An LP-based algorithm for the data association problem in multitarget tracking☆
Introduction
A basic question in the field of (air) surveillance systems is: how to keep track of all observed targets using information from all available sensors? Sensors, like radar or infra red, detect and locate objects (targets) that are present within their coverage. Each time a detection (also called measurement or plot) takes place, several features are measured, such as the location of the object (expressed in range, bearing and elevation) and Doppler shift (a measure for the velocity in the direction of the sensor).
In this work, we restrict ourselves to a single scanning surveillance radar, providing two-dimensional positional information (i.e. range and bearing). Furthermore, we assume that the radar's measurements are processed after each scan, i.e. a full rotation of the radar antenna.
The so-called Data Association Problem (DAP) is now to find out which sensor detections originate from which target. More precisely, when assuming that for each scan at most one measurement can be assigned to a target, a problem arises if more than one measurement in a scan is a candidate to be assigned to the same target. Even more, a single measurement may be a candidate to be associated with two or more targets. The DAP aims at finding an assignment of measurements to targets such that a measurement is associated with at most one target and such that a target receives at most one measurement per scan. Notice that, in operational circumstances, the DAP has to be solved after each scan, for all collected scans of measurements.
Complicating factors when solving instances of the DAP are the presence of spurious plots (false alarms), noise contaminated measurements, (deliberate) target maneuvres, a sensor detection probability <100% which leads to missed detections, and hard real-time requirements. Each received plot either has to be associated to an existing target, or has to be labeled as a new target or a false alarm. A track is defined as a sequence of measurements which are assigned to the same target.
The DAP is illustrated by Fig. 1. The circle represents the sensor's coverage, the dots are sensor plots that are received in sensor scans A–F. Interconnected dots represent a track, i.e. the plots are assumed to originate from the same target. Track A2–B1–C2–D2–F3 has a missed detection for scan E. Plots A1 and E1 are false alarms.
Notice that each individual plot is considered to be a potential target and is initialized as a tentative track. A track is declared “confirmed” if there is sufficient confidence that the track represents a true target, e.g. when the track consists of a minimum number of plots.
The goal of this work is threefold:
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to present a compact, integer programming (IP) formulation of the DAP,
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to propose an algorithm that is capable of solving a series of instances of the DAP, and
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to test this algorithm on randomly generated instances.
Let us now review some of the literature.
A well-known DAP solution method is Multiple Hypothesis Tracking (MHT), based on a formulation by Reid [2]. The algorithm enumerates possible assignments (plot to existing track, plot to new track and plot to false alarm) and represents them in a hypothesis tree. For each hypothesis a likelihood of truth is calculated, after which the most probable assignment is chosen as the solution for the DAP. Notice that the solution found after scan k is not necessarily a part of the solution for scan k+1. Other less likely hypotheses are maintained, since they might eventually be used when future plots are received. MHT deals with the combinatorial explosion of possible assignments by using a number of pruning techniques, for instance based on clustering (see e.g. [3]).
Instead of MHT's enumerative approach, alternative methods view DAP as an integer programming problem. More specific, the problem is described as a multidimensional assignment problem (see e.g. [4] for an early reference), a problem in the field of combinatorial optimization.
An early contribution in this direction dates back to 1977, when C.L. Morefield suggested to apply 0–1 integer programming to the DAP [5]. More recent work on this subject is reported in [6], [7], [8], [9], [10], [11], [12]. In these papers the DAP is not only formulated as a multidimensional assignment problem, in addition a number of ways to solve the problem, notably by using Lagrangian relaxation, are suggested.
This work (see also [13]) is organized as follows. Section 2 is devoted to an IP formulation of the DAP; we propose a compact formulation of the integer program. In Section 3, we describe the heuristic algorithm, and in Section 4, we present some computational results. These are discussed in Section 5 and Section 6 contains the conclusions.
Section snippets
A compact IP formulation of the DAP
In this subsection, we propose a compact formulation of the integer programming formulation of the DAP. This compact formulation follows the formulation of the axial multidimensional assignment problem as suggested in e.g. [14] and avoids multiple summation signs and multiple indices. In Section 2.1, we introduce some notation; in Section 2.2 the decision variables are defined, and the (linear) constraints are introduced. Finally, Section 2.3 develops the objective function of the model.
A solution method for the DAP
Of course, in practice model (9) has to be solved after each scan. This requirement has certain implications for the way an instance of the DAP is solved. First, there is a need to be efficient. More concrete, the time to solve an instance should not surpass the rotation time of the radar; thus, a solution is required fast. This motivates a heuristic approach. Second, there is a difference in importance when considering the assignment of plots in early scans versus the assignment of plots in
Experiments
The layout of the experiments is as follows. The simulated sensor is a single scanning surveillance radar that provides 2D positional data (i.e. range and bearing) within a range of . We constructed 35 scenarios divided over 4 groups called A, B, C and D. Groups A, B and C consist of 10 scenarios each while group D consists of five scenarios. In each scenario targets are observed for . The duration of a scan is . The measurement noise is set to and for range
Discussion
The experiments show that there is a distinction between scenarios of groups A, B and C on the one hand and scenarios of group D on the other hand. Indeed, 4225 out of the 4500 (94%) IP problem instances from scenarios of groups A, B and C are solved by their LP relaxation, i.e. the Simplex algorithm immediately yields integer valued solutions. For the remaining 275 problem instances, the GRP, recovers a feasible (integer valued) solution from the fractional valued solution of the LP
Conclusions
In this section, we formulate our conclusions. We have modeled the data association problem as an integer programming problem. We have described an LP-based algorithm using a K-scan sliding window to solve instances of the data association problem. This algorithm consists of solving the LP-relaxation of the associated integer programming formulation, and next applying a greedy rounding procedure called GRP. From the computational experiments that we performed it follows that
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GRP yields
Acknowledgements
The authors are indebted to J.N. Driessen and H.W. de Waard at Thales Naval Nederland for their valuable comments on this paper.
Patrick P.A. Storms is a consultant at Acklin B.V. (http://www.acklin.nl) and is specialized in intelligent software agent systems. Formerly he was a designer of naval Command & Control systems at Thales Naval Systems Nederland (http://www.thales-nederland.nl).
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Patrick P.A. Storms is a consultant at Acklin B.V. (http://www.acklin.nl) and is specialized in intelligent software agent systems. Formerly he was a designer of naval Command & Control systems at Thales Naval Systems Nederland (http://www.thales-nederland.nl).
Frits C.R. Spieksma is a Professor at the Department of Applied Economics of the Katholieke Universiteit Leuven. His research interests concern combinatorial optimization problems and their applications (http://www.econ.kuleuven.ac.be).