Multiscan association as a single-commodity flow optimization problem G. Battistelli 1 , L. Chisci 1 , F. Papi 1 , A. Benavoli 2 , A. Farina 2 1 DSI, Universit` a di Firenze, Florence, Italy e-mail: {battistelli,chisci}@dsi.unifi.it 2 Engineering Division, SELEX Sistemi Integrati, Rome, Italy e-mail: {abenavoli,afarina}@selex-si.com Abstract—Multiscan data association can significantly enhance tracking performance in critical radar surveillance scenarios involving multiple targets, low detection probability, high false alarm probability, evasive target maneuvers and finite radar resolution. Unfortunately, however, this approach is affected by the curse of dimensionality which hinders its real-time application for tracking problems with short scan periods and/or long association windows and/or many measurements. In this paper it is shown how the formulation of the multiscan association as a single commodity flow optimization problem allows a relaxation of the association problem which, on one hand, guarantees close- to-optimal association performance and, on the other hand, implies a significant reduction of the computational load. I. I NTRODUCTION It is well known that in critical radar scenarios, like e.g. GMTI (Ground Moving Target Indicator) radar tracking, in- volving many targets with low detection probability moving in a highly cluttered environment, a considerable improvement in tracking performance can be achieved by using multiscan, in place of single scan, data association [1]. In fact, the memory of multiscan association allows the partial recovery of association errors which represent the main error source in multitarget tracking. On the other hand, the complexity of multiscan association grows exponentially with the size (number of scans) of the association window. More precisely let M be the number of measurements per scan, T the number of tracks and S the number of scans in the association window, then multiscan association amounts to a linear binary programming problem with “in the order of” TM S variables, i.e. O(TM S ), and O(MS ) constraints. To avoid the “curse of dimensionality”, Lagrangian relaxation techniques have been proposed [4]. In [3], a novel relaxation technique, by which it is possible to represent the association problem as a multi-commodity flow optimization problem on a suitable graph involving O(MS ) nodes and O(M 2 TS ) arcs, has been proposed. This solution, quite efficient in practice, does not completely avoid the curse of dimensionality as it involves the solution of a linear binary programming problem for which the integrality property of the solution is not guaranteed. In this paper, a step further is carried out by reformulating multiscan data association as a single-commodity flow optimization. This allows to solve the problem, though in an approximate and sub-optimal way, in polynomial time. On the other hand, simulation results obtained from realistic case studies will demonstrate the effectiveness of the proposed approximation. II. FORMULATION OF THE MULTISCAN DATA ASSOCIATION PROBLEM In this section, the multiscan association problem over a window of S scans (in short S-D Assignment) is described in detail. To this end, let T be the number of tracks 1 . It is assumed that for each track n =1, 2,...,T an estimate ¯ x n of the track state at the beginning of the window is available. Further, for each k =1, 2,...,S, let us denote by Z k = {z 1,k , z 2,k ,..., z M k ,k } the set of measurements obtained at scan k (M k being the cardinality of Z k ). Given the T tracks and the S sets of measurements Z k for k =1, 2,...,S, the objective of S-D Assignment is to assign a sequence of S measurements to each track, where the k-th element of such a sequence is either taken from Z k or represents a missed detection. Among all feasible assignments, an optimal one is found by minimizing a suitably defined cost. In this connection, let c(m 1 ,m 2 ,...,m S ,n) denote the cost of associating a certain sequence (m 1 ,m 2 ,...,m S ) to the track n =1, 2,...,T . Here, each variable m k , for k =1, 2,...,S, takes its value in the set {0, 1,...,M k } and refers either to the m k -th measurement of the set Z k (when m k > 0) or to a missed detection (when m k = 0). The cost to be minimized is typically the total negative log-likelihood and takes the form c(m 1 ,m 2 ,...,m S ,n)= S  k=1 c k (m 1 ,m 2 ,...,m k ,n) where c k (m 1 ,m 2 ,...,m k ,n) is the cost of adding m k to the partial sequence (m 1 ,m 2 ,...,m k−1 ). In order to derive an expression for each c k (m 1 ,m 2 ,...,m k ,n), some preliminary definitions are needed. Given a measurement z and a state x, let Λ(z, x) be the likelihood that the measurement z originates from a target with state x. Further, consider some filtering mechanism (e.g, the extended Kalman filter or a Sequential Monte Carlo filter) 1 In this paper, for the sake of brevity, the track deletion and track initialization problems are not considered. However, many solutions to these problems are available in the literature; see, e..g., [1] and the references therein.