry-
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sky scenario, the focus and difficulty of correctly tracking the target are how to deal with the anomaly of the weight caused by the star. In order to solve 405
this problem, the existing algorithms start from the aspects of penalty weights
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and weight redistribution and propose various algorithms for solving weight anomalies, such as PGM-PHD, CGM-PHD, CPGM-PHD, etc. However, these
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algorithms are good at dealing with weight anomalies caused by tight spaces, and can not solve the problem of derived new targets caused by stars in the starry410
sky scenario. This paper proposes a simple and effective combined penalized weights based on the idea of penalizing weights, which makes the proposed
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algorithm maintain a very good tracking effect in different complexity tracking environments.
We constructed four typical scenarios with different background complexity,
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different target signal-to-noise ratios, and different tracking difficulty. Detailed comparative experimental analysis was carried out with three typical methods,
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GM-PHD, N-SCAN, and GM-PHD-IW. The experimental results show that the GM-PHD algorithm does not have any discriminating power for stars. When there are stars around the target, the algorithm will indiscriminately identify
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the star as the derived new target of the real target, so its tracking ability is the worst of the four algorithms. The N-SCAN algorithm and the GM-PHD-IW algorithm have better tracking capabilities in a simple tracking scenario but are
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not very stable. And in complex tracking scenarios, the tracking capabilities of these two algorithms may drop dramatically. The algorithm proposed in this 425
paper shows strong stability and accuracy in both simple and complex tracking scenarios.
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Acknowledgment This work has been partially supported by the Ministry of education Chunhui project(Grant No: Z2016149), the Key scientific research fund of Xihua U-
niversity (Grant No: Z17134), Xihua University Key Laboratory Development
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Program (Grant No: szjj2017-065), Xihua University Graduate Innovation Fund
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Research Project (Grant No: ycjj2018044) and Sichuan science and technology program (Grant No: 2019YFG0108).
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References
[1] Y. Bar-Shalom, P. K. Willett, X. Tian, Tracking and data fusion, YBS
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publishing Storrs, CT, USA:, 2011.
[2] Y. Wang, H. Meng, Y. Liu, X. Wang, Collaborative penalized gaussian mixture phd tracker for close target tracking, signal processing 102 (2014)
440
ur na
1–15.
[3] R. Mahler, D. Hall, J. Llinas, Random set theory for target tracking and identification (2001).
[4] D. J. Daley, D. Vere-Jones, An introduction to the theory of point process-
Jo
es: volume II: general theory and structure, Springer Science & Business Media, 2007.
445
[5] R. P. Mahler, Multitarget bayes filtering via first-order multitarget moments, IEEE Transactions on Aerospace and Electronic systems 39 (4) (2003) 1152–1178.
25
[6] R. Mahler, Phd filters of higher order in target number, IEEE Transactions on Aerospace and Electronic systems 43 (4) (2007) 1523–1543. 450
[7] B.-N. Vo, W.-K. Ma, The gaussian mixture probability hypothesis density filter, IEEE Transactions on signal processing 54 (11) (2006) 4091–4104. [8] T. Zajic, R. P. Mahler, Particle-systems implementation of the phd mul-
of
titarget tracking filter, in: Signal Processing, Sensor Fusion, and Target
Recognition XII, Vol. 5096, International Society for Optics and Photonics, 2003, pp. 291–299.
ro
455
[9] Y. Wang, Z. Jing, S. Hu, J. Wu, Detection-guided multi-target bayesian
-p
filter, Signal Processing 92 (2) (2012) 564–574.
[10] B. Ristic, D. Clark, B.-N. Vo, B.-T. Vo, Adaptive target birth intensity
460
re
for phd and cphd filters, IEEE Transactions on Aerospace and Electronic Systems 48 (2) (2012) 1656–1668.
lP
[11] H. Zhang, J. Wang, B. Ye, Y. Zhang, A gm-phd filter for new appearing targets tracking, in: 2013 6th International Congress on Image and Signal Processing (CISP), Vol. 2, IEEE, 2013, pp. 1153–1159.
465
ur na
[12] K. Panta, D. E. Clark, B.-N. Vo, Data association and track management for the gaussian mixture probability hypothesis density filter, IEEE Transactions on Aerospace and Electronic Systems 45 (3) (2009) 1003–1016.
[13] S. Reuter, B.-T. Vo, B.-N. Vo, K. Dietmayer, The labeled multi-bernoulli filter, IEEE Transactions on Signal Processing 62 (12) (2014) 3246–3260.
Jo
[14] B.-T. Vo, B.-N. Vo, Labeled random finite sets and multi-object conjugate
470
priors, IEEE Transactions on Signal Processing 61 (13) (2013) 3460–3475.
[15] B.-N. Vo, B.-T. Vo, D. Phung, Labeled random finite sets and the bayes multi-target tracking filter, IEEE Transactions on Signal Processing 62 (24) (2014) 6554–6567.
26
[16] H. Zhang, J. Yang, H. Ge, L. Yang, An improved gm-phd tracker with track 475
management for multiple target tracking, in: 2015 International Conference on Control, Automation and Information Sciences (ICCAIS), IEEE, 2015, pp. 185–190. [17] F. Yang, Y. Su, W. Zhang, X. Yao, A track extraction method based
480
of
on target topology for phd filter, in: 2016 31st Youth Academic Annual Conference of Chinese Association of Automation (YAC), IEEE, 2016, pp.
ro
449–453.
[18] B.-N. Vo, B.-T. Vo, R. P. Mahler, A closed form solution to the probability hypothesis density smoother, in: 2010 13th International Conference on
485
-p
Information Fusion, IEEE, 2010, pp. 1–8.
[19] B.-N. Vo, B.-T. Vo, R. P. Mahler, Closed-form solutions to forward–
re
backward smoothing, IEEE Transactions on Signal Processing 60 (1) (2011) 2–17.
lP
[20] M. Yazdian-Dehkordi, Z. Azimifar, Novel n-scan gm-phd-based approach for multi-target tracking, IET Signal Processing 10 (5) (2016) 493–503. 490
[21] M. Yazdian-Dehkordi, Z. Azimifar, M. Masnadi-Shirazi, Competitive gaus-
ur na
sian mixture probability hypothesis density filter for multiple target tracking in the presence of ambiguity and occlusion, IET Radar, Sonar & Navigation 6 (4) (2012) 251–262.
[22] M. Yazdian-Dehkordi, Z. Azimifar, M. A. Masnadi-Shirazi, Penalized gaus-
495
sian mixture probability hypothesis density filter for multiple target track-
Jo
ing, Signal Processing 92 (5) (2012) 1230–1242.
[23] H. Zhang, H. Ge, J. Yang, Improved gaussian mixture phd for close multitarget tracking, in: 2014 IEEE 7th Joint International Information Technology and Artificial Intelligence Conference, IEEE, 2014, pp. 311–315.
27
500
[24] H. Zhang, H. Ge, J. Yang, Y. Yuan, A gm-phd algorithm for multiple target tracking based on false alarm detection with irregular window, Signal Processing 120 (2016) 537–552. [25] X. Zhou, Y. Tang, J. Yang, Z. Xie, S. Chen, Penalized gaussian mixture probability hypothesis density tracker with multi-feature fusion, in: 2014 IEEE International Conference on Robotics and Biomimetics (RO-
of
505
BIO 2014), IEEE, 2014, pp. 1415–1420.
ro
[26] H. D. Hesar, H. A. Moghaddam, A. Safari, P. Eftekhari-Yazdi, Multiple sperm tracking in microscopic videos using modified gm-phd filter, Machine
510
-p
Vision and Applications 29 (3) (2018) 433–451.
[27] R. P. Mahler, ” statistics 101” for multisensor, multitarget data fusion,
re
IEEE Aerospace and Electronic Systems Magazine 19 (1) (2004) 53–64. [28] R. Mahler, An introduction to multisource-multitarget statistics and ap-
lP
plications, Lockheed Martin, 2000.
[29] B.-N. Vo, W.-K. Ma, A closed-form solution for the probability hypothesis 515
density filter, in: 2005 7th International Conference on Information Fusion,
ur na
Vol. 2, IEEE, 2005, pp. 8–pp.
[30] D. Schuhmacher, B.-T. Vo, B.-N. Vo, A consistent metric for performance evaluation of multi-object filters, IEEE transactions on signal processing
Jo
56 (8) (2008) 3447–3457.
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(a) Tracking trajectory of the GM-PHD
(b) Number estimation of the GM-
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PHD
(d) Number estimation of the N-SCAN
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(c) Tracking trajectory of the N-SCAN
(e) Tracking trajectory of the GM-PHD-IW (f) Number estimation of the GM-
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PHD-IW
29 (g) Tracking trajectory of our algorith- (h) Number estimation of our algorithm
m Figure 9: Tracking trajectory and number estimation on data set A.
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(b) Number estimation of the GM-PHD
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(a) Tracking trajectory of the GM-PHD
(d) Number estimation of the N-SCAN
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(c) Tracking trajectory of the N-SCAN
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(e) Tracking trajectory of the GM-PHD-IW (f) Number estimation of the GM-PHD-IW
(g) Tracking trajectory of our algorithm
(h) Number estimation of our algorithm
30 Figure 10: Tracking trajectory and number estimation on data set B.
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(b) Number estimation of the GM-PHD
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(a) Tracking trajectory of the GM-PHD
(d) Number estimation of the N-SCAN
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(c) Tracking trajectory of the N-SCAN
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(e) Tracking trajectory of the GM-PHD-IW (f) Number estimation of the GM-PHD-IW
(g) Tracking trajectory of our algorithm
(h) Number estimation of our algorithm
31 Figure 11: Tracking trajectory and number estimation on data set C.
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(b) Number estimation of the GM-PHD
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(a) Tracking trajectory of the GM-PHD
(d) Number estimation of the N-SCAN
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(c) Tracking trajectory of the N-SCAN
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(e) Tracking trajectory of the GM-PHD-IW (f) Number estimation of the GM-PHD-IW
(g) Tracking trajectory of our algorithm
(h) Number estimation of our algorithm
32 Figure 12: Tracking trajectory and number estimation on data set D.
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(b) OSAC curve on data set B
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(a) OSAC curve on data set A
(c) OSAC curve on data set C
(d) OSAC curve on data set D
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Figure 13: The OSA distance of the compared algorithms on the four data sets.
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