Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 629023, 9 pages

http://dx.doi.org/10.1155/2015/629023

## GM-PHD Filter Combined with Track-Estimate Association and Numerical Interpolation

^{1}School of Computer Science, Xi’an Polytechnic University, Xi’an 710048, China^{2}School of Human Ecology, University of Texas at Austin, Austin, TX 78712, USA

Received 29 April 2015; Revised 19 June 2015; Accepted 21 June 2015

Academic Editor: Muhammad N. Akram

Copyright © 2015 Jinguang Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

For the standard Gaussian mixture probability hypothesis density (GM-PHD) filter, the number of targets can be overestimated if the clutter rate is too high or underestimated if the detection rate is too low. These problems seriously affect the accuracy of multitarget tracking for the number and the value of measurements and clutters cannot be distinguished and recognized. Therefore, we proposed an improved GM-PHD filter to tackle these problems. Firstly, a track-estimate association was implemented in the filtering process to detect and remove false-alarm targets. Secondly, a numerical interpolation technique was used to compensate the missing targets caused by low detection rate. At the end of this paper, simulation results were presented to demonstrate the proposed GM-PHD algorithm is more effective in estimating the number and state of targets than the previous ones.

#### 1. Introduction

Multitarget tracking uses multisensor integration to provide a variety of observation data through comprehensive processing and to obtain multitarget state estimation, which has a wide range of civilian and military applications. Mahler introduced the theory of random finite set into the problem of multitarget tracking [1, 2]. Based on this theoretical framework, Mahler presented the probability hypothesis density (PHD) to replace the multitarget posterior probability density for multitarget tracking [3]. The PHD is also known as “first-order statistical moment,” which can recursively estimate the number and state of targets, avoid the problem of data association, and effectively reduce the time complexity. However, the recursion of the PHD filtering involves multiple integrals which may not have the closed solution. Under nonlinear and non-Gaussian conditions, the PHD filter does not have its analytical form [4].

Subsequently, several researchers enhanced the PHD filter by designing the particle PHD filter (also known as the SMC-PHD) suitable for nonlinear and non-Gaussian conditions [5, 6] and the Gaussian mixture PHD (GM-PHD) filter suitable for linear Gaussian condition [7, 8]. However, The PHD recursion propagates cardinality information with the mean of the cardinality distribution. It effectively approximates the cardinality distribution by Poisson distribution. Since the mean and the variance of Poisson distribution are equal, when the number of targets present is large, the estimated error of the PHD filter is large. To solve this problem, PHD filter can be extended to the cardinality probability hypothesis density (CPHD) filter as seen in [9]. The posterior intensity and cardinality posterior probability distribution of multitarget state sets in the CPHD filter can be obtained through recursion at the same time. Subsequently, The Gaussian mixture cardinality probability hypothesis density (GM-CPHD) filter was given in [10, 11]; and the sequential Monte Carlo cardinality probability hypothesis density (SMC-CPHD) filter was presented in [12].

Since the PHD filter can only obtain the target state estimation, it cannot determine which track the state estimation belongs to. To solve this problem, Lin et al. determined whether the estimated state at current time belongs to a previous track through track-estimate association at adjacent time [13]. In [14], the multiple hypotheses tracking approach [15] was used to form tracks among targets in SMC-PHD filter after clutters were removed from measurements. In [16], a new multitarget tracking filter was proposed based on GM-PHD filter, in which the Gaussian items were labeled to distinguish estimates.

In addition, the clutter rate and the detection rate have important influences on effectiveness of tracking algorithms. Juang and Burlina showed the connection between the clutter rate and the false alarm rate [17]. When the clutter rate and the detection rate are unknown, Mahler et al. devised the process of PHD or CPHD to adaptively learn the clutter rate and the detection profile during the filtering and obtained the recursive closed-analytical-formula of PHD or CPHD by mixing the Beta distribution and the Gaussian distribution [18]. It is well known that the GM-PHD filter can track the target accurately only when the clutter rate is low and the detection rate is high. In many practical applications, however, these conditions are hardly satisfied. An excessive clutter rate may cause the number of targets to be overestimated, while a low detection rate may cause the number of targets to be underestimated. All these conditions can lead to estimation errors or poor filtering performance.

In this paper, we proposed a new algorithm (GM-PHD-TANI) by incorporating track-estimate association and numerical interpolation into GM-PHD filter for solving the multitarget tracking problems under the conditions of a high clutter rate and a low detection rate. The algorithm mainly consists of two stages: track-estimate association and numerical interpolation. In the first stage, most clutters can be removed by the track-estimate association to lessen the overestimate problem. In the second stage, undetected targets can be made up to some extent through numerical interpolations so that the underestimate problem can be alleviated.

This paper is organized as follows: Section 2 is the problem formulation, including single target filtering problem, the theory of random finite set, and multitarget tracking model; Section 3 includes the PHD filter and GM-PHD filter; Section 4 is the proposed GM-PHD-TANI algorithm; and the experimental results and analysis are given in Section 5.

#### 2. Problem Description

##### 2.1. Single Target Bayesian Filter

Assume that, in the state space , is the state vector and is the dimension of the state vector. The state transition equation iswhere is a process noise with a Gaussian distribution whose mean is zero and covariance is .

In the measurement space , the measurement vector is and the dimension of the measurement vector is . The measurement equation iswhere is the measurement noise with a Gaussian distribution whose mean is zero and covariance is .

The probability density function of the state , decided by measurements , is the posterior probability density function. According to the prior distribution, , the posterior probability density function can be deduced via the Bayes theory; that is,

contains the relevant information about the state at time . The estimated information can be obtained by minimizing the mean square error of state estimation or by maximizing the posterior probability .

##### 2.2. Multitarget Bayesian Filter Based on RFS

In the RFS (Random Finite Set) theory, both the value of an element and the cardinality of a set are random. In a multitarget tracking problem, the birth, spawning, and disappearance of a target and the total number and the states of all targets change over time. Because of a false alarm, miss detection, clutters, and measurements received by sensors are changeable over time as well. Thus, the state set and the measurement set of the targets are a set of random variables.

Assume that, at time , is the multitarget state set, is the number of targets, is the measurement set, and is the number of measurements. Among and , denotes the th target’s state, and denotes the th target’s measurement:where and denote the state space and the measurement space, respectively. The measurement set includes clutters. Assume the probability density of clutter is , its number obeys Poisson distribution whose mean is , the state of a target continues from time to with probability and disappears with probability , and the evolution of each target is independent. According to the RFS theory, the evolution of a single target can be described as a RFS . The new birth target set at time is expressed as , and the spawned target derived from the survival target is denoted as . The state model of multiple targets is

The received measurement set is the mixture of the target RFS, , with the clutter RFS, ; that is,

Let . The multitarget posterior density function can be obtained recursively from the optimal multitarget Bayesian filtering through the following formulas: where is a proper measure factor in space [6]. Equations (7) involve multiple integrations in space , which are often difficult, sometimes even infeasible, to calculate. Their computational complexity dramatically increases with the increase of the target number. Thus, reasonable approximations become inevitable in practice.

#### 3. Gaussian Mixture Probability Hypothesis Density Filter

As an extension of the Bayes filtering under the framework of RFS, the PHD filter uses the PHD form of multitarget posterior probability density, that is, its first-order moment of estimation, to replace the posterior probability density for estimating target’s number and state recursively. In space , the probability distribution of RFS is , and the first-order moment is denoted as , which is a negative function called the intensity function. For a region in space ,

If denotes the posterior intensity of the multitargets at time , then the predicted intensity at time iswhere is the intensity of the birth target, is the intensity of the spawned target, is the state transition probability density of a single target, and is the survival probability of a target from time to .

The target number at one prediction step iswhere is the number of birth targets; is the number of survival targets; and is the number of spawned targets.

The PHD is updated as follows when the set of measurements is received:where is the intensity of clutter, is the likelihood function of a single measurement, and is the probability of detection. Correspondingly, the target number is updated as follows:

The Gaussian mixture form of a PHD is formed by Gaussian components, that is, . In the linear Gaussian system, the posterior intensity at time can be constructed with Gaussian components in a linear combination with different weights:where is the number of Gaussian components, and , , and , respectively, denote weight, state, and covariance of the th Gaussian component.

Assume that the posterior intensity of multitargets at time is expressed in the following Gaussian mixture form:The PHD of multitargets at time can be also expressed in a Gaussian mixture form:Then, the posterior PHD of multitargets at time can be achieved as follows:

#### 4. GM-PHD Filter with Track-Estimate Association and Numerical Interpolation

##### 4.1. Implementation of GM-PHD

Assume that the state model of targets and the measurement model of sensors are linear Gaussian; that is,where is the state transition matrix, is the covariance matrix of process noise, is the measurement matrix of the sensors, and is the covariance matrix of measurement noise. The PHD of birth targets and spawned targets are written in the following Gaussian mixture forms: where , , , and are the numbers of Gaussian components, weights, mean, and covariance of birth targets, and , , , , , and are the numbers of Gaussian components, weights, state transition matrix, noise disturbing items, mean, and covariance of spawned targets.

Based on the posterior intensity at time in (14), the posterior intensity at time can also be predicted bywhere is given in (18) and is the intensity of the survival target which can be expressed aswhereThe intensity of the spawned target, , is

According to (20), the predicted PHD can be written in a Gaussian mixture form:Then, the posterior intensity function at time also follows a Gaussian mixture form:where are the measurements obtained at time and

A large number of the Gaussian components will bring a heavy burden into the computation. Usually, a merging and pruning stage is needed to deal with this problem [8]. In order to find targets in the Gaussian components, we set a weight threshold (usually chosen as 0.5). If the weight is larger than , the corresponding Gaussian component is regarded as a target; that is,

##### 4.2. Track-Estimate Association

By associating a single target track at the previous time with the state of the target at the current time, the overestimation of the target number in the GM-PHD can be restrained. First, RFS at the current time is obtained according to the conational GM-PHD filter, and the track set at the previous time is already generated, and then track-estimate association can be done. Due to the randomness of clutter, the error state generated by the clutter also has randomness, so we can eliminate the error of state through the existing track.

Figure 1 is an illustration of the track-estimate association from the time to . In Figure 1, are the end nodes of tracks in formed at time . The end node is the same as a state vector where denote the location of the th target and denote the velocity of the th target. denotes the end time of tracks at time . It should be pointed out that the end time is not always equal to because some tracks may not be associated with the estimates at previous time. denotes a time interval from the end time of the th track to time ; that is, , . is the interruption time of the th track. is the predicting vector corresponding to the end time state of each track at time , and . If targets move with a constant speed, the prediction of the th track can be calculated by