A feedback approach to multitarget multisensor tracking with

A feedback approach to multitarget multisensor tracking with application to bearing-only tracking G. Battistelli, L.Chisci, S. Morrocchi, F. Papi Dip. Sistemi e Informatica Universit`a di Firenze Firenze, Italy. {battistelli,chisci}@dsi.unifi.it Abstract – A novel approach to multitarget multisensor tracking, exploiting the feedback connection of a PHD (Probability Hypothesis Density) smoother and multisensor multidimensional data association, is presented. The PHD smoother is used to initialize target tracks while the feedback from the hard data association makes the PHD smoother, and hence the overall tracker, less sensitive to missed detections and false alarms. An application to bearing-only tracking is investigated in order to demonstrate the potentials of the proposed approach for tracking problems wherein the state of a target is observable from multiple sensors but not from a single one. Keywords: Multitarget multisensor tracking; PHD filter; multidimensional association; bearing-only tracking.

1

Introduction

Multitarget and/or multisensor tracking is a problem of great practical relevance in a wide range of civilian and military applications. It is worth pointing out that, besides its relevance, tracking is also an extremely difficult and, hence, challenging estimation problem due to a number of critical issues (e.g., missed detections, false alarms, unknown origin of measurements, unknown and time-varying number of targets, high density of targets and/or clutter, highly maneuvering targets, finite sensor resolution, etc.) that are often encountered in practice. At the current state-of-art, there are essentially two mainstream approaches to multitarget multisensor tracking. (1) The classical approach [1]-[3] adopts a divideand-conquer strategy trying to decouple the overall multitarget multisensor problem into multiple independent single-target problems. This decoupling is usually accomplished by means of two auxiliary procedures: track formation, which is responsible for detecting new targets and initializing their tracks (initialization) as well as for recognizing disappeared targets and

A. Farina, A. Graziano Engineering Division SELEX Sistemi Integrati Rome, Italy {a.farina,a.graziano}@selex-si.com

terminating their tracks (termination); data association, which aims at finding out the unknown source, either a target or clutter, of the available measurements. Thanks to the combined action of track formation and data association, it is then possible to use a separate, possibly nonlinear, filter for estimating the kinematic state of any detected target. (2) The random set approach [4]-[5] regards both targets and measurements as random sets, i.e. objects in which randomness is not only in the assumed values but also in the number of elements. In the random set framework, multitarget multisensor tracking amounts to recursively estimating the random target set, i.e. the set of the states of all targets that are present at a given time in the scenario, based on the measurement sets, i.e. the sets of all measurements collected from all sensors, up to that time. In particular, among random set tracking techniques the so called Probability Hypothesis Density (PHD) filter [5] has received considerable attention in view of its computational tractability even for large-scale tracking problems involving a high number of targets and/or measurements. In engineering terms, the PHD filter recursively estimates the density of targets in the state space so that the integration of the estimated density over a certain region provides the expected number of targets in that region. Hence, the PHD filter directly estimates the number and spatial distribution of targets while does not directly provide explicit estimates of the kinematic states of the individual targets, which can however be obtained by suitable peak detection and/or clustering techniques. The main advantage of the PHD filter is that it does not require data association and can, therefore, efficiently cope with highly crowded scenarios (e.g., road traffic monitoring). A further strength-point of the PHD approach is that it does not require any external track formation logic or prior knowledge on the number of targets, which is actually on-line estimated. On the converse, drawbacks of the PHD filter are that: (1) it does not provide explicit

estimates of the target states; (2) it usually produces inaccurate estimates of the target number for low detection probability 𝑃𝑑 and/or high false alarm probability 𝑃𝑓 𝑎 ; (3) it is highly sensitive to uncertainties on parameters of the scenario like, e.g., the above mentioned 𝑃𝑑 and 𝑃𝑓 𝑎 . Classical (traditional) methods have complementary pros and cons. In fact they directly provide estimates of target states and can be made more robust with respect to uncertainties on the scenario by exploiting multiscan/multihypothesis logics. On the other hand, they are characterized by high computational burden implied by the data association task. As a further drawback, they also need an external track formation logic which can be difficult to realize in certain circumstances such as, for instance, in multisensor tracking problems where full state observability is only guaranteed with multiple sensors but not with a single one (e.g. multistatic passive radar, bearing-only and range-only tracking). The above considerations on the complementary issues of PHD and classical approaches to multitarget multisensor tracking as well as the goal to tackle difficult tracking problems (like, e.g., bearing-only tracking) for which the existing techniques do not provide yet satisfactory performance, has motivated the work described in this paper. More precisely, this work has concerned the development of a hybrid tracker following an idea first introduced in [6, 7] and later pursued in [8] for the single-sensor case. The tracker proposed in this paper, named PHD-SDA, combines a PHD smoother and multisensor multiscan association (also called Multisensor S-Dimensional Association and referred to hereafter by the acronym MsSDA) in a feedback fashion so as to provide robust and efficient multitarget multisensor tracking. It is worth pointing out that a fundamental novelty of PHD-SDA with respect to the hybrid trackers in [6, 7] is the exploitation of feedback and that, moreover, the present work generalizes the feedback approach of [8] to the multisensor case.

∙ 𝒴𝑡 =

𝑁 ∪

𝒴𝑡,ℎ : set of measurements at time 𝑡;

ℎ=1

∙ 𝒴𝑡1 :𝑡2 =

∪

𝒴𝑘 : set of measurements in the time

𝑡1 ≤𝑘≤𝑡2

interval ∪ [𝑡1 , 𝑡2 ]; ∙ 𝒴:𝑡 = 𝒴𝑘 : set of measurements up to time 𝑡. 𝑘≤𝑡

In a random set framework, the problem consists therefore of estimating 𝒳𝑡 given 𝒴:𝑡 . According to the Bayesian approach, this would amount to propagating in time the multitarget density 𝑓 (𝒳𝑡 ∣𝒴:𝑡 ), defined via finite-set statistics (FISST) [4], which represents a full probabilistic characterization of the target set 𝒳𝑡 given the observations 𝒴:𝑡 . Unfortunately, however, the resulting multitarget Bayes recursion is computationally intractable in most practical cases. Hence, multitarget tracking algorithms usually employ simpler characterizations of the target set. The PHD filter, for instance, updates the so called probability hypothesis density (PHD) or intensity function 𝐷𝑡∣𝑡 (x) such that, for any region 𝒮 in the state space, the expected number of targets in 𝒮 can be obtained by ∫ 𝑛 ˆ 𝑡∣𝑡 (𝒮) = 𝐷𝑡∣𝑡 (x) 𝑑x, 𝒮

i.e. by integration of 𝐷(⋅) over 𝒮. Conversely, a traditional multitarget tracker typically updates, by some formation logic, an estimate 𝑛 ˆ 𝑡∣𝑡 of the target number and, for each target track 𝑖 = 1, 2, . . . , 𝑛 ˆ 𝑡∣𝑡 , the probability density function (PDF) 𝑝𝑖𝑡∣𝑡 (x) of the corresponding target state. Hereafter, the PDF 𝑝𝑖𝑡∣𝑡 (⋅) or some sufficient statistics will be referred to, by some abuse of terminology, as “target track” or more simply “track”. It is worth pointing out that from the PHD 𝐷𝑡∣𝑡 (⋅) it is possible to extract the estimated number of targets by integration, i.e. ∫ 𝑛 ˆ 𝑡∣𝑡 = 𝐷𝑡∣𝑡 (x) 𝑑x, (1) IR𝑛𝑥

2

Multitarget multisensor tracking

The objective of multitarget multisensor tracking is to estimate, at any sampling time and based on the available sensors measurements up to that time, the unknown, possibly time-varying, number of targets present in the scene and, for each detected target, the kinematic state. The following notations are introduced: ∙ 𝑛𝑡 : unknown number of targets at time 𝑡; ∙ 𝑥𝑖𝑡 ∈ IR𝑛𝑥 (𝑖 = 1, . . . , 𝑛𝑡 ): state of target 𝑖 at time 𝑡; } △ { ∙ 𝒳𝑡 = 𝑥1𝑡 , 𝑥2𝑡 , . . . , 𝑥𝑛𝑡 𝑡 : target set at time 𝑡; ∙ 𝑁 : number of sensors; ∙ 𝒴𝑡,ℎ = {y1,𝑡,ℎ , y2,𝑡,ℎ , . . . , y𝑚𝑡,ℎ ,𝑡,ℎ } (ℎ = 1, 2, . . . , 𝑁 ): set of measurements of sensor ℎ at time 𝑡;

as well as the target tracks by suitable clustering and/or peak extraction techniques [4, 15]. On the other hand, { }𝑛ˆ 𝑡∣𝑡 from 𝑛 ˆ 𝑡∣𝑡 and 𝑝𝑖𝑡∣𝑡 (⋅) it is straightforward to con𝑖=1 struct the corresponding PHD function by 𝐷𝑡∣𝑡 (x) =

𝑛 ˆ 𝑡∣𝑡 ∑

𝑝𝑖𝑡∣𝑡 (x)

(2)

𝑖=1

Since the transformation { }𝑛ˆ 𝑡∣𝑡 𝐷𝑡∣𝑡 (⋅) ←→ 𝑛 ˆ 𝑡∣𝑡 , 𝑝𝑖𝑡∣𝑡 (⋅)

𝑖=1

can be easily carried out in both directions, one can devise efficient hybrid tracking schemes that combine the PHD filter and traditional association/filtering algorithms so as to exploit the complementary positive

features of both approaches. Hybrid schemes have been proposed in [6, 7, 8] for the single-sensor case; next section will present a multisensor extension of [8], named PHD-SDA. Since the proposed tracker will exploit smoothing, it is convenient to define the relevant estimated quantities for two arbitrary time instants 𝜏 and 𝑘, i.e.: ∙ 𝐷𝜏 ∣𝑘 (⋅): PHD of the target set 𝒳𝜏 given 𝒴:𝑘 ; ∙𝑛 ˆ 𝜏 ∣𝑘 : estimate of the target number 𝑛𝜏 given 𝒴:𝑘 ; ∙ 𝑝𝑖𝜏 ∣𝑘 (⋅): PDF of the state 𝑥𝑖𝜏 of target 𝑖 given 𝒴:𝑘 . Notice that the infinite-dimensional PHD and PDF functions can actually be finitely parameterized in different ways, e.g. via sets of particles [14] or as Gaussian mixtures [15].

3 3.1

Closed-loop PHD-SDA tracker Architecture

The rationale of this approach is that a PHD smoother and SDA can be jointly exploited according to the feedback scheme of fig. 1 for efficient multitarget multisensor tracking. As it can be seen from the figure, the resulting closed-loop PHD-SDA tracker essentially consists of six processing blocks. The input-output functionality of each block is described hereafter. ∙ The PHD smoother provides, at each time 𝑡, the intensity 𝐷𝑡−𝑆∣𝑡 (⋅), where 𝑆 ≥ 1 is a suitably chosen smoothing lag. Such a smoothed PHD is obtained from the corrected PHD 𝐷𝑡−𝑆∣𝑡−𝑆 (⋅), provided by the PHD reconstruction block, by exploiting the measurements 𝒴𝑡−𝑆+1:𝑡 and the target dynamics, with no birth and death, over the time interval [𝑡 − 𝑆, 𝑡]. More details on the operations of this block will be given in subsection 3.2. ∙ The track extraction block extracts from 𝐷𝑡−𝑆∣𝑡 (⋅) the estimated number of targets 𝑛 ˆ 𝑡−𝑆∣𝑡 via integration and the tracks 𝑝𝑖𝑡−𝑆∣𝑡 (⋅) for 𝑖 = 1, 2, . . . , 𝑛 ˆ 𝑡−𝑆∣𝑡 by means of some clustering/peak extraction technique. ∙ The track merging block merges the tracks { }𝑛ˆ 𝑡−𝑆∣𝑡 𝑝𝑖𝑡−𝑆∣𝑡 (⋅) from the track extraction block 𝑖=1 { }𝑛ˆ 𝑡−𝑆 with the one-step delayed tracks 𝑝𝑖𝑡−𝑆 (⋅) 𝑖=1 from the track filtering & termination block. The merged }𝑛𝑡−𝑆 { ˆ 𝑡−𝑆 , tracks, denoted as 𝑝𝑖𝑡−𝑆 (⋅) 𝑖=1 with 𝑛𝑡−𝑆 ≥ 𝑛 are passed to the association. The aim of merging is to add, to the existing tracks, possible new tracks detected by the PHD smoother, while avoiding the duplication of tracks. ∙ The SDA block processes the measurements 𝒴𝑡−𝑠+1:𝑡 { }𝑛𝑡−𝑆 and the merged tracks 𝑝𝑖𝑡−𝑆 (⋅) 𝑖=1 in order to find measurement-to-track associations, i.e. pairings {( 𝑖 )}𝑛𝑡−𝑆 𝑖 denotes the 𝒴𝑡−𝑠+1:𝑡 , 𝑝𝑖𝑡−𝑆 (⋅) 𝑖=1 where 𝒴𝑡−𝑠+1:𝑡 subset of measurements assigned to track 𝑝𝑖𝑡−𝑆 (⋅) over the association window [𝑡 − 𝑠 + 1 : 𝑡]. Further, it also provides the subset of unassigned measurements at time 𝑡 − 𝑆 + 1, denoted as 𝒴 𝑡−𝑆+1 , to the PHD recon-

struction block. Some details on the multisensor SDA problem and its solution will be given in subsection 3.3. ∙ The track filtering & termination block filters each track 𝑝𝑖𝑡−𝑆∣𝑡 (⋅) with the associated measurements 𝑖 𝒴𝑡−𝑠+1:𝑡 and the target dynamics. Further, it eliminates unlikely tracks according to some termination criterion [2]. The filtered and surviving (confirmed) { }𝑛ˆ 𝑡−𝑆+1 tracks at time 𝑡 − 𝑆 + 1, denoted as 𝑝𝑖𝑡−𝑆+1 (⋅) 𝑖=1 with 𝑛 ˆ 𝑡−𝑆+1 ≤ 𝑛𝑡−𝑆 , are passed to the PHD reconstruction block and are also fed back, via a delay of one time step, to the merging block in order to be used at time 𝑡 + 1. Further, the filtered target tracks at time 𝑡, denoted as 𝑝ˆ𝑖𝑡 (⋅) for 𝑖 = 1, . . . , 𝑛 ˆ 𝑡−𝑆 , are used for displaying the target tracks at the current time. ∙ The PHD reconstruction block constructs the PHD 𝐷𝑡−𝑆+1∣𝑡−𝑆+1 (⋅) induced by the confirmed tracks { 𝑖 }𝑛ˆ 𝑡−𝑆+1 𝑝𝑡−𝑆+1 (⋅) 𝑖=1 and by the unassigned measurements 𝒴 𝑡−𝑆+1 . Then, 𝐷𝑡−𝑆+1∣𝑡−𝑆+1 (⋅) is fed back, via a unit delay, to the PHD smoother. Details on the PHD reconstruction will also be given in subsection 3.2. A few remarks on the above described scheme are in order. ∙ The use of a multistep window of length 𝑆, in both the PHD smoothing and association, allows to make the overall tracker more robust with respect to the confirmation of false tracks and the termination of true tracks. On the other hand, this implies a delay of 𝑆 steps in both track confirmation and termination. Hence, the choice of 𝑆 ≥ 1 must trade off robustness versus responsiveness and computational efficiency. ∙ Track initialization represents a critical task in multisensor tracking applications wherein the full observability of the target state requires multiple sensors (e.g. bearing-only, range-only, passive multistatic radar tracking). The solution adopted in the proposed PHDSDA scheme is to initialize target tracks by means of a PHD smoother with track extraction. ∙ As it can be seen from fig. 1, the PHD-SDA scheme incorporates several feedback loops in order to allow mutual correction among the various blocks. In particular: the connection from the track filtering & termination block to the track merging block allows to avoid the loss of tracks due to missed detections; the connection from the PHD reconstruction block to the PHD smoother allows to correct the PHD function on the basis of the hard association outcome; the connection from the SDA block to the PHD reconstruction block provides to the latter unassociated measurements that are used to update the birth intensity. ∙ The focus in this paper is on the feedback architecture of the tracking system (cf. fig. 1), i.e. on the overall interconnection of the various blocks rather than on the specific algorithmic implementation of the blocks. From this viewpoint, it is worth to say that

𝒴𝑡−𝑠+1:𝑡

PHD smoother

𝐷𝑡−𝑆∣𝑡 (⋅)

track extraction

PHD tracks

track merging

𝑧 −1

merged tracks

𝑧 −1

𝐷𝑡−𝑆+1∣𝑡−𝑆+1 (⋅)

PHD reconstruction

confirmed tracks unassigned measurements

SDA

measurement-to-track associations

track filtering & termination

Figure 1: Block diagram of the closed-loop PHD-SDA tracker the tracker of fig. 1 has a modular structure in which each processing block can be realized in different ways. For instance, the PHD smoother can be replaced by a CPHD (Cardinalized PHD) smoother. Further, different PHD/CPHD implementations (either particle filter or Gaussian mixture) and/or SDA solvers and/or track extraction, merging, filtering and termination algorithms can be used. Hereafter, subsection 3.2 will describe the multisensor PHD recursions used for the PHD smoothing and correction while subsection 3.3 will formulate the multisensor association problem solved by the SDA block.

3.2

Multisensor PHD smoother

Before discussing the operations performed in the PHD reconstruction and PHD smoother blocks, it is convenient to recall the recursion of the multisensor PHD filter. To this end, the following notations are needed: ∙ 𝑓 (x∣𝝃): single-target state transition PDF originated by the target dynamics; ∙ ℓℎ (x∣y), ℎ = 1, 2, . . . , 𝑁 : likelihood function originated by the ℎ-th sensor measurement relationship; ∙ 𝑃𝑠,𝑡 (x): survival probability of a target at state x and time 𝑡; ∙ 𝑃𝑑,ℎ (x), ℎ = 1, 2, . . . , 𝑁 : probability that a target at state x is detected by sensor ℎ; ∙ 𝑏𝑡 (x): birth density at time 𝑡 in the single-target state space; ∙ 𝑐ℎ (y), ℎ = 1, 2, . . . , 𝑁 : clutter (false alarm) density for sensor ℎ in the single-measurement space. Notice that all the above defined quantities can actually depend on the time instant 𝑡 but for some quantities such a dependence has not been explicitly indicated for the sake of notational simplicity. In the single-sensor case, under reasonable assumptions [5] on the target

dynamics as well as on the measurement generation, an elegant and tractable recursion exists for the propagation of the PHD, i.e., the first-order statistical moment of the multitarget posterior. When multiple sensors are available, FISST calculus can still be applied to derive rigorous formulas for the multisensor PHD filter, but the resulting recursion turns out to be computationally intractable and some approximation has to be used instead. In this connection, a possible approach is the iterated-corrector approximation [5, p. 594], wherein the PHD corrector step is applied multiple times in succession, one for each available sensor. According to such an approximation, the multisensor PHD recursion takes the following form: ∫ 𝐷𝑡∣𝑡−1 (x) = 𝑏𝑡 (x) + 𝑃𝑠,𝑡 (𝝃)𝑓 (x∣𝝃)𝐷𝑡−1∣𝑡−1 (𝝃)𝑑𝝃 (3) 𝐷𝑡∣𝑡 (x) =

𝑁 ∏

[ℎ]

Λ𝑡 (x) 𝐷𝑡∣𝑡−1 (x)

(4)

ℎ=1 [ℎ]

where the correction factors Λ𝑡 (⋅) are iteratively computed as [ℎ]

Λ𝑡 (x) = [1 − 𝑃𝑑,ℎ (x)] ∑ 𝑃𝑑,ℎ (x)ℓℎ (x∣y) + ∫ [ℎ−1] 𝑃𝑑,ℎ (𝝃)ℓℎ (𝝃∣y)𝐷𝑡∣𝑡−1 (𝝃)𝑑𝝃 y∈𝒴𝑡,ℎ 𝑐ℎ (y) + [ℎ] 𝐷𝑡∣𝑡 (x)

=

[ℎ] [ℎ−1] Λ𝑡 (x) 𝐷𝑡∣𝑡 (x)

(5) (6)

for ℎ = 1, 2, . . . , 𝑁 , with the initialization [0]

𝐷𝑡∣𝑡 (x) = 𝐷𝑡∣𝑡−1 (x) .

(7)

Notice that, with respect to the general PHD recursion derived in [5], the target spawning has not been included in (3).

As anticipated, the multisensor PHD recursion (3)(6) is exploited in both the PHD reconstruction and the PHD smoother blocks. In particular, in the PHD reconstruction block, an SDA-induced PHD of the target set 𝒳𝑡−𝑆+1 is constructed from the confirmed tracks 𝑝𝑖𝑡−𝑆+1 (⋅), 𝑖 = 1, . . . , 𝑛 ˆ 𝑡−𝑆+1 , and from the unassigned measurements 𝒴 𝑡−𝑆+1 by letting

for 𝑗 = 𝑡 − 𝑆 + 1, . . . , 𝑡, where the correction factors [ℎ] Λ𝑡 (⋅) are iteratively computed as in (5)-(6) with the initialization ∫ [0] 𝐷𝑗∣𝑗 (x) = 𝒟𝑗∣𝑗−1 (x, 𝜻) 𝑑𝜻 .

After such a recursion has been carried out, the smoothed PHDs 𝐷𝑡−𝑆∣𝑗 (⋅), 𝑗 = 𝑡 − 𝑆 + 1, . . . , 𝑡 can 𝑛 ˆ 𝑡−𝑆+1 ∑ be computed via marginalization as 𝑏 𝐷𝑡−𝑆+1∣𝑡−𝑆+1 (x) = 𝐷𝑡−𝑆+1∣𝑡−𝑆+1 (x)+ 𝑝𝑖𝑡−𝑆+1 (x) ∫ 𝑖=1 𝐷𝑡−𝑆∣𝑗 (x) = 𝒟𝑗∣𝑗 (𝝃, x) 𝑑𝝃 . 𝑏 where 𝐷𝑡−𝑆+1∣𝑡−𝑆+1 (⋅) accounts for the possible birth of new targets at time 𝑡 − 𝑆 + 1 and is obtained as 3.3 Multisensor SDA 𝑏 𝐷𝑡−𝑆+1∣𝑡−𝑆+1 (x) =

𝑁 ∏

[ℎ]

Λ𝑡−𝑆+1 (x) 𝑏𝑡−𝑆+1 (x) .

ℎ=1 [ℎ]

The correction factors Λ𝑡−𝑆+1 (⋅) are computed with a recursion similar to (5)-(6) where only the measurements belonging to 𝒴 𝑡−𝑆+1 are considered and the [0] initialization (7) is replaced by 𝐷𝑡−𝑆+1∣𝑡−𝑆+1 (x) = 𝑏𝑡−𝑆+1 (x). As to the PHD smoother block, here the multisensor PHD recursion is exploited in order to derive a fixedpoint PHD smoother that provides the smoothed PHD 𝐷𝑡−𝑠∣𝑗 (⋅) for 𝑗 = 𝑡 − 𝑆 + 1, . . . , 𝑡 given an initial PHD 𝐷𝑡−𝑆∣𝑡−𝑆 (⋅) (provided by the PHD reconstruction block at time 𝑡 − 1 as discussed above) and the measurement sets 𝒴𝑡−𝑠+1:𝑡 . To this end, following a classical strategy used in fixed-point smoothing of discrete-time signals [10, p. 170], an auxiliary state vector 𝜻 𝑗 is introduced with dynamics 𝜻 𝑗+1 = 𝜻 𝑗 ,

𝐽𝑆𝑖

belonging to the set

𝑗 = 𝑡 − 𝑆, . . . , 𝑡 − 1

initialized at time 𝑡 − 𝑆 by setting 𝜻 𝑡−𝑆 = x𝑡−𝑆 . It is immediate ( ) to see that the single-target augmented state x𝑗 , 𝜻 𝑗 admits the augmented state transition PDF ℱ(x, 𝜻∣𝝃, 𝝎) = 𝑓 (x∣𝝃) 𝛿(𝜻 − 𝝎)

where 𝛿(⋅) denotes the Dirac delta. Let now 𝒟𝑗∣𝑗 (x, 𝜻) denote the PHD in the singletarget augmented state space at time 𝑗. Then, under the simplifying assumption1 that in the interval [𝑡 − 𝑆, 𝑡] the target number is constant (i.e., 𝑏𝑗 (x) = 0 and 𝑃𝑠,𝑗 (x) = 1 for 𝑗 = 𝑡 − 𝑆 + 1, . . . , 𝑡), one can apply the multisensor PHD recursion and write ∫ 𝒟𝑗∣𝑗−1 (x, 𝜻) = 𝑓 (x∣𝝃) 𝒟𝑗−1∣𝑗−1 (𝝃, 𝜻) 𝑑𝝃 𝒟𝑗∣𝑗 (x, 𝜻) = =

In this section, the multisensor multiscan data association problem, referred to by the acronym MsSDA, is briefly described and discussed. The interested reader can refer to the textbooks [1]-[3] and the many references therein for more details. Let 𝑛𝑡−𝑆 be the number of tracks and suppose that, for each track 𝑖 = 1, 2, . . . , 𝑛𝑡−𝑆 , a PDF 𝑝𝑖𝑡−𝑆 (⋅) of the track state at the beginning of the association window is available (of course, in the proposed tracker such quantities are provided by the track merging block). Then, given the 𝑛 ¯ 𝑡−𝑆 tracks and the 𝑆𝑁 measurement sets 𝒴𝑡−𝑆+𝑘,ℎ (𝑘 = 1, 2, . . . , 𝑆 and ℎ = 1, 2, . . . , 𝑁 ), the aim is to assign to each track 𝑖 a vector of indices ⎡ 𝑖 ⎤ 𝐽1 ⎢ 𝐽2𝑖 ⎥ [ 𝑖 ]′ ⎢ ⎥ 𝑖 𝑖 𝐽𝑖 = ⎢ . ⎥ , where 𝐽𝑘𝑖 = 𝑗𝑘,1 , 𝑗𝑘,2 , . . . 𝑗𝑘,𝑁 ⎣ .. ⎦

𝑁 ∏

[ℎ]

Λ𝑡 (x) 𝒟𝑗∣𝑗−1 (x, 𝜻)

ℎ=1 1 Such an assumption could be relaxed but the resulting PHD smoother recursion would be quite cumbersome. So, due to space constraints, the general case is not discussed here.

△

𝒥 =

𝑆 ∏ 𝑁 ∏

{0, 1, . . . , 𝑚𝑡−𝑆+𝑘,ℎ }

(8)

𝑘=1 ℎ=1

where 𝑖 ∙ 𝑗𝑘.ℎ = 0, if at scan 𝑡 − 𝑆 + 𝑘 no measurement from sensor ℎ is assigned to track 𝑖; 𝑖 ∙ 𝑗𝑘,ℎ > 0, if at scan 𝑡 − 𝑆 + 𝑘 measurement 𝑖 y𝑗𝑘,ℎ ,𝑡−𝑆+𝑘,ℎ ∈ 𝒴𝑡−𝑆+𝑘,ℎ is assigned to track 𝑖.

Among all feasible assignments, an optimal one is found by minimizing a suitably defined cost. Let 𝑐(𝐽; 𝑖) denote the cost of associating a certain sequence 𝐽 ∈ 𝒥 to the track 𝑖 = 1, 2, . . . , 𝑛𝑡−𝑆 . As well known [3, 12], such a cost can be defined as the negative log-likelihood ratio 𝐿𝑖 (𝐽) 𝑐(𝐽; 𝑖) = − log (9) 𝐿𝑓 𝑎 (𝐽) where 𝐿𝑖 (𝐽) is the joint likelihood that the sequence of measurements corresponding to 𝐽 originated from track 𝑖, while 𝐿𝑓 𝑎 (𝐽) is the likelihood that all measurements of such a sequence are false. Given the track

2000

target 2 birth t=1 death t=39

1500 coordinate y (m)

PDF 𝑝𝑖𝑡−𝑆 (⋅) and the sequence of measurements corresponding to 𝐽, explicit formulas for computing 𝑐(𝐽; 𝑖) in a recursive fashion can be found in [3, 12]. Let us introduce the binary association variables 𝑎(𝐽; 𝑖) for all multiscan multisensor indices 𝐽 ∈ 𝒥 and tracks 𝑖 = 1, 2, . . . , 𝑛𝑡−𝑆 . Then the MsSDA (Multisensor SDA) problem can be formulated as follows:

target 1 birth t=1 death t=59

𝑛𝑡−𝑆 △

𝑎

∑ ∑

𝑐(𝐽; 𝑖) 𝑎(𝐽; 𝑖)

(10)

subject to the linear constraints: 𝑛𝑡−𝑆

∑ 𝑖=1

∑

𝑎(𝐽; 𝑖) ≤ 1

𝑘 = 1, 2, . . . , 𝑆;

𝐽∈𝒥 :𝑗𝑘,ℎ =𝑗

∑

𝑎(𝐽; 𝑖) = 1

ℎ = 1, 2, . . . , 𝑁 ; 𝑗 = 1, 2, . . . , 𝑚𝑡−𝑆+𝑘,ℎ (11) 𝑖 = 1, 2, . . . , 𝑛𝑡−𝑆 (12)

𝐽∈𝒥

Notice that (11) imposes that each measurement y𝑗,𝑡−𝑆+𝑘,ℎ is assigned to at most one track while (12) imposes that exactly one multiscan multisensor index 𝐽 ∈ 𝒥 is assigned to each track. The resulting MsSDA problem (10)-(12) is clearly a binary integer programming problem with 𝑂(𝑛𝑚𝑁 𝑆 ) variables and 𝑂(𝑚𝑁 𝑆) constraints. Unfortunately, this problem has been shown to be NP-hard for dimension 𝑁 𝑆 + 1 > 2. As a consequence, unless suitably small instances of the problem are considered, SDA cannot be solved exactly and several computationally tractable approximations have been devised. In particular, [11, 12] proposed a novel relaxation technique that is based on the idea of representing the association problem as a multi-commodity flow optimization problem on a suitable graph: each track corresponds to a commodity and the minimum cost path needs to be determined for each commodity. This latter technique has been adopted in the simulation experiments of Section IV for algoritmic implementation of SDA. To this end, since the arrangements of [11, 12] only pertained to the single-sensor case, a novel multisensor extension of such a relaxation technique has been developed. The details are omitted here for the sake of brevity and will be presented in a future paper.

4

Application tracking

to

bearing-only

The aim of this section is to assess the performance of the proposed hybrid PHD-SDA multitarget multisensor tracker for the challenging problem of bearing-only tracking. To this end, the algorithm is tested on the scenario of fig. 2 with the simulation parameters reported in table 1. In the simulation experiments, the processing blocks of PHD-SDA (see fig. 1) have been

target 4 birth t=20 death t=100

0 0

𝑖=1 𝐽∈𝒥

target 5 birth t=1 death t=59

1000

500

min 𝑉 (𝑎) =

target 3 birth t=20 death t=100

500

1000

1500 2000 coordinate x (m)

2500

3000

Figure 2: Simulated scenario: black square are sensor locations implemented as follows: ∙ particle representation of the PHD function [14] in the PHD smoother and correction blocks; ∙ clustering/peak extraction algorithm in [9] for track extraction; ∙ multisensor extension of the multicommodity algorithm proposed in [11, 12] for the SDA block; ∙ locally linearized particle filter [13, section 3.5.4] for track filtering. The proposed feedback PHD-SDA tracker of fig. 1 has been compared with the following trackers: ∙ OLPHD-SDA (Open Loop PHD-SDA), i.e. feedforward connection of the PHD smoother, track extraction and MsSDA blocks of fig. 1; ∙ PHDS, i.e. PHD smoother with track extraction; ∙ PHDF, i.e. standard PHD filter with track extraction. For each target, appearing and disappearing at a different time in the scenario, the motion follows a nearlyconstant velocity model. It is assumed that targets are observed independently by 𝑁 bearing sensors, each being characterized by the measurement equation ) ( 𝜃𝑡ℎ = 𝑎𝑡𝑎𝑛2 𝑥𝑡 − 𝑥ℎ , 𝑦𝑡 − 𝑦 ℎ + 𝑣𝑡ℎ (13)

where: (𝑥𝑡 , 𝑦𝑡 ) is the target position at (time 𝑡;) 𝑎𝑡𝑎𝑛2 (⋅, ⋅) stands for 4-quadrant arctangent; 𝑥ℎ , 𝑦 ℎ is the position of sensor ℎ and 𝑣𝑡ℎ is a zero-mean white measurement noise with covariance 𝜎𝜃2 . In addition, each sensor has non unit detection probability 𝑃𝑑 and generates clutter measurements whose number is Poisson-distributed with mean value 𝑛𝑐 and positions are uniformly distributed in the measurement space (see fig. 3). The Monte Carlo simulation results are displayed in figs. 4-8. Specifically: figs. 4 and 5 show the statistics (mean and std. deviation) of the estimated number of targets; figs. 6 and 7 plot the OSPA (Optimal SubPattern Assignment) metric [16] for two different values of cutoff; fig. 8 plots the position RMSE. Notice that: ∙ the PHD smoother greatly reduces the std. deviation of the target number estimate (figs. 5 and 7) but does

−2

Sensor 2 Sensor 3

40 60 Simulation Time (s)

80

100

2 0 −2 0

Sensor 4

20

Number of Targets

5 0

Sensor 5

True Number PHD−SDA OL PHD−SDA=PHDS PHDF

6

0

20

40 60 Simulation Time (s)

80

4 3 2

100

1

2

0

0

0

20

40 60 Simulation Time (s)

80

100

−2 0

20

40 60 Simulation Time (s)

80

100

0

20

40 60 Simulation Time (s)

80

100

Figure 4: Mean of the estimated number of targets

2 0 −2

2 0 −2 0

20

40 60 Simulation Time (s)

80

100

[𝑚/𝑠] [∘ ]

[𝑠]

𝐷𝑒𝑠𝑐𝑟𝑖𝑝𝑡𝑖𝑜𝑛 𝑚𝑒𝑎𝑛 𝑡𝑎𝑟𝑔𝑒𝑡 𝑠𝑝𝑒𝑒𝑑 𝑎𝑧𝑖𝑚𝑢𝑡ℎ 𝑠𝑡𝑑. 𝑑𝑒𝑣. 𝑑𝑒𝑡𝑒𝑐𝑡𝑖𝑜𝑛 𝑝𝑟𝑜𝑏. 𝑎𝑣𝑔.# 𝑐𝑙𝑢𝑡𝑡𝑒𝑟 𝑚𝑒𝑎𝑠. 𝑎𝑠𝑠. 𝑠𝑐𝑎𝑛𝑏𝑎𝑐𝑘 𝑠𝑎𝑚𝑝𝑙𝑖𝑛𝑔 𝑝𝑒𝑟𝑖𝑜𝑑 # 𝑀 𝑜𝑛𝑡𝑒 𝐶𝑎𝑟𝑙𝑜 𝑡𝑟𝑖𝑎𝑙𝑠

0.8 0.6 0.4 0.2

Figure 3: Measurement realization for each sensor. Circles and ‘+’ represent target and, respectively, clutter measurements. 𝑃 𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑣𝑚 = 15 𝜎𝜃 = 1 𝑃𝑑 = 0.8 𝑛𝑐 = 3 𝑆=3 𝑇𝑠 = 1 𝑁𝑀 𝐶 = 100

PHD−SDA OL PHD−SDA=PHDS PHDF

1 STD. deviation

Sensor 1

2

0

0

20

40 60 Simulation Time (s)

80

100

Figure 5: Standard deviation of the estimated number of targets

hibit peaks only at target births and PHDF exhibits no peaks. This is due to the fact the PHD-SDA is multiscan in both track initialization and termination while OLPHD-SDA and PHDS are multiscan only for track initialization and PHDF is single-scan.

Table 1: Simulation parameters

not provide significant advantages in terms of target localization accuracy (figs. 6 and 8); ∙ the OLPHD-SDA tracker, thanks to the track refinement by means of the hard association, yields better performance than the PHD smoother alone (figs. 6 and 8); ∙ the presence of feedback in the PHD-SDA tracker provides remarkable performance improvements with respect to OLPHD-SDA in terms of both estimated number of targets (figs. 5 and 7) and target tracking accuracy (figs. 6 and 8); ∙ the enhanced performance and robustness achieved by the introduction of a 𝑆-steps window is paid with a reduced responsiveness in track initialization and termination (fig. 4); ∙ the closed-loop PHD-SDA exhibits significant OSPA peaks (figs. 6 and 7 ) in correspondence of both target birth and deaths while OLPHD-SDA and PHDS ex-

5

Conclusions

A novel multitarget multisensor tracking technique based on the feedback connection of a PHD (Probability Hypothesis Density) smoother and of multisensor SDA (S-Dimensional Association) has been presented. This improves existing multitarget trackers based on PHD [6, 7] thanks to the exploitation of feedback and also extends previous work [8] to a “truly multisensor” setting wherein target initialization necessarily requires measurements from multiple sensors (e.g. bearing-only and range-only tracking). The soft association approach of the PHD smoother allows initialization of the target tracks while the feedback from the SDA’s hard association allows to refine the estimated target tracks, thus overcoming the well known pitfalls of the PHD update. The effectiveness of the proposed tracker has been demonstrated by simulation experiments concerning a bearing-only tracking case study.

40

50 PHD−SDA OL PHD−SDA PHDS PHDF

30 25 20

0

20

40 60 Simulation Time (s)

80

100

Figure 6: OSPA distance for order 2 and cutoff=40 350 PHD−SDA OL PHD−SDA PHDS PHDF

OSPA Distance

300 250 200 150 100 50 0

30

20

10

15 10

PHD−SDA OL PHD−SDA PHDS PHDF

40 position RMSE

OSPA Distance

35

0

20

40 60 Simulation Time (s)

80

100

Figure 7: OSPA distance for order 2 and cutoff=500

Acknowledgments This work has been partially supported by SELEX Sistemi Integrati.

References

0

0

20

40 60 Simulation Time (s)

80

100

Figure 8: position RMSE averaged over all five targets filter”, IEEE Trans. on Aerospace and Electronic Systems, vol.43, n.2, pp. 556-570, 2007. [8] F. Papi, G. Battistelli, L. Chisci, S. Morrocchi, A. Farina and A. Graziano: “Multitarget tracking via joint PHD filtering and multiscan association”, Proc. 12th Int. Conf. on Information Fusion, pp. 1163-1170, Seattle, USA, 2009. [9] M. Tobias and A.D. Lanterman: “Techniques for birthparticle placement in the probability hypothesis density particle filter applied to passive radar”, IET Radar Sonar Navigation, vol. 2, n. 5, pp. 351-365, 2008. [10] B.D.O. Anderson and J.B. Moore: Optimal filtering, Prentice Hall, 1979. [11] G. Battistelli, L. Chisci, F. Papi, A. Benavoli and A. Farina: “Multiscan association as a multi-commodity flow optimization problem”, Proc. 2008 IEEE Radar Conference, Rome, Italy, May 2008.

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