A Relaxation Algorithm for the Passive Sensor Data Association Problem

Abstract

This paper is concerned with the problem of associating measurements from multiple angle-only sensors in the presence of clutter, missed detections and unknown number of targets. The measurement-target association problem is formulated as one of maximizing the joint likelihood function of the measurement partition. Mathematically, this formulation of the data association problem leads to a generalization of the multi-dimensional matching (assignment) problem, which is known to be NP-complete when the number of sensors S ≥ 3, i.e., the complexity of an optimal algorithm increases exponentially with the size of the problem. The new solution to the optimization problem developed in this paper is a Lagrangian relaxation technique that successively solves a series of generalized two-dimensional assignment problems, with the worst case complexity of max [O(S n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ), O(M)], where n is the number of reports from each sensor and M is the number of possible measurement-target associations. The dual optimization problem is solved via an accelerated subgradient method. A useful feature of the relaxation approach is that the resulting dual optimal cost is a lower bound on the feasible cost and, hence, provides a measure of how close the feasible solution is to the (perhaps unknowable) optimal solution. For the passive sensor data association problem, the feasible solution costs are typically within 1% of their corresponding dual optimal costs. The algorithm is illustrated via several examples.