Generalized Murty's Algorithm With Application to Multiple Hypothesis

Generalized Murty’s Algorithm With Application to Multiple Hypothesis Tracking Evan Fortunato, William Kreamer, Shozo Mori, Chee-Yee Chong, Gregory Castanon BAE Systems, Advanced Information Technologies, Burlington, MA, U.S.A. {evan.fortunato,bill.kreamer,shozo.mori,chee.chong,greg.castanon}@baesystems.com This work supported by DARPA/IXO and AFRL/IFKE under Contract No. FA8750-05-C-0115 Approved for public release; distribution is unlimited Abstract – This paper describes a generalization of models was derived in [19] for i.i.d.-Poisson cases, and for Murty’s algorithm generating ranked solutions for general i.i.d. but non-Poisson cases in [20-21]. The recursive MHT algorithm described in [5] can be classical assignment problems. The generalization extends the domain to a general class of zero-one integer linear viewed as a repeated application of track-to-report bipartite programming problems that can be used to solve multi- assignment hypothesis generation and evaluation. An frame data association problems for track-oriented algorithm that selects only a single best track-to-report multiple hypothesis tracking (MHT). The generalized assignment hypothesis and propagates it forward can be Murty’s algorithm mostly follows the steps of Murty’s viewed as an extreme form of approximate recursive MHT ranking algorithm for assignment problems. It was algorithm, called the zero-scan algorithm. Such a singleimplemented in a hybrid data fusion engine, called All- best hypothesis selection can be formulated [6] as a classical Source Track and Identity Fusion (ATIF), to provide a k- assignment problem [7,8]. One approach of implementing best multiple-frame association hypothesis selection recursive MHT is by sequential application of k-best capability, which is used for output ambiguity assessment, bipartite assignment algorithms [9,10], so as to preserve multi-frame optimal data association solutions through the hypothesis space pruning, and multi-modal track outputs. recursion. It was found out [11] that application of Murty’s Keywords: Generalized Murty’s algorithm, data k-best algorithm to the bipartite, track-to-report assignment association hypothesis evaluation, k-best multiple frame is indeed very effective. assignment (MFA), multiple target tracking (MHT), trackOn the other hand, track-oriented MHT [12-14] is oriented MHT, All Source Track and Identify Fuser (ATIF). based on batch processing of data association hypothesis generation and selection in which a single best (multidimensional or multi-frame data assignment) hypothesis is 1 Introduction sought by a {0,1} integer programming [4] method, We are concerned with a generalization of Murty’s Lagrangian relaxation algorithms [15,16], or other algorithm that selects k-best solutions for bipartite techniques. The hybrid data fusion algorithm developed by assignment problems [1]. This generalization is to extend BAE Systems, called All Source Track and Identity Fusion applicable problems to a general class of {0,1} linear (ATIF), is based on the track-oriented MHT approach, but programming problems, and was motivated by the need to uses another kind of relaxation method for multi-frame data association [17] to process track reports as well as assess the quality of the solutions in track-oriented multiple measurement reports (thus called “hybrid”). hypothesis tracking. Thus, we need an effective algorithm to In difficult tracking situations, such as high target select not only the best but also the second, the third, and in density situations, the best multi-frame data association general down to the k-best multiple-frame data association hypothesis may only be slightly better than many other hypotheses. hypotheses. To quantitatively identify such situations, we The goal of multiple target tracking is to estimate the need the k-best (multiple) hypotheses to probabilistically states of an unknown number of dynamical objects, called assess the ambiguity of the data association solution(s). targets, using imperfect reports (or measurements) with Moreover, multi-frame hypothesis selection is used as the unknown origins, extraneous returns, and generally lessmain hypothesis management technique in track-oriented than-one probability of detecting any target [2,3]. Under MHT to control the growth of the hypothesis space. In high such a situation, it is necessary to decide or hypothesize (as target density situations, k-best (multiple) hypotheses a form of delayed decision) the origin of each report since selection is proven to be very useful in preserving seemingly state estimation is meaningful only when conditioned by a unlikely but potentially vital hypotheses. data association hypothesis. Multiple hypothesis tracking A k-best multi-frame hypothesis selection algorithm (MHT) techniques were devised to generate, evaluate, and was developed in [18] for the Lagrangian relaxation multimaintain multiple data association hypotheses. The earliest frame assignment algorithm described in [15]. The objective works on MHT algorithms produced a batch-processing of this paper is to describe a new k-best hypothesis selection algorithm in [4], and its recursive counterpart in [5]. algorithm for the particular (different) relaxation algorithm Bayesian hypothesis evaluation with general non-Gaussian used in ATIF, i.e., the linear programming algorithm where

1 0    0 A=  1  0    0

the integer constraints are relaxed to interval constraints. Moreover, as shown in Section 3, the new k-best hypothesis selection algorithm, described in this paper, can be applied to a slightly wider class of {0,1} integer programming optimization problems with a nonlinear objective function. The next section, Section 2, will define the class of problems to which our algorithm for ranking k-best solutions can be applied. It is followed by the algorithm description in Section 3. Section 4 describes the applications of our k-best hypothesis selection algorithm implemented in ATIF. Section 5 states our conclusions.

2

A Class of 0-1 Integer Programming Problems

Let n be a positive integer and consider a {0,1} integer programming problem, Minimize f ( x ) subject to x ∈{0,1}n and Ax = b

(1)

with an arbitrary functional1 f : {0,1}n → ( −∞, ∞) defined on the solution space that is the set of all the n -tuples of numbers either 0 or 1, i.e., {0,1}n , where the constraint m×n

matrix A is a zero-one matrix, i.e., A ∈ {0,1} , with a given positive integer m , and b is an m -vector whose elements are all 1s. Eqn. (1) defines a class of problems that is a proper extension of the bipartite assignment problems to which Murty’s k-best algorithm is applied, since any bipartite assignment can be defined by (1) with a linear objective function2 f ( x ) = cT x with3 c ∈ ( −∞, ∞)n and the constraint matrix A is expressed as

1

0 1 0

1 0

0 1

0

0

0 0

1 0

1

0

1

0 1 0

0 1

0 0

0

1

1

0 0

1 0

0 1

1

0

0

0 0    1 0  0   1 

(2)

Our problem is to find the best, the second-best, …, and up to k-best solutions, with a given integer k, to a problem for minimizing an arbitrary objective function f with an arbitrary {0,1}-valued matrix A , i.e., a finite sequence, x1 , x2 ,..., xk of k distinct feasible solutions, such and Axi = b for i = 1,..., k , and f ( x1 ) ≤ f ( x2 ) ≤ ≤ f ( xk ) . We assume that there is at least one feasible solution, i.e., an x ∈ {0,1}n such that Ax = b . We should note that that

xi ∈ {0,1}n

x ∈ {0,1}n is feasible if and only if (i) for every row j of A , there exists a column i such that x[i ] = A[ j , i ] = 1 , and

(ii) for any (i1 , i2 ) ∈ {1,..., n}2 such that i1 ≠ i2 , there is no row j of A such that4 A[ j, i1 ] = A[ j , i2 ] = x[i1 ] = x[i2 ] = 1 . This excludes any possibility of all-zero row of matrix A . We also assume that, for each i = 1,..., n , there exists a row j of A such that A[ j , i ] = 1 , to exclude any unnecessary anomaly. Constraint redundancy may not hurt us significantly. Nonetheless, since any pair of non-zero, {01}-vectors is linear dependent if and only if they are identical, any redundancy can be removed easily.

3

Generalized Murty’s Algorithm

Following Murty’s original paper [1], we will use “stage” in place of the more commonly used word “step,” in defining our algorithm. First we introduce some notations to clearly define successive decomposition of the feasible solution space at each stage, which is the essential concept of Murty’s ranking algorithm. Let S = {x ∈ {0,1}n | Ax = b} be the set of all feasible The functional f may take the value +∞ but such a case is excluded from this paper to avoid unnecessary complication. By {0,1}n , we mean the set of all the n -dimensional (column) vectors whose components are either 0 or 1. 2 T By X we mean the transpose of a vector or matrix X , so that T x y is the inner product when x and y are vectors of the same dimension. 3 Again the objective coefficient c[i ] ( i = 1,.., n ) may be in ( −∞, ∞ ] , but such a case is avoided for this paper. Eqn. (2) expresses a bipartite assignment problem with a square cost matrix but it can be easily generalized to cover rectangular cost matrix cases. 1

solutions, and for any pair ( I INC , I EXC ) of index sets, i.e., I INC ⊆ {1,..., n} (inclusion set) and I EXC ⊆ {1,..., n} (exclusion set), let us define a restricted feasible solution set S ( I INC , I EXC ) ⊆ S by  S ( I INC , I EXC ) =  x ∈ S 

4

x[i ] = 1 for all i ∈ I INC and   (3) x[i ] = 0 for all i ∈ I EXC 

In this paper, we denote the i -component of vector x by x[i ] , and the (i , j ) -element of matrix A by A[i , j ] .

In the first stage, the original problem is solved using an appropriate algorithm5, and since we assume S ≠ ∅ , we have at least one best solution xˆ1 ∈ {0,1}n with a non-empty index set Iˆ1 = {i ∈{1,..., n} | xˆ1[i ] = 1} = {iˆ(1,1) ,..., iˆ(1,n ) } . Then 1

we decompose the set of all feasible solutions as

S =∪

n1 +1 =1

(1, ) (1, ) S ( I INC , I EXC ) where

(1, ) (1, ) empty, we should find a best solution in S ( I INC , I EXC ). Then, we make a list of triples, each of which, ( xˆ , IˆINC , IˆEXC ) , consists of an optimal solution xˆ , and inclusion and exclusion index sets that define the restricted solution space S ( IˆINC , IˆEXC ) in which xˆ is an optimal solution. Then the list of such triples ( xˆ , IˆINC , IˆEXC ) is sorted

according to the values of objective function f ( xˆ ) , as (1, )  I INC = {iˆ(1,1) ,..., iˆ(1, −1) } if >1, ∅ otherwise  (1, )  I EXC = {iˆ(1, ) } if 0 , while hypothesizing its non-detection in the k -th frame if j = 0 . A data association hypothesis8 λ on ( yκ )κk =1 is a set of non-overlapping, non-empty tracks9. Under standard assumptions, including no-split-ormerged-report assumption, it is known [3,4,12-16] that each data association hypothesis λ on ( yκ )κk =1 can be probabilistically evaluated as    P (λ | ( yκ )κk =1 ) = C −1  ∏ L(τ )   ∏ γ FA (k , j )    τ ∈λ   j∉∪τ ∈λ {τ ( k )} 

(5)

where C is the normalizing constant, L(τ ) is the track likelihood function10, and γ FA (k , j ) is the density of false alarms contained in the k -th frame evaluated at the value ykj of the j -th report. Taking a negative logarithm of eqn. (5), the maximization of the a posteriori probability P (λ | ( yκ )κk =1 ) can be transformed into the optimization problem of eqn. (1). The set of all the tracks plus the set of all the report ykj for which γ FA (k , j ) > 0 is mapped into an index set {1,..., n} . Each hypothesis λ is mapped into a solution x ∈ {0,1}n . x[i ] = 1 if and only if the i -th track is included in hypothesis λ when index i represents the track. x[i ] = 1

the k -th frame and hypothesis λ assigns report ykj as a false alarm. By the negative-logarithm conversion, the track likelihood and the false alarm density functions in (5) are transformed into the coefficient vector c ∈ ( −∞, ∞)n that makes the objective function in (1) a linear functional as f ( x ) = cT x . Thus the k-best hypothesis selection problem is transformed into a k-best {0,1} integer linear programming problem. In each stage of the generalized Murty’s algorithm defined in Section 3, a set of integer linear programming problems, each of which is equivalent to a multi-frame assignment problem (said to be an NP-hard problem in [15] if it is over three or more frames), must be solved. In the ATIF implementation, each {0,1} linear programming problem is relaxed to a [0,1] linear programming problem, obtained by replacing the solution space {0,1}n by a unit hypercube [0,1]n . Most of the time, a [0,1] linear programming solver returns a {0,1} solutions that are also optimal within the {0,1} problem. When it does not, we use a version of branch-and-bound method to obtain an optimal {0,1} solution, or an approximate {0,1} solution if necessary. ATIF uses a very effective combination of a linear programming solver and a branchand-bound supplement. The k-best hypothesis selection implemented in ATIF is used in several ways: Ambiguity Assessment: When we select enough number of hypotheses and evaluate them by their likelihoods or unnormalized a posteriori probabilities defined in eqn. (5), we may approximate the a posteriori probability for each selected hypothesis by ignoring tail hypotheses. We can estimate the quality of the approximation by measuring the difference in the approximate probabilities of the high-ranked hypotheses as we increase k in k-best hypothesis selection. In some cases, even when reporting a best or a few best hypotheses, it may be important to see how much confidence we can put on those hypotheses probabilistically.

7

κ = 1,..., k . Tracks τ 1 and τ 2 are said to overlap if τ 1 (κ ) = τ 2 (κ ) > 0 for some κ . 10

The track likelihood is defined as an appropriate product of the track-to-measurement likelihood functions and a priori target density when it is initiated, and is a positive number.

k=5 A Posteriori Probability

We use the term track as originally used in [4] and [5]. It is sometimes called a track hypothesis to emphasize the fact that it is a hypothesized trace of detections from a single target. 8 Hereinafter we call a data association hypothesis simply a hypothesis. It is sometimes referred to as a global hypothesis, “global” emphasizing the fact that the hypothesis is concerned with not only a single target but all the targets, while each track is considered as a local hypothesis concerning only with a single target. 9 k A track τ on ( yκ )κ =1 is said to be empty if τ (κ ) = 0 for all

k=10 k=20

k=50 Ranked Hypotheses

Fig. 2: Approximate A Posteriori Hypothesis Probability

Fig. 2 illustrates the approximate calculation of a posteriori probability of each data association hypothesis. This example was taken from a simulation using a ten-target scenario with extremely high target density of targets moving together with almost constant velocities. An approximate a posteriori probability of each data association hypothesis over five frames is calculated by our k-best hypothesis selection algorithm. Hypothesis 1 means the best hypothesis, hypothesis 2 the second best, and so forth, in the figure. The curves are parameterized by the number k of selected best hypotheses and show the approximate probabilities when ignoring the tail hypotheses. Fig. 3 shows the convergence of the ambiguity assessment in terms of approximate a posteriori probability of each of the best to the 5th best hypothesis.

A Posteriori Probability

Best Hypothesis

2nd Best Hypothesis 3rd Best Hypothesis 4th Best Hypothesis

5th Best Hypothesis

Hypothesis Ranking

Fig. 3: Convergence of Approximate A Posteriori Hypothesis Probability

C P U T im e (se c)

Fig. 4 shows the CPU time to compute the k-best hypotheses by a 2GHz-X86-based PC. The figure shows a linear-plus-fixed-overhead type performance.

Number of Selected Hypotheses k

Fig. 4: CPU Time by K-Best Hypothesis Selection Track Pruning: In a track-oriented MHT algorithm, tracks are expanded and propagated forward when recursively processing each frame of data, but data association hypotheses are not maintained or propagated explicitly. Whenever necessary, a set of data association hypotheses are constructed as a collection of nonoverlapping sets of tracks. Thus the hypothesis space that

each track-oriented MHT algorithm maintains is the space of the tracks on each past cumulative frame. Without an appropriate set of hypothesis management, the hypothesis space may grow rapidly, generally exponentially with the number of frames. Therefore, an appropriate algorithm for pruning tracks is an essential part of any track-oriented MHT algorithm. A set of tracks defined on a collection of all the past cumulative frames, as well as hypotheses defined on a collection of all the past cumulative frames, are naturally ordered by the predecessor-successor relationship defined through restriction and extension of functions. In this partial ordering, both the set of tracks and the set of hypotheses form a tree structure11. The basic hypothesis management technique of trackoriented MHT is the so-called n-scan pruning, or mid-level track pruning. The conventional n-scan pruning is based on a single best hypothesis selection of a hypothesis λ formed on frames up to frame K , and prunes all the tracks τ that do not share the predecessor tracks that are formed on frames up to frame K − n and are predecessors of tracks in the best hypothesis λ . Equivalently, every hypotheses λ formed on frames up to frame K are pruned away if the predecessor of λ is not the predecessor of the best hypothesis λˆ . Using this pruning strategy, any track initiated at frames between frames K − n + 1 and Κ are protected, as well as preserving some variations of tracks included in the best hypotheses (variations sharing the predecessors). Using this pruning, “ambiguities” are resolved according to their “ages,” after having been given a fixed time for additional data to disambiguate the ambiguities. In some “confusing” situation, we may need to maintain additional hypotheses instead of forcing to choose one of “comparable” hypotheses. For this purpose, we can extend the conventional n-scan pruning based on k-best hypothesis selection instead of the single best hypothesis. The k-besthypothesis, n-scan pruning is to retain all the tracks except for the tracks that do not share any of the predecessors of the tracks included in at least one of the k-best selected hypotheses. By using a large enough k, it is possible to defer ambiguity resolution beyond the n of the n-scan pruning, usually referred to as the depth. Fig. 5 illustrates the effects of the k-best-hypothesis, nscan pruning, in terms of the number of retain tracks as a function of the number k of the selected hypotheses and the depth n of the n-scan pruning. The dramatic increase in the number of tracks as the number k of hypotheses is reflected by a large number of tracks generated by an extremely high target density, over the five frames of track reports.

11

In the sense that an immediate predecessor of each track (or hypothesis) is always unique if it exists.

N um b er o f R eta in ed T ra cks

Pruning Level = 4

Pruning Level = 3

Pruning Level = 2 Pruning Level = 1

given chances to grow while their likelihoods may be very small and vulnerable to be pruned). Fig. 6 shows the effect of the pruning level used by the mid-level hypothesis selection, using the same sample data used for Figs. 2 to 6. In the figure, the effect of the midlevel hypothesis selection is measured by the number of retained tracks. The dramatic increase of the number of retained tracks is mainly due to the fact that all the newly initiated tracks under the evaluation level are included in the set of retained tracks.

Pruning Level = 0

Number of Retained Tracks

Evaluation Level = 2

Number of Selected Hypotheses k

Fig. 5: K-Best-Hypothesis, N-Scan Pruning Mid-Level Hypothesis Selection: ATIF system uses a unique application of the generalized k-best hypothesis selection algorithm. The application may be called mid-level hypothesis selection. Consider selection of hypotheses based only on a partial evaluation, i.e., a sub collection ( yκ )κk '=1 of cumulative frames with k ' < k , when frame k is the current frame. Each partial hypothesis λ on ( yκ )κk '=1 can then be evaluated by a revised objective function g ( x ) = max{ f ( x ) | x is a successor of x on ( yκ )κk =1} (6)

where f ( x ) = cT x is the objective function transformed from the hypothesis evaluation of eqn. (5) through the negative-logarithm transformation, x is the {0,1} representation of a hypothesis λ on ( yκ )κk =1 , and x is the {0,1} representation (with an appropriate dimension) of the predecessor hypothesis λ on ( yκ )κk '=1 . Although the objective function g defined by (6) is a nonlinear function, we can apply our k-best solution selection algorithm as described in Section 3, and moreover, each sub-problem appearing in this k-best optimization process can be solved by the same integer program solving function used to obtain k-best hypotheses on ( yκ )κk =1 . Once a specific number of predecessor hypotheses λ on ( yκ )κk '=1 is selected, we may use these selected k-best partial hypotheses λ for track pruning as described before. Benefits of using this mid-level k-best hypothesis selection, as opposed to the leaf-node level k-best hypothesis selection (described above), are two folds: (1) whenever a mid-level hypothesis is selected, effectively, we select multiple leafnode hypotheses. As a consequence, in effect, we can select more leaf-node hypotheses with a fewer number of “k” for the k-best hypothesis selection, and (2) since newly generated tracks (below the mid-level, called the evaluation level) are included in every mid-level hypothesis, all the tracks initiated below the evaluation level are protected (or

Evaluation Level = 1

Evaluation Level = 0

Number of Selected Hypotheses k

Fig. 6: Effect of Pruning Level

Multi-Modal Track Outputs: A common practice for track-oriented MHT system such as ATIF is to output a set of tracks in a single-best hypothesis. Using the k-best hypothesis selection, we may output multiple hypotheses with probabilistic ambiguity assessments, thus addressing the so-called hypothesis hopping problem. However, interpreting outputs of multiple sets of hypotheses, each of which is a set of tracks, may not be easily understood by the users. An alternative output may be a form of track trees. As mentioned above, a collection of all the tracks defined on all the past cumulative frames, ( y1 ) , ( y1 , y2 ) , …., ( yκ )κk '=1 ,…, ( yκ )κk =1 , forms a tree. Each sub-tree12 with a root at the first detected report represents a track hypothesized to originate from the same target. Let τ 1 ,...,τ M be M tracks

on ( yκ )κk =1 represented by leaves of such a track tree. Then we may define a target state probability distribution of the target state s (tk ) at the time tk of the k -th frame conditioned by the track tree, by its density M

P ( s(tk ) | τ 1 ,...,τ M ) = ∑Wi P ( s (tk ) | τ i )

(7)

i =1

12

We may call such a sub-tree with a unique non-empty root a track instead of a track tree since the tree as a whole represents a single detected target, while each individual track is called a track hypothesis instead.

where P ( s(tk ) | τ i ) is the target state probability density function conditioned only by a track τ i , and Wi is the probability weight, i.e., Wi > 0 and

∑

M

W =1. i =1 i

The probabilistic weights (Wi )iM=1 can be obtained by normalizing the set of track probabilities, while each track probability is defined as the sum of all the probabilities of hypotheses that contain each track. A set of multiple hypotheses and their a posteriori probabilities, expressed by eqn. (5), are obtained by the k-best hypothesis selection algorithm and used for this weight calculation.

References [1] Katta G. Murty, An algorithm for ranking all the assignments in order of increasing cost, Operations Research, Vol. 16, No. 3, pp. 682 – 687, May-June, 1968. [2] Yaakov Bar-Shalom, and Xi-Ron Li, MultitargetMultisensor Tracking: Principles and Techniques, Storrs, CT, YBS Publishing, 1995. [3] Samuel S. Blackman, and Robert Popoli, Design and Analysis of Modern Tracking Systems, Artech House, Norwood, MA, 1999. [4] Charles L. Morefield, Application of 0-1 Integer Programming to Multi-Target Tracking Problems, IEEE Transaction on Automatic Control, Vol. AC-23, No. 3, pp. 302-312, June 1977. [5] Donald B. Reid, An Algorithm for Tracking Multiple Targets, IEEE Transaction on Automatic Control, Vol. AC24, No. 6, pp. 843-854, Dec., 1979.

(a) Single-Modal Output

(b) Multi-Modal Output

(c) Aggregated Multi-Modal Ouput

Fig. 7: Multi-Modal Track Tree Output Fig. 7 illustrates the multi-modal output concept. In the figure, a sum-of-Gaussian distribution is shown as an example of a multi-modal probability distribution in (b), which is compared with the conventional single-hypothesis, single-modal output in (a) and an approximate multi-modal output obtained by aggregating some of the modes in (c).

5

Conclusion

A generalization of Murty’s algorithm for ranking k-best bipartite assignment problems, was described, and its application to track-oriented MHT systems was discussed. Generalization of the algorithm is two-fold: (i) from bipartite (two-dimensional) assignment problem to a general {0,1} integer linear constraint, including the multi-frame (or multi-dimensional) assignment problems, and (ii) from linear objective functions to a general, potentially, nonlinear objective functions. The generalized Murty’s algorithm was used in a hybrid data fusion system, called ATIF [17], to perform several functions. These functions include output ambiguity assessment, track pruning, and multi-modal track outputs. The generalized Murty’s algorithm relies on an effective integer programming solving algorithm, just as Murty’s algorithm needs an effective bipartite assignment algorithm such as Munkres or [7] or Jonker-Volgenant-Castañón algorithm [6,8]. Thus, any improvement in the integer programming algorithm can potentially enhance the performance of any track-oriented MHT system using our generalized Murty’s algorithm.

[6] David A. Castañón, New assignment algorithms for data association, Proceedings of SPIE Symposium on Signal and Data Processing of Small Targets, ed. by Oliver E. Drummond, Vol. 1698, pp. 313-323, August, 1992. [7] James Munkres, Algorithms for Assignment and Transportation Problems, Journal of the Society for Industrial and Applied Mathematics Vol. 5, No. 1, March, 1957. [8] Roy Jonker, and Ton Volgenant, A shortest augmenting path algorithm for dense and sparse linear assignment problems, Computing Vol. 38, No. 11, pp. 325340, Nov., 1987. [9] Roy Danchick, and George .E. Newnam, A fast method for finding the exact N-best hypotheses for multitarget tracking, IEEE Transactions on Aerospace and Electronic Systems, Vol. 29, No. 2, pp. 555-560, 1993. [10] Ingemar J. Cox, and Matt L. Miller, On finding ranked assignments with application to Multitarget tracking and motion correspondence, IEEE Transactions on Aerospace and Electronic Systems, Vol. 32, No. 1, pp. 486-489, 1995. [11] Ingemar J. Cox, Matt L. Miller, Roy Danchick, and George .E. Newnam, A Comparison of two algorithms for determining ranked assignments with application to multitarget tracking and motion correspondence, IEEE Transactions on Aerospace and Electronic Systems, Vol. 33, No. 1, pp. 295-301, 1997. [12] Thomas G. Allen, Thomas Kurin, and Robert B. Washburn, Parallel computer structures for multiobject

tracking algorithms on associative processors, Proc. American Control Conf., Seattle, WA, June, 1986. [13] Thomas Kurien, Issues in the design of practical Multitarget tracking algorithms, in Multitarget-multisensor tracking: advanced applications, ed. by Yaakov BarShalom, Chap. 3, pp. 43 – 83, Artech House, 1990. [14] Samuel S. Blackman, Multiple hypothesis tracking for multiple target tracking, IEEE Aerospace and Electronic Systems Magazine, Vol. 19, No. 1, Part 2: Tutorial, pp. 5 – 18, January 2004. [15] Krishna R. Pattipati, Somnath Deb, Yaakov BarShalom, and Robert B. Washburn, A new relaxation algorithm and passive sensor data association, IEEE Transactions on Automatic Control, Vol. 37, No. 2, pp. 198 – 213, February 1992. [16] Aubrey Poore, and Nenad Rijavec, A Lagrangian relaxation algorithm for multidimensional assignment problems arising from Multitarget tracking, SIAM Journal of Optimization, Vol. 3, No. 3, pp. 544 – 563, August 1993. [17] Stefano Coraluppi, Craig Carthel, Mark Luettgen, and Susan Lynch, All-Source Track and Identity Fusion, Proc. National Symposium on Sensor and Data Fusion, San Antonio, TX, June 2000. [18] Robert L. Popp, Krishna R. Pattipati, and Yaakov BarShalom, m-Best S-D assignment algorithm with application to multiple target tracking, IEEE Transactions on Aerospace and Electronic Systems, Vol. 37, No. 1, pp. 22 – 39, January 2001. [19] Shozo Mori, Chee-Yee Chong, Edison Tse, and Richard P. Wishner, Tracking and classifying multiple targets without a priori identification, IEEE Transactions on Automatic Control, Vol. AC-31, No. 5, pp. 401-409, May 1986. [20] Shozo Mori, and Chee-Yee Chong, Data association hypothesis evaluation for i.i.d. but non-Poisson multiple target tracking, Proc. SPIE Symposium on Signal and Data Processing of Small Targets, Vol. 5428, pp. 224 – 236, Orlando, FL, April 2004. [21] Shozo Mori, and Chee-Yee Chong, Evaluation of data association hypotheses: non-Poisson i.i.d. cases, Proc. 7th International Conference on Information Fusion, pp. 1133 – 1140, Stockholm, Sweden, July 2004.