Multisensor Tracking Problem - CiteSeerX

The Multitarget/Multisensor Tracking Problem Alexander Toet, Huub de Waard 11 April 1995

 CALMA Report CALMA.TNO.WP31.AT.95d

1

CONTENTS

Page

SUMMARY

3

SAMENVATTING

4

1 INTRODUCTION

5

2 THE MULTITARGET/MULTISENSOR TRACKING PROBLEM

5

3 THE MULTIDIMENSIONAL ASSIGNMENT FORMULATION

7

4 CONCLUDING REMARKS

11

REFERENCES

13

The Multitarget/Multisensor Tracking Problem A. Toet and H. de Waard

SUMMARY Various combinatorial optimization techniques are currently available. Most of these techniques have not been thourougly tested on realistic problems. The EUCLID (EUropean Cooperation for the Long term In Defence) CALMA (Combinatorial Algorithms for Military Applications) RTP (Research and Technology Project) 6.4 project has the main objective to investigate the relevance of various existing algorithmic optimization techniques to the actual solution of complex combinatorial problems arising in military applications. It has recently been shown that the data association problems arising both in multitarget/multisensor tracking and in multisensor data fusion can be formulated as multidimensional assignment problems (Poore, 1994). This report briefly reproduces the derivation of this formulation. No attempt is made to solve the multitarget/multisensor tracking and multisensor data fusion problems here. However, it is suggested that the assignment formulation turns these problems into good candidates for the testbed that is required for the last part of the CALMA study, which will be a comparative evaluation of the different optimization techniques that were studied and/or developed in the earlier stages of this project. The outcome of this comparative study may lead to a better understanding of the applicability of these methods.

Het Volgen van Meerdere Doelen A. Toet en H. de Waard

SAMENVATTING Er zijn tegenwoordig verscheidene combinatorische optimalizatietechnieken beschikbaar. De meeste daarvan zijn niet volledig getest op realistische problemen. Het EUCLID (EUropean Cooperation for the Long term In Defence) CALMA (Combinatorial Algorithms for Military Applications) RTP (Research and Technology Project) 6.4 project heeft als voornaamste doel het onderzoeken van de bruikbaarheid van verschillende bestaande combinatorische optimalizatie technieken voor het oplossen van complexe combinatorische problemen zoals die voorkomen in militaire omgevingen. Recentelijk werd aangetoond dat de data associatie problemen die zowel in multitarget/multisensor tracking en in multisensor fusie voorkomen geformuleerd kunnen worden als een multidimensionaal toewijzings probleem. Dit rapport geeft een beknopte afleiding van deze formulering. De multitarget/multisensor tracking en multisensor data fusie problemen zelf worden in dit rapport niet opgelost. Er wordt slechts voorgestelt dat de herformulering van deze problemen als toewijzingsproblemen ze goed geschikt maakt om te dienen als een testomgeving waarin de verschillende optimalizatietechnieken, die in eerdere stadia van het CALMA project werden ontwikkeld en bestudeerd, met elkaar kunnen worden vergeleken. De uitkomst van een dergelijke excercitie kan mogelijk inzicht verschaffen in de praktische toepasbaarheid van deze technieken.

1 INTRODUCTION In military operations, problems in planning and scheduling often require feasible and close to optimal solutions with limited computing resources and within very short time periods. Various approaches to combinatorial problem solving have emerged over the last three decades. Most of these have only been tested on a few simplified problems (e.g. the “Traveling Salesman Problem”). It is presently not known to what extent they can solve complex realistic problems as those arising in military appplications, and what the required computing resources are. The EUCLID (EUropean Cooperation for the Long term In Defence) CALMA (Combinatorial Algorithms for Military Applications) RTP (Research and Technology Project) 6.4 project has the main objective to investigate the relevance of the various existing algorithmic techniques to the actual solution of some of the most complex combinatorial problems arising in military applications. In the context of the CALMA project, the application of various existing algorithmic optimization techniques to some realistic problems of military relevance may serve to gain a better understanding of what makes a combinatorial problem solving approach adequate on some problems and inadequate on others. The various techniques may for instance be compared with respect to - the quality of the solution, - the runtime performances, - the tradeoff between quality and computing time, - the memory requirements, - the ease of implementation and the flexibility e.g. with respect to progressively adding new constraints to the problem, and - the stability with respect to small perturbations in the data. This report briefly reproduces the derivation of a multidimensional assignment formulation of the data association problems arising both in multitarget/multisensor tracking and in multisensor data fusion. This formulation was first presented by Poore (1994). It turns the abovementioned problems into optimization problems that may serve as testbeds to compare the relative performance of the combinatorial optimization techniques studied in the CALMA project.

2 THE MULTITARGET/MULTISENSOR TRACKING PROBLEM One of the key functions performed by a surveillance system is to keep track of all the targets of interest within the coverage region of its sensors. For military surveillance systems, the coverage region generally involves several thousand cubic miles containing several hundred targets. Typical sensors used in these surveillance systems (e.g. radars) cannot provide perfect information about the targets. In general, sensor measurements of the

targets tend to be ambiguous (it is not clear to which target a measurement corresponds), incorrect (reports of false targets), and imprecise (random errors in the measurements composing a report). The design of a multitarget/multisensor tracking algorithm for such surveillance systems poses a difficult problem. Multitarget tracking has many applications, both in military and in civilian areas. For instance, application areas include ballistic missile defense (reentry vehicles), air defense (enemy aircraft), air traffic control (civil air traffic), ocean surveillance (surface ships and submarines), and battlefield surveillance (ground vehicles and military units). The central problem of multitarget tracking is that of data association — the problem of determining from which target, if any, a particular measurement originates (Bar-Shalom and Fortman, 1988; Blackman, 1986; Reid, 1979; Waltz and Llinas, 1990). The problem is especially difficult in situations where there are missing reports (probability of detection less than unity), unknown targets (requiring track initiation), and false reports (from clutter). Current methods for multitarget tracking generally fall into two categories: sequential and deferred logic. Methods based on sequential logic include nearest neighbour, one-to-one or few-to-many assignments, and all-to-one assignments (Bar-Shalom and Fortman, 1988). For track maintenance, the nearest neigbour method is valid in the absence of clutter when there is little or no track contention, i.e. when there is little chance of misallocation. Problems involving one-to-one or few-to-one assignments are generally formulated as (2D) assigment or multi-assignment problems, for which there are some excellent algorithms (Bertsekas, 1991; Bertsekas and Casta˜non,1992; Jonker and Volgenant, 1987). This methodology can be performed real-time, but can result in a large number of partial and incorrect assignments in dense and high contention scenarios, and thus incorrect track identification. The difficulty is that decisions, once made, are irrevocable, so that there is no mechanism to correct misassociations. The use of all observations in a scan or within a neighbourhood of a predicted track position to update a track has been successful in tracking a few targets in dense clutter (Bar-Shalom and Fortman, 1988). Deferred logic techniques consider several data sets or scans of data all at once in making data associations. At one extreme is batch processing, in which all observations (from all time) are processed together. This is computationally too intensive for real-time applications. The other extreme is sequential processing. The principal deferred logic method used to track large numbers of targets in low to moderate clutter is called multiple hypothesis tracking (MHT). This method builds a tree of posibilities, assigns a likelihood score to each track, develops an intricate pruning logic, and then solves the data association problem by an explicit enumeration scheme. The use of these enumeration schemes to solve this NP-hard combinatorial optimization problem in real-time is inevitably faulty in dense scenarios, since the time required to solve the problem optimally can grow exponentially in the size of the problem. Another important aspect in surveillance systems is the growing use of multisensor data fusion (Bar-Shalom, 1990; Blackman, 1990; Deb et al., 1993; Waltz and Llinas, 1990), in which one associates reports from multiple sensors together. Once matched, this more

varied information has the potential to greatly enhance target identification and state estimation (Blackman, 1990). The central problem is again that of data association and the principal method employed is multiple hypothesis tracking. Similar data association problems occur in other fields of research, such as edge grouping and contour segmentation in image analysis (e.g. Cox et al., 1993). Some general classes of data association problems in multitarget tracking and multisensor data fusion have recently been formulated as multidimensional assignment problems (Pattipati et al., 1990; Poore, 1994). The objective function is derived from a composite negative log posterior or likelihood function for each of the track reports. This formulation covers the popular multiple hypothesis tracking method introduced by Reid (1979) and modified by Kurien (1990) to include maneuvering targets and terminations, and the work of Deb et al. (1993) on centralized multisensor data fusion. The only known method to solve the NP-hard multidimensional assignment problems is branch-and-bound. However, this approach is too computationally intensive to produce real time solutions, especially in dense scenarios. Since the objective function is generally noisy due to various sources of errors (e.g. plant noise, observation errors, and uncertainty about the exact values of various probability parameters), one only needs to solve these problems to or just below the noise level in the problem. The goal of algorithm development is therefore the computation of high quality and near optimal solutions. Several algorithms based on Lagrangian relaxation have been and continue to be developed (Poore and Rijavic, 1993, 1994). These algorithms are near optimal, provide a lower and upper bound on the optimal value, and work real time for many scenarios.

3 THE MULTIDIMENSIONAL ASSIGNMENT FORMULATION In tracking, a common surveillance problem is to estimate the past, current, or future state of a collection of objects (e.g. airplanes), moving in three dimensional space, from a sequence of measurements made of the surveillance region by one or more sensors. The objects will be called targets. The dynamics of these targets are generally modeled from physical laws of motion. However, there may be noise in the dynamics and certain parameters of the motion may be unknown. At time t = 0 one or more sensors are turned on to observe the region. In an ideal situation measurements are taken at a finite sequence of times ftk gnk=0 , where 0 = t0 < t1 <


< tn . Due to the finite amount of time required for a sensor to sweep the surveillance region, measurements are generally made asynchronously, i.e. not at the same time, so that a time tag is associated with each measurement. At each time tk the sensor produces a sequence of measurements Z (k) = fzikk gMik =k 1 where each zikk is a vector of noise contaminated measurements. The actual type of measurement is sensor dependent. For example, a 2D radar measures range and azimuth of each potential target, a 3D radar measures range, azimuth, and elevation, a 3D radar with Doppler measures these together with the time derivative of the range, and a 2D passive sensor measures the azimuth and elevation angle. Some of the measurements may be false, and the number of targets and which measurement emanates from which target are not known a priori. The

problems are then (a) to determine the number of targets, which measurements go with which targets and which are false (i.e. the data association problem), and (b) to estimate the state of each target given a sequence of measurements that emanate from that target. As part of posing the data association problem one must estimate the state of a potential target given a particular sequence of measurements from the data sets fZ (k )gN k=1 . Thus, the two problems of data association and state estimation are inseparable parts of the same problem. In the surveillance example, the data sets are measurements made of the surveillance region. To allow for more general types of data such as tracks and target descriptions, the elements in the data sets will be called reports (Reid,1979). Let Z (k ) denote a data set of Mk reports fzikk gMik =k 1 and let Z N denote the cumulative data set of N such sets defined by

Z (k) = fzikk gMik k 1

Z N = fZ (1); : : : ; Z (N )g

and

=

(1)

respectively. In multisensor data fusion and multitarget tracking the data sets Z (k ) may represent different classes of objects. For track initiation in multitarget tracking the objects are measurements that must be partitioned into tracks and false alarms. For track maintenance, one data set will be tracks, and the remaining data sets will be measurements which are either (a) assigned to existing tracks, (b) marked as false measurements, or (c) used to initiate new tracks. Now it will be defined what is meant by a partition of the cumulative data set Z N in Equation 1. Since this partition is to be independent of the actual data in the data set Z N , the partition will be defined on the set of indices in Z N . Let

I N = fI (1); I (2); : : :; I (N )g

where

denote the indices in the data sets (1). A partition partitions Γ is defined by

I (k) = fik gMik k 1 =

of I N and the collection of all such

= f 1; : : : ; n j 8i : i 6= ;g ;

i \ j = ; for i 6= j ;

(3) (4)

( )

I Γ

=

(2)

[

n( )

j ;

(5)

f j satisfies (3) ? (5)g :

(6)

N

=

j =1

Here, i in Equation 3 represents a track, so that n( ) denotes the number of tracks (or elements) in the partition . A 2 Γ is called a set partitioning of the indices I N if properties (3)–(5) are valid, a set covering of I N if property (4) is omitted but the other two properties (3) and (5) are retained, and a set packing if (5) is omitted but (3) and (4) are retained. A partition 2 Γ of the index set I N induces a partition of the data set Z N via

Z = fZ ; : : : ; Z n g 1

( )

where

Z i = ffzikk gik 2 i gNk 1 :

Sn( )

=

(7)

Clearly, Z i \ Z j = ; for i 6= j and Z N = j =1 Z j . Each Z j will be called a track of data. The definition of a partition in Equations 3 and 7 implies that each actual report belongs to at most one track of reports Z i in a partition Z of the cumulative data set.

The combinatorial optimization problem that governs a large number of data association problems in multitarget tracking and multisensor data fusion is generally posed as

(

Max

P (Γ = j Z N ) j 2 Γ P (Γ = 0 j Z N )

)

(8)

where

ZN

Γ Γ

0 P (Γ = j Z N )

represents N data sets (1), is a partition of the indices of the data (Eqs. 2 and 3), is the finite collection of all such partitions (3), is a discrete random element defined on Γ , is a reference partition, and is the posterior probability of a partition being true given the data Z N .

The objective function in the optimization problem (8) can be converted into an equivalent linear one by making some independence assumptions for the density function p(Z N j Γ =

) and the probability PΓ (Γ = ) in Bayes’ formula N P (Γ = j Z N ) = p(Z j Γ p=(Z N)P) Γ(Γ = ) :

These assumptions are respectively

Y

p(Z i j Γ = ) ;

(9)

p(Z N j Γ = )

=

p(Z i j Γ = )

=

p(Z i j Γ = !) 8 ; ! 2 Γ ;

(11)

PΓ (Γ = )

=

C

(12)

i 2

Y

n( ) i=1

G( i) ;

(10)

where

C G

is a constant independent of the partition 2 Γ , and is a probability distribution on the set of tracks i in Equation 3.

A probabilistic framework that illustrates each of these assumptions was presented by Poore (1994). Since for each 2 Γ , Z corresponds to a partition of the data into n( ) feasible tracks of data, Equation 10 says that the n( ) tracks of data, Z 1 ; : : : ; Z n( ) , are independent if

is the true track. (There will of course be dependence between reports within a single track.) Equation 11 states that these tracks are independent across all partitions of the data. Substitution of Equation 10 in Bayes’ formula (Eq. 9) gives

P (Γ = j Z N )

=

N p(Z N ) p(Z j Γ = ) PΓ(Γ = )

1

(13)

=

1

2n( ) 3 Y 4 p(Z i )5 P (Γ = )

(14)

p(Z N ) i 1 C Y p(Z ) G( ) i p(Z N ) i2 i =

=

(15)

Equation 13 is Bayes’ formula. If PΓ (Γ = ) = C , a constant, over all partitions 2 Γ , then Equation 14 is proportional to the likelihood function. Otherwise, the expansion (15) follows from Equation 12. Thus, Equation 15 includes both the likelihood function with the idenfication C = 1 and G( i ) = 1 and the posterior function for a more general probability distribution G on the tracks. This equation will now be transformed into an assignment formulation . For notational convenience in representing tracks, a zero index is added to each of the index sets I (k ) (k = 1; : : : ; N ) in Equation 2, a dummy report z0k to each of the data sets in Equation 1, while

i Z i

i ; : : : ; iN ) (16) 1 N = Zi :::in  (zi ; : : : ; ziN ) where ik and zikk can now adopt the values 0 and z0k respectively. The dummy report z0k serves several purposes in the representation of missing data, false reports, initiating and terminating tracks. If Z i is missing an actual report from the data set Z (k ), then i = (i1 ; : : : ; ik?1 ; 0; ik 1; : : : ; iN ) and Z i = fzi1 ; : : : ; zikk??1 ; 0; zikk 1 ; : : : ; ziNN g. A false report zikk (ik > 0) is represented by i = (0; : : : ; 0; ik ; 0; : : : ; 0) and Z i = fz01 ; : : : ; z0k?1; zikk ; z0k 1; : : : ; z0N g in which there is but one actual report. The partition

0 of the data in which all reports are declared to be false reports is defined by Z = fZ0:::0 ik 0:::0  z01 ; : : : ; z0k?1; zikk ; z0k 1; : : : ; z0N j ik = 1; : : : ; Mk ; k = 1; : : : ; N g : =

( 1

1

1

+

+

1

1

+1

+

+

0

(17) Let each data set Z (k ) represent a scan of measurements. A track that initiates on scan m > 1 will contain only the dummy report z0k from each of the data sets Z (k) for each k = 1; : : : ; m ? 1. Likewise, a track that terminates on scan m only has the dummy report from each of the data sets for k > m.

The cumlative data set Z N can be seen as a set of equality constraints with the help of a binary 0-1 variable  (i1 ; : : : ; iN ) 2 , zi1:::iN = 10;; ifotherwise which transforms Equations 3 and 17 into ?X

(M1 ;:::;Mk 1 ;Mk+1 ;:::;MN )

?

(i1 ;:::;ik 1 ;ik+1 ;:::;iN )=(0;:::;0)

zi :::iN = 1 for ik = 1; : : : ; Mk and k = 1; : : : ; N : 1

(18)

Equation 15 can be written as a likelihood ratio:

P (Γ = j Z N )  L  P (Γ = 0 j Z N )

Y 2

(i1 :::iN )

Li :::iN 1

(19)

where

p(Zi :::iN ) G(Zi :::iN ) : 0 p(Zo:::0 ik 0:::0 )G(Zo:::0 ik 0:::0 )

Li :::iN = QN 1

1

1

k=1;ik 6=

Let

ci :::iN = ? ln Li :::iN ; 1

so that

? ln

"

(21)

1

P ( j Z N ) P ( 0 j Z N )

#

X

=

2

(i1 :::iN )

(20)

ci :::iN : 1

(22)

The optimization problem (8) can now be written as the following N-dimensional assignment problem: Min

M1 X

i1 =0






X

MN

iN =0

ci :::iN zi :::iN 1

1

(23)

subject to

P M2




PMiNN 0 zi :::iN = 1 ; i1 = 1; : : : ; M1 ; PM PMk? PMk PMN i 0


ik? 0 ik 0


iN 0 zi :::iN = 1 ; for ik = 1; : : : ; Mk and k = 2; : : : ; N ? 1 , PM PMN ? i 0


iN ? 0 zi :::iN = 1 ; iN = 1; : : : ; MN ; zi :::iN 2 f0; 1g for all i1 ; : : : ; iN ; i2 =0 1 1=

1 1=

=

1

+1

1

1=

+1 =

=

1

1

1=

1

1

where c0:::0 is arbitrarily defined to be zero.

The complexity of the optimization problem (23) makes its formulation and solution formidable. In fact, it is NP-hard (Garvey and Johnson, 1979). The computation of all the cost coefficients can be a time consuming task. For example, six scans of measurements with one hundred measurements per scan requires the computation of one trillion cost coefficients for track initiation. Thus preprocessing is essential to reduce the complexity. One class of preprocessing methods for sensor fusion and tracking is called gating (Bar-Shalom and Fortman, 1988; Blackman, 1986). The idea of gating is to compute only those cost coefficients that are feasible for the underlying physical problem, thereby removing unlikely pairings of measurements. Another commonly used complexity reducing technique is that of clustering (Bar-Shalom, 1990; Blackman, 1990). This method decomposes the problem into a number of smaller independent problems.

4 CONCLUDING REMARKS The data association problems arising both in multitarget/multisensor tracking and in multisensor data fusion are actual real-time defence industry problems with a large number of military and civilian appplications. The multidimensional assignment formulation of

these problems (Pattipati et al., 1990; Poore, 1994) turns them into good candidates for the testbed that is required for the last part of the CALMA study, which will be a comparative evaluation of the different optimization techniques that were studied and/or developed in the earlier stages of this project. The outcome of the comparative study may lead to a better understanding of the applicability of these optimization methods.

REFERENCES

Bar-Shalom, Y. (Ed., 1990) Multitarget-multisensor Tracking: Advanced Applications. Artech House, Norwood, MA. Bar-Shalom, Y. and Fortman, T.E. (1988). Tracking and Data Association. Academic Press, Boston. Bertsekas, D.P. (1991). Linear Network Optimization: Algorithms and Code. MIT Press, Cambridge, Mass. Bertsekas, D.P. and Casta˜non, D.A. (1991). A forward/reverse auction algorithm for asymmetric assignment problems. Computational Optimization and Applications 1, pp. 277–298. Blackman, S.S. (1986). Multiple Target Tracking with Applications. Artech House, Norwood, MA. Blackman, S.S. (1990). Association and fusion of multiple sensor data. In: Bar-Shalom, Y. (Ed.), Multitarget-multisensor Tracking: Advanced Applications. Artech House, Norwood, MA, pp. 187–218. Cox, I.J., Rehg, J.M. and Hingorani, S. (1993). A Bayesian multiple hypothesis approach to contour grouping and segmentation. Int. J. of Computer Vision 11, pp. 5–24. Deb, S., Pattipati, K.R., and Bar-Shalom, Y. (1993). A multisensor-multitarget data asociation algorithm for heterogeneous systems. IEEE Tr. on Aerospace and Electronic Systems 29, pp. 560-568. Garvey, M.R. and Johnson, D.S.(1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Co., CA. Jonker, R. and Volgenant, T. (1987). A shortest augmenting path algorithm for dense and sparse linear assignment problems. Computing 38, pp. 325–340. Kurien, T. (1990). Issues in the designing of practical multitarget tracking algorithms. In: BarShalom, Y. (Ed.), Multitarget-multisensor Tracking: Advanced Applications. Artech House, Norwood, MA, pp. 43–83. Pattipati, K.R., Deb, S., and Bar-Shalom, Y. (1990). Passive multisensor data association using a new relaxation algorithm. In: Bar-Shalom, Y. (Ed.), Multitarget-multisensor Tracking: Advanced Applications. Artech House, Norwood, MA, pp. 43–83. Poore, A.B. (1994). Multidimensional assignment formulation of data association problems arising from multitarget and multisensor tracking. Computational Optimization and Applications 3, pp. 27–57. Poore, A.B. and Rijavic, N. (1993). A Lagrangian relaxation algorithm for multidimensional assignment problems. In: Drummond, O.E. (Ed.), Proc. of the 1991 SPIE Conf. on Signal and Data Processing of Small Targets, vol. 1481, pp. 345–356. Poore, A.B. and Rijavic, N. (1994). Partitioning multiple data sets: multidimensional assignments and Lagrangian relaxation. To appear in: DIMACS Series in Discrete Mathematics and Theoretical Computer Science. Reid, D.B. (1979). An algorithm for tracking multiple targets. IEEE Tr. on Automatic Control AC-24, pp. 843–845.

Waltz, E. and Llinas, J. (1990). Multisensor Data Fusion. Artech House, Boston.

Soesterberg, 11 April 1995

Dr. A. Toet

30 woorden abstract Er wordt een beknopte afleiding gegeven van de manier waarop het probleem van het volgen van meerdere doelen, met gebruikmaking van meerdere sensoren, kan worden herschreven als een optimalizatie probleem (en wel een meer-dimensionaal toewijzingsprobleem). In deze formulering kan dit probleem goed dienen als een testomgeving waarin de verschillende optimalizatietechnieken, die in eerdere stadia van het CALMA project werden ontwikkeld en bestudeerd, met elkaar kunnen worden vergeleken. Descriptors Target Acquisition

Identifiers multiple hypothesis testing multiple target tracking multisensor fusion optimization

MANAGEMENT UITTREKSEL Er zijn tegenwoordig verscheidene combinatorische optimalizatietechnieken beschikbaar. De meeste daarvan zijn niet volledig getest op realistische problemen. Het EUCLID (EUropean Cooperation for the Long term In Defence) CALMA (Combinatorial Algorithms for Military Applications) RTP (Research and Technology Project) 6.4 project heeft als voornaamste doel het onderzoeken van de bruikbaarheid van verschillende bestaande combinatorische optimalizatie technieken voor het oplossen van complexe combinatorische problemen zoals die voorkomen in militaire omgevingen. Het volgen van meerdere doelen heeft veel toepassingen, zowel in de militaire als ook in de civiele sfeer. Toepassingen zijn o.a. de verdediging tegen raketaanvallen (reentry vehicles), luchtverdedigingssytemen (tegen vijandige vliegtuigen), lucht verkeerscontrole (civiele luchtvaart), bewaking van zeegebieden (voor oppervlakte verkeer en duikboten), en slagveld bewaking (de detectie van voertuigen en militaire eenheden). Een van de belangrijkste functies van een surveillance systeem is het volgen van alle interessante doelen binnen het door de sensoren bestreken gebied. Voor militaire systemen beslaat dit gebied vaak enkele duizenden kubieke kilometers die honderden doelen kunnen bevatten. Doorgaans geven de gebruikte sensoren (bijv. radar) geen perfecte informatie over de doelen. De sensor metingen zijn vaak ambigu (het is niet duidelijk van welk doel een meting afkomstig is), incorrect (een meting correspondeert met een vals alarm), en onnauwkeurig (random fouten in de meetwaarden). Deze problemen maken het ontwerpen van algoritmen voor het volgen van meerdere doelen met gebruikmaking van meerdere sensoren tot een complexe taak. Recentelijk werd aangetoond dat de data associatie problemen, die zowel in multitarget/multisensor tracking en in multisensor fusie voorkomen, geformuleerd kunnen worden als een multidimensionaal toewijzings probleem. Dit rapport geeft een beknopte afleiding van deze herformulering. Het probleem zelf wordt in dit rapport niet opgelost. Er wordt slechts voorgestelt dat de herformulering van deze problemen als toewijzingsproblemen ze goed geschikt maakt om te dienen als een testomgeving waarin de verschillende optimalizatietechnieken, die in eerdere stadia van het CALMA project werden ontwikkeld en bestudeerd, met elkaar kunnen worden vergeleken. De uitkomst van een dergelijke excercitie kan mogelijk inzicht verschaffen in de praktische toepasbaarheid van deze technieken.