A Track Management System for the PMHT Algorithm
Tod Luginbuhl, Yan Sun and Peter Willett
Abstract – The probabilistic multi-hypothesis tracking (PMHT) algorithm is a powerful and computationally efficient multi-target tracking algorithm whose computational efficiency is such that its load grows linearly with the number of targets. However, the formulation of the PMHT algorithm requires a priori knowledge of the number of objects to track, and it does not allow the number of objects to change over time. This seriously limits its utility in practical applications. To address these issues, a track management system is developed that is consistent with the PMHT algorithm. This track management system is an overlay to the PMHT algorithm, and handles new track generation, track deletion and clutter adaptation. The combined algorithm can be initialized with minimal a priori information about the clutter. Keywords: Track management, Multi-target tracking, PMHT.
1 Introduction While the PMHT algorithm is a multi-target tracking algorithm, it assumes that the number of tracks is known a priori [1]. This is almost never true in practice. The reality is that tracks appear and disappear as objects come into and go out of the range of a sensor. Tracks may also coalesce when two or more objects become unresolvably close in the sensor’s measurement space. In each of these cases, the number of tracks must be adjusted. Another issue for the PMHT algorithm is the correct characterization of the clutter distribution (i.e., the distribution of false measurements). Typically the clutter distribution is not static over time and must be adapted using the observations for the PMHT algorithm to work properly. The objective of this paper is to present an algorithmic structure to support the PMHT algorithm and address these
This research was funded by ONR. T. Luginbuhl is with the Naval Undersea Warfare Center, Newport, RI 02841-1708 USA. EMAIL:
[email protected] Y. Sun and P. Willett are with the University of Connecticut, Storrs, CT 06269 USA. EMAIL:
[email protected]
issues. In summary, a track management system is developed to support the PMHT algorithm in real tracking applications that addresses the following issues: 1. detection and initiation of new tracks; 2. merging of track estimates when two or more tracks coalesce; 3. removal of tracks that correspond to objects that have moved outside the sensor’s range; and 4. adaption of the clutter distribution. Conceptually, the track management algorithm works with the PMHT algorithm in the following manner. Assume that the PMHT algorithm has estimated the parameters of some set of tracks on the current batch of data. A new scan of data is received from the sensor and is added to the batch; the oldest scan of data is dropped from the batch. This new batch of data is processed by the PMHT algorithm which updates all the track estimates. Line or curve integrals are performed using the Radon transform on the new batch of data and on the PMHT algorithm’s mixture PDF for the measurements. The Radon transform of the mixture PDF is used to normalize (“whiten”) the Radon transform of the new batch of data; this effectively removes all known tracks from the batch. The normalized Radon transform is then thresholded to find new tracks. If any new tracks are found, the initial parameter estimates for these tracks are extracted from the curves corresponding to the points detected in the normalized Radon transform of the batch. Assuming new tracks are found, these new tracks are added to the PMHT algorithm’s list of tracks, and the batch is then re-processed by the PMHT algorithm. The resulting track list is then checked to see if any tracks need to be merged. If any tracks are merged, then the PMHT algorithm reprocesses the batch to update the track estimates. The resulting tracks are then examined to see if any tracks need to be deleted. If any tracks are removed from the PMHT algorithm track list, the batch is reprocessed by PMHT algorithm. Following
references [2] and [3], all measurements that have a high probability of being clutter in the batch are added to a histogram of the clutter. This clutter histogram is then used to update the clutter measurement PDF. The next section states the statistical assumptions used in the PMHT algorithm and in this paper. To simplify the discussion, a one-dimensional tracking problem for linear, Gaussian targets is considered. Detection and initiation of new tracks is addressed in Section 3 using the Radon and Hough Transforms. Section 4 discusses merging one or more tracks using the merging procedure for mixture probability density functions (PDFs) described in [4]. Old tracks are removed from the PMHT algorithm’s track list based on the effective number of measurements assigned to a track. This idea is developed in Section 5. An approach to the problem of clutter adaptation is given in Section 6.
2 Notation and Statistical Assumptions A scan is a collection of sensor measurements obtained during a given time interval. A batch is a collection of successive scans. The PMHT algorithm is assumed to operate on a sliding batch comprised of scans. The set of target state vector sequences for the batch at time is represented by for The collection of real, target state vectors in scan is denoted by for where represents the number of tracks in the batch at time The target state vectors and are assumed to be statistically independent for all For each target the state vector is assumed to evolve according to a first order Markov process. In the case of linear, Gaussian statistics, the PDF of is given by
is given by 6 mF3N where h represents the output matrix for track ! at time and jn represents the measurement error covariance matrix for track ! at time I target
and Gaussian, the PDF of a measurement coming from
at time
Since the origin of each sensor measurement is unknown, for each measurement, there is an unknown, discrete assignment variable The assignment variables and are statistically independent for all and Let represent the set of unknown, discrete assignment variables for the sensor scan at time and let the represent the set of unknown, discrete assignment variables for the batch at time The set is treated as the “missing information” in the expectation-maximization (EM) method used to derived the PMHT algorithm [1]. The PMHT algorithm assumes each measurement has a non zero probability of coming from each target. Consequently, the PMHT algorithm models the PDF of each measurement with a mixture PDF:
o@Q p*I H$ ofQ o ab Tq 1 c rq 1 ef s t]o>Q I uVvwDs @ u
xzy { 6 Measurements correspondstate sequences for all d+ ing to clutter are assume to be statistically independent for all d" 1 ef When the target measurement process is linear
!
where the track measurement probabilities satisfy for all and
*
!5
~
(3)
, (
{x , " > (4) |}g~ The track corresponding to !* represents the PDF of the clutter; hence, 6 } 7 P = g}k ? 6 } 7 P CH }H ? The clutter PDF is assumed to be a mixture of uniform PDFs on 7 *IHP UZ[]\W y{ H } 6 } 7 P CH }F ?
~ 7F7 P' }H ??F }F ? (5) | where }HU% }F k }Fk }HF @ }Fv *g@ |8y ; }FU% > }Hr 7 ~ ; ? S }Hr ~ ;]g |y } is a~ partition of the sensor measurement space and 7 ? is the uniform PDF defined on 7 *I \ (the selection of and will be discussed in Section 6). The PDF of all the measurements in the batch is then given by
7 J4 = ? 9N 0-
