Crame ´ r – Rao lower bound for tracking multiple targets B. Ristic, A. Farina and M. Hernandez Abstract: The derivation and computation of the theoretical Crame ´r–Rao lower bounds for multiple target tracking has traditionally been considered to be a notoriously difficult problem. The authors present a simple and exact solution based on the assumption that raw sensor data (before thresholding) are available. The multi-target tracking problem can then be formulated as recursive Bayesian track-before-detect estimation. The advantage of this formulation is that it is identical to nonlinear filtering, for which the exact posterior Crame ´r–Rao bound is already known. The paper presents several numerical examples in support of the theoretical findings. 1 Introduction Target tracking in general involves nonlinear filtering with measurements of uncertain origin. Problems of this type do not have closed form optimal solutions and in practice are implemented as (suboptimal) approximations. In the absence of the closed form solution, for some thirty years there have been attempts to derive the theoretically best achievable tracking error performance, in the form of the Crame ´r – Rao lower bound (CRLB). Such a bound would be extremely important as it could be used for many practical purposes, such as tracking algorithm assessment; perform- ance prediction; tracking system design; sensor allocation and scheduling (see [1] and references therein). A complete history of the developments of the CRLB for target tracking would involve more than fifty publications and hence we present only the key developments. A review of the pre-1989 formulations of the CRLB for nonlinear filtering is presented in [2]. The key modern reference for this subject, however, is [3], where the authors present very simple and elegant recursions for the computation of the information matrix (the inverse of the CRLB). The implicit assumption in these early developments of the CRLB for nonlinear filtering was that there is a single target and there is no uncertainty in the origin of the measurements (i.e. probability of detection P d ¼ 1; prob- ability of false alarm P fa ¼ 0). It was discovered for the first time in [4] that the effect of measurement uncertainty can be approximated by scaling the information matrix with a constant factor less than unity (this factor was referred to as the information reduction factor (IRF) subsequently). The IRF approach to measurement uncertainty has since then been refined in several publications, culminating in [5]. More recently, the enumeration method for the computation of the CRLB when P d < 1 and P fa ¼ 0 was proposed [6]. This method yields a very accurate bound [6], but its computational load grows exponentially with time. Recently it has been proved [1] that CRLBðIRFÞ < CRLB ðEnumerationÞ; meaning that the IRF method yields an overly optimistic bound. In the case of multiple targets, Daum proposed two ways of computing the CRLB. The first one [7] is based on a ‘genie’ which knows up to a certain degree the correct measurement associations. By changing the degree of genie’s knowledge (starting from the complete knowledge as the trivial case) one can obtain a set of optimistic CRLBs, manageable for computation. The second approach by Daum [8] is based on the symmetric measurement equations (SME) [9]. The idea is to transform a multitarget tracking problem into a (single-target) nonlinear filtering problem, and then to find the CRLB for nonlinear filtering. This approach, unfortunately, does not lead anywhere because the SME transformation creates a nonlinear filtering problem with a highly non-Gaussian noise, for which there is no simple and elegant CRLB bound. Alternative bounds for multiple target tracking are proposed in [10–13]. Their evaluation, unfortunately, is in general extremely complicated and requires approximations. Consequently none of these bounds have been widely used in practice. This paper presents a simple and exact method for the computation of a theoretical Crame ´r–Rao bound for multiple target tracking. The main idea is to work with the raw sensor data (before thresholding), as it is done in the track-before-detect approach [14]. Thresholding in general is performed to reduce the data flow and hence the computational load of the tracker, but in doing so it inevitably causes the loss of information. In addition, thresholding artificially introduces the problem of measure- ment-to-track association, caused by the uncertainty in the measurement origin (this is a direct consequence of P d < 1 and P fa > 0). Since our CRLB is computed for the raw sensor data, it represents the truly best achievable performance and we refer to this bound as the lowest or the ‘ultimate’ CRLB. There are two assumptions in our approach: (i) we know in advance the number of targets and (ii) all targets exist during the observation period. These two assumptions must be kept, because the concept of birth and death of targets really belongs to detection theory and its q Australian Crown copyright, 2004 IEE Proceedings online no. 20040532 doi: 10.1049/ip-rsn:20040532 B. Ristic is with DSTO, ISR Division, 200 Labs, PO Box 1500, Edinburgh SA 5111, Australia and is temporarily with IRIDIA, Universite ´ libre de Bruxelles, Av. F. Roosevelt 50, CP 194/6, 1050 Bruxelles, Belgium A. Farina is with Alenia Marconi Systems, Chief Technical Office, Via Tiburtina Km.12.400, 00131 Rome, Italy M. Hernandez is with QinetiQ Ltd, The Advanced Processing Centre, St Andrew’s Road, Malvern WR14 3PS, UK Paper first received 27th November 2003 and in revised form 17th March 2004 IEE Proc.-Radar Sonar Navig., Vol. 151, No. 3, June 2004 129