Fig. 1 shows the connection diagram for a single-phase motor with external capacitance C in one winding. (The notation is the same as that used in [l].) For simplicity of analysis, we assume the motor is a three-phase machine but connected single-phase as shown.

to the corresponding motor terminals a, bm, an, and bn. We can write

Here t , is a single-phase voltage source connected

1, + Z c i , = Lam

and

- - -e , = ebm

i , = 0

from the connections. From the theory of the motor, using symmetrical components,

t a m = t a l m + za2m + zoom

= -Z1ial - Z2ia2 - Zoi,~

where Z1, 2 2 , and 20 are the motor sequence impedances. Thus

0, = -zci, - z l i a l - z 2 i a 2 - zoi,o

Z O ] C O -

Also i, = icl + ic2 + id = 0.

symmetrical component relations icl = aia1, ic2 = a2i,2, and id = iao = ibo we can finally write,

4, = (2; - a2b)ia1 + (2; - a2Z& (1)

From these last three equations and the

electromagnetic torque ([l], p. 135, eq. (31)) as a function of s, etc., for the different selected C values. For example, the proposed technique allows one to predetermine the effect of the capacitance value, and therefrom its weight and volume, upon the electrical and mechanical performance of the motor, both during starting and at rated load. In this manner tradeoffs in cost, performance, weight, and volume can be predetermined.

RICHARD ?: SMITH 402 Yosemite Dr. San Anlonio, TX 78232

REFERENCES

[l] Smith, R. ‘I (1982) Analysis of Electrical Machines. Elmford, N Y Pergamon, 1982

Efficient Gating in Data Association with Multivariate Gaussian Distributed States

We describe an efficient algorithm for evaluating the (weighted bipartite graph of) associations between two sets of data with Gaussian error, e.g., between a set of measured state vectors and a set of estimated state vectors. First a general method is developed for determining, from the covariance matrix, minimal

d-dimemional error ellipsoids for the state vectors which ahways overlap when a gating criterion is satisfied. Clrcuoscribing boxes, or dqanges, for the data ellipsoids are then found and whenever they overlap the association probability is conputed For efficiently determining the intersections of the d-ranges a multidimemional search tree method is used to reduce the overall scaling of the evaluation of associations. Very few associations that lie outside the

predetermined error threshold or gate are evaluated The search

method developed is a fixed “Mahaladis distance” search Elnpirical testing for variously distributed data in both three and

eight dimensions indicate that the scaling is significantly reduced

from N 2 , where N is the size of the data set. Computational loads for many large scale (N > 10 - 100) data association tasks may therefore be significantly reduced by this or related methods

and Manuscript received March 16, 1990, revised August 11,1991 and

IEEE Log No. 9107206.

This work was sponsored in part by the National Research Council and also in part by the Strategic Defense Initiative Organization.

The authors would like to thank David Book, Miguel Zuniga, and Mike Picone for their useful conversations.

+zs = 6(a2z1 - a Z o ) i a l + ( a 2 2 - a 2 ~ 0 ) 1 a 2 (2) October 24,1991.

where 2; = 21 + ZC, etc.

Equations (1) and (2) are best handled numerically.

and given motor impedances, we can vary slip s over the range 1.0 to 0 and thus generate

Usually ZS = 1.0 LOO. For selected values of C,

the data needed to plot curves of lial, ( i b l , average 0018-9251/w$3.00 @ 1992 IEEE

CORRESPONDENCE 909

1

I . INTRODUCTION: ALGORITHMIC SCALING AND DATA ASSOCIATION

As demand for practical solutions to larger scale data fusion or data association tasks increases, issues of computational complexity of the algorithmic approaches used become topics of more serious interest. Theoretically well-defined and well-grounded analytical models for data association problems may exist without having efficiently computable solutions. When such situations arise it is desirable to determine efficient approximation procedures that, if possible, are well conditioned such that certain bounds on errors can be determined.

When data association tasks consist of merging, or fusing, two or more large sets of data that both represent a common underlying reality, or ground truth, such issues of the computational complexity of the algorithms do arise. We assume that a general task of data association is to take two independent sets of state estimates, e.g., measurements and predicted state values, that each represent a common underlying set of objects and to determine the associations between the elements from the distinct sets of data. Fusion, or subsequent assignment of the elements from the independent representations to a single representation depends on the determination of these associations.

Trpically in discussions of computational complexity, analysis is made with respect to the underlying graph theoretic structure of the problem at hand. The purpose of this is to determine if the general probIem has ever arisen in other applications. In fact, this is indeed the case with some data association problems. The graph structure associated with the fusion problem described here is a bipartite graph, since by assumption there are two sets of data, independently arrived at. From each element a E d there may be an association with an element b E B represented by an edge wab E E. On a graph, I'{d,Z?,E} this corresponds to vertices of A connected to vertices off? with weighted edges, &b, where the weight represents the weight of association (see Fig. 1). One may think of the association weights forming an association matrix, R. In the process of fusion or data association there are commonly constraints that describe feasible joint events [l], i.e., each element of one set may be assigned to at most only one element of the other set and vice-versa. In such cases the joint event is properly termed a match or, defined on a bipartite graph, a bipartite match. If this match has maximum cardinality, generally corresponding to the constraint that a feasible joint event is a one-to-one mapping (always possible with false alarms), then it is called aperfect match (see Fig. 2).

multidimensional state value with an associated error estimate, i.e., a random variable. Commonly the state value is an estimated mean value and the error

In general the representation of objects is a

estimation vertices

~ ~ ' I n d e p c n d e n l subgraphs

assOeialion weighlr (edges)

Fig. 1. Weighted bipartite graph.

0

0 0

Fig. 2. Perfect matches.

is represented as an associated covariance matrix defining a Gaussian probability distribution. When a probabilistic interpretation is given to the association weights, as we assume, the merging may be done in a variety of ways that have particular probabilistic interpretations. The nearest neighbor approach, where the largest association weight at each node of one set alone determines the assignment, essentially assumes that there is virtually only one individual pairwise association of importance for every element, a condition of sparsity of the association matrix, 0. The nearest neighbor approach to data fusion is the simplest of greedy heuristics. Another approach, the optimum weighted bipartite match corresponds to the most probable, or modal, joint event, and is also used as a fusion heuristic.

In dense environments where individual elements appear to have many significant associations, an example of a more appropriate approach to this data association problem is described by Bar-Shalom [l] and referred to as JPDA (joint probabilistic data association). In this and related approaches each association weight is taken as an individual pairwise association probability or marginal association probability. The relative probability of one match with respect to any other is taken as the product of th eindividual pairwise association probabilities (the edge weights) of the match. (Since objective functions defined on a graph are usually summed weights, this corresponds to summing over the logarithms of the pairwise association probabilities, logwab). The expected probability that a and b refer to the same object is determined by summing the joint probabilities of the perfect matches in which a and b are paired

910 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 28, NO. 3 JULY 1992

-~ - - -- __I_- - - -

and normalizing by the sum of joint probabilities over all possible perfect matches. For a simple N x N association matrix (e.g., no false alarms), R, this may be alternatively stated as

(Pab) = f l i j *perM(i, j) /prfl (1)

where thepermanent [2] of R is defined as

which is similar to the definition of the determinant with the exception that the absolute value of the Levi-Civita symbol is used to represent the sum over all permutations, CT, of the indices l . . . N, and the permanent-minor, or perM(i,j), is determined by taking the permanent of the matrix M(i,j), generated by crossing out the ith row and j th column of the matrix R.

Valiant [3] has shown that the evaluation of the permanent of a general 0-1 matrix (similar to the validation matrix for JPDA [l]), equivalent to counting the number of perfect matches, is NP-hard. Complete evaluation of the permanent of a general association matrix is equivalent to enumeration of all possible perfect matches, a significantly more difficult problem. While the association matrices only form a subclass of general matrices, Valiant’s result appears to imply that the exact solution of the JPDA is indeed NP-hard. Approximation methods, however, do exist and generally reduce the computational scaling significantly. These methods themselves then become the important subjects of consideration in terms of computational scaling.

Determining computational tractability in approaches to tracking for large-scale real-time problems is an exercise of putting practical upper bounds on algorithmic scaling. Clearly, any algorithm will scale at least linearly with the size of the input data. We would maintain, perhaps arbitrarily, that for large-scale problems quadratic algorithmic scaling forms a practical upper bound.

that reduces the scaling of JPDA (or any other approach that computes all the joint probabilities). The permanent may be evaluated recursively via a development through permanent-minors in the same manner as a determinant is evaluated by a Laplacian development through minors. For sparse matrices with O ( k N ) elements this evaluation method appears to scale as Nk. Clearly, by thresholding matrix elements to guarantee sparsity, one may both approximate the permanent and put a limit on the scaling behavior of the evaluation algorithm. Gating [4] is the general problem in data fusion of determining

Thresholding the elements of fl is one approach

significant association probabilities between two large sets of linear, multidimensional pattern elements, e.g., thresholding the association probability overlap integrals between a set of measured state vectors and a set of estimated state vectors, both having associated error distributions. As a precursorial coarse assignment method for multitarget tracking, it reduces the computational complexity of the exact JPDA [l], or any other similarly complex data association method. One can see how this result is partially achieved by noting that as elements of s1 are thresholded to zero, the corresponding edges on the bipartite graph disappear, possibly resulting in independent subgraphs that represent a set of clusters over which the data association may be performed independently. (This is not to imply that we are introducing a cluster-based gating method, which generally scale poorly).

Computing the full association matrix between two large data sets of size N scales as N 2 in time. Also, computing the association probability overlap integral between two errored vectors is generally a computationally expensive procedure. For sparse systems, those where there are only order N significant associations, where significance is determined by a “gate”, this becomes a problem since a lot of work is performed relative to the significant pieces of information garnered. Two outstanding problems are determining a good or optimally scaling association algorithm and a method of evaluation that computes a minimum number of associations that are likely to satisfy the gating criterion. Solution of these problems contributes important computational labor savings to many data association tasks.

II. DATA THRESHOLDING CRITERION FOR MULTIVARIATE NORMAL DISTRIBUTIONS

An association probability is essentially the probability overlap integral between two (distributed) states. Significant associations exist, roughly speaking, for near-neighbors in state space. If a probability distribution for a state is not local, e.g., cannot be bounded by a convex region of the state space, then it will generally be necessary to evaluate a large number of overlap integrals and consideration of efficiently scaling algorithms may not be worthwhile. On the other hand, if the probability distributions do have these geometric properties, then preprocessing of the distributed states and the application of an efficiently scaling multidimensional search tree algorithm can significantly improve gating efficiency.

In order to eliminate the need to individually consider every pairwise association it is necessary to determine if the thresholding imposed by the gating criterion on the pairwise associations of errored vectors induces or can be decomposed to a thresholding criterion on the individual distributed

CORRESPONDENCE 911

1

states themselves. If a thresholding criteria can be so determined so as to define finite volumes in state space for each state, then the existence of a significant association implies the geometric intersection of these volumes (or at least we can so define the induced thresholding criterion). The determination of the intersections of finite geometric objects can be done efficiently by the use of data structures known as multidimensional search trees [5, 6, 71. It is shown that for independent Gaussian random variables there is a geometric property that allows this.

An outline of the gating procedure required to implement an efficient search is as follows. A standard deviation threshold (fixed Mahalanobis distance), or gate size, is chosen which each associated pair must satisfy. This induces a standard deviation thresholding for the independent elements, e.g., for each of the measured states and the estimated states. The resulting volume representing each errored vector is described by an ellipsoid in d-dimensional space. Detailed below is a justification showing that when the probability of association satisfies the gating criterion then the two ellipsoids of the associated data elements overlap. This is done by determining the thresholding criterion on each of the data element distributions that is induced by the gating criterion and that has this property. The geometric property we use turns out to be a consequence of a geometric inequality reminiscent of a triangle inequality.

representing a Gaussian random variable, as a pairing of a mean state vector and a covariance matrix. For example an estimated distributed state, A = {a ,A} , has an estimated mean state a and a covariance matrix A; a measured distributed state, B = {b ,B} , has a measured state b and a covariance matrix B. Their probability distributions in d-dimensional state space r are

We begin by defining a distributed state,

e-(1/2)(r-a)TA-'(r-a) and

m /2 \

PA(r) =

\JI e- (1/2)(r- b)TB-'(r- b)

rn =

respectively, where A and B are positive definite and symmetric.

The probability of association between the two object representations is proportional to the overlap integr a1

e-(l/2)(r-a)rA-'(r-a) e- (1/2)(r- b)TB-' (c- b) ddr rn

(4)

PAB =

which, when integrated over state space r reduces to

e - (1/2)(a- b)T(At B)-l(a- b)

PAB = (5 )

The probability of association, AB, represents the probability density of measured states in the estimated-state state space (or vice-versa). For a given object, the corresponding probability that b will lie within the ellipsoidal surface centered at a

EAB@') I (a - b')T(A + B)-'(a - b') = TAB (6)

is the probability, or rate, of correct associations

EAB e - (1/2)(a- b'f(AtB)-'(a- b') ddb'. (7) J J ( ~ T ) ~ I A + BI

PAB =

The ellipsoidal gate [4] is determined by choosing a threshold TAB or a probability threshold PAB which are related by

PAB = em(&E, d ) (8)

where we define

(9)

Ellipsoidal surfaces, EA for the estimated position and EB for a measurement of the position, may also be defined by T A and like,

and EB I (r" - b)TB-'(r'' - b) = y~ (11)

within which the actual object represented may be expected to lie with probabilities

EA e- (1/2)(r-a)'A-'(r-a) ddr m P A = J " PA(r)ddr = J

and

EB EB e-(1/2)(r-b)TB-'(r-b) ddr rn P B = J pB(')ddr = J

respectively.

is satisfied ellipsoids EA and EB overlap, we note that each must be at least greater than or equal to ?AB since as one distribution, say PA, tends to a Dirac &function then the gating criterion will only be satisfied if a falls within ellipse EB, or

To determine ?A and T B such that when the gate

(a - b)TB-l(a - b) 5 TAB (14)

and vice-versa if p~ tends to a Dirac 6-function. In other words ?A 2 TAB and YB 2 ?AB are necessary conditions to guarantee overlap.

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 28, NO. 3 JULY 1992

1 -

The ellipsoidal gating condition

(a - b)T(A + B)-'(a - b) 5 Y A B (15)

can be interpreted to mean that b lies within ellipsoid EAB centered around a, or vice versa. A hyperplane tangent to the ellipsoid along any particular coordinate

any coordinate frame) is a distance from the center (a simple proof is

detailed in the Appendix). Since a and b lie within the ellipsoid EAB with a at the center, the projection of (a - b) along the same coordinate direction satisfies

Because the diagonal elements of the matrices A and B are positive definite, we also know that

which gives rise to the result

implies overlap but the reverse implication is not necessarily true and the probability overlap integrals corresponding to each overlapping pair of ellipsoids must be evaluated and compared to ?AB).

As a final note, we point out that since

(a - b)i I J3 (22)

cannot be guaranteed in general, one cannot assume that any arbitrary finite gate chosen for one of the two sets will always determine ellipsoids that overlap the mean position of associated elements from the other set. This implies that without particular knowledge of the datasets a volumetric intersection approach is the only way to guarantee finding all associations that satisfy the gating criterion (see Fig. 3).

Ill. BOXES AND TREES

Efficient methods for determining the intersection of two volumes rely on comparisons of common reference coordinate values that bound a hyper-rectangular volume. These bounding coordinate values define circumscribing,c oordinate aligned (isothetic) boxes or d-ranges in d-dimensions for each ellipsoid. A box oriented along the rectangular coordinate described by the expression (a - r)TA-l(a - r) = y is determined by two tangent hyperplanes for each dimension {pf)}, defined by equations

This result implies that all projections of the ellipsoids E A and E B , This can only be true if the ellipsoids EA and EB themselves overlap.

function

at a and

There is another way to understand this result. The that circumscribes a d-ellipsoid

f(r) = (r - a ) T ~ - ~ ( r - a) + (r - b)TB-'(r - b)

(19)

has a minimum value at a point r = r, that is exactly f(r,) = (a - b)T(A + B)-'(a - b). This allows the expression for all r

(a - bf(A + B)-'(a - b) 5 (r - a)TA-'(r - a)

+ (r - bfB-'(r - b) (20)

which reflects an essential geometric property of the space of Gaussiandistributed, independent random variables. From this result follows the interpretation that at this point r,, both

are true, i.e., the ellipses EA and EB overlap by both containing the point r,.

This proves the sufficiency condition that w a s sought; we therefore find that 7.4 = TAB and are necessary and sufficient conditions to guarantee overlap of E A and EB if the gating criterion is satisfied. It should also be noted that since any Y A or greater than 7~ is also sufficient, the condition "(A = YB = TAB is the minimum and therefore optimal criterion, resulting in the minimum number of possible tests for gate satisfaction. (Satisfaction of the gating criterion

= 7 A B

(pp);6;k = 6ik(Uk f m) { i , k = 1, ..., d }

(23)

with no summation implied (a simple proof is detailed in the Appendix). Overlapping box pairs are cheaply determined and for each of the overlapping box pairs a probability of association is calculated and thresholded.

Circumscribing error ellipsoids with multidimensional ranges permits the application of efficient search algorithms [5, 6, 71 called multidimensional search trees. In our implementation of a multidimensional tree search-structure the set of ddimensional boxes is divided (roughly) in half at each node of a ternary tree on one of the d coordinates. At each node the set is partitioned into a set of boxes which lie entirely to the left of the partitioning hyperplane, a set of boxes which lie entirely to the right of the partitioning hyperplane and a set of boxes which are intersected by the partitioning hyperplane, cycling through the d coordinates for partitioning at progressively deeper nodes as in the usual fashion for multidimensional search trees. The method for searching the tree then simply involves the determination of which set the search box would be assigned according to the partitioning hyperplane at each node. For example, if the search box lies entirely

913 CORRESPONDENCE

k 9

2 0 + / ! 2 1 1 ?I2 2 1 3 2 1 J l l i 210

\=\urnher of Ohlectc

Fig. 4. Volumetric intersect gating.

to the left of the partitioning hyperplane, then the subtree of boxes which lie entirely to the right of it can be ignored. Similarly, if the search box lies entirely to the right of the partitioning hyperplane, then the subtree of boxes which lie entirely to the left of it can be ignored. Only in the case where the partitioning hyperplane intersects the search box must all subtrees be examined. (Note that the set of boxes intersected by the partitioning hyperplane represent a subproblem of reduced dimensionality.) The depth of the tree is roughly logz N, where N is the number of d-ranges in the tree. In the worst case a single search may require N1-@Id) + k comparisons (where k is the average number of elements intersecting the search volume and d is the dimensionality of the data).

Tests were performed on datasets ranging in size from 1K to 64K, K = 1024, and for data dimensions d = 3 and d = 8 (see Figs. 3 and 4 for uniformly distributed data). The mean positions of the boxes as well as their individual coordinate ranges were drawn independently from uniform distributions such that the average number of intersections per search query remained constant (five) independent of dataset size.

914

Additional tests confirmed that the algorithm is relatively insensitive to the distribution of the mean positions; specifically, tests were performed using clustered and nonisotropic distributions along the following lines.

1) Anisotropic. The distribution of the boxes projected onto each coordinate is uniform, but the density of the projection onto coordinate k is twice that of the projection onto coordinate k - 1.

2) Clustered. uniformly distributed.

3) Inverse-Square. Boxes are distributed in the (-1, +l)d hypercube by drawing each mean position uniformly from the hypercube and dividing by the cube of its magnitude.

The volumes of the boxes in each search structure was selected so as to assure an average of five intersections per query. In the case of the inverse-square distribution, holding this variable constant implied that a few query boxes and many intersections while most had only one (itself). Scaling results similar to that for the uniform case were obtained.

architecture with an instruction rate of about sixteen MIPS. In our test cases for three and eight dimensions average scaling performance was consistent with a poly-log linear scaling, N(log, N ) 3 for three dimensions and N(log, N)4 for eight dimensions. We also note that the scaling performance was also consistent with a power-law scaling, N1.3 for three dimensions and N1.47 for eight dimensions. It appears that these two interpretations are numerically indistinguishable for the dataset sizes considered.

Clusters of n/2d boxes are

These tests were carried out on a SUN4

IV. SUMMARY AND CONCLUSIONS

Efficient gating may be done by only evaluating the association probabilities of those distributed state pairs whose coordinate aligned circumscribing boxes, or d-ranges, overlap. This is a result of the geometric properties of the space of Gaussiandistributed, independent random variables. The importance of this point lies in the fact that independent Gaussian random variables do not form a metric space and the geometric properties of this space are not generally well characterized. To be precise, the so-called Mahalanobis distance

d2({a,A}, {b,B}) = (a - b)T(A + B)-’(a - b) (24)

does not obey a triangle inequality, i.e., it satisfies

d({a,A), {b,B)) I d({a,A), {r,R)) + d({r,R}, {b,B)) (3)

for arbitrary r only when R = 0. (In fact, since both the difference and sum of independent Gaussian random variables results in a summed covariance,

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 28, NO. 3 JULY 1992

~

-1--- ~

- --

i.e., {a ,A} f {bB} ={ (a+b) , (A+B)} , {A,A}- {b,B} = {(a- b),(A +B)}, it does not even form a vector space). If the space of Gaussiandistributed states have no appropriate geometric properties the efficiently scaling algorithms we consider would not apply, and without a proper characterization of the geometric properties appropriate application of efficient algorithms are difficult. By determining such a characterization, the gating problem is converted to a volumetric intersection problem, for which efficient algorithms are hown. This, then, constitutes an efficient solution method to the “fmed Mahalanobis distance” search problem.

A simple implementation of the method utilizes boxes that are simply determined from the data vector a, its covariance matrix A, and the gating criterion y, to be defined by the set of hyperplanes

{ p $ ) } , ( p f ) ) i b i k = bik(ak f a) {i,k = l , * - * , d }

(26) with no summation implied. The efficient determination of overlaps between boxes can be easily done with a search tree algorithm. In our test cases average scaling performance was consistent with a poly-log linear scaling (indistinguishable from some power-law significantly less than 2) for the total association evaluation task, where N is the size of the data set. The variability of scaling appeared to vary only with the dimensionality of the data.

scaling of the gating process is reduced by use of a multidimensional tree based search algorithm; and, the ellipsoidal volumes defined for each distributed state are the minimal volumes that can be so defined, minimizing the number of computationally expensive evaluations of the association probability.

the most efficient method for determining ellipsoid intersections. As the dimensionality increases it may be worthwhile to investigate more sophisticated approaches to determining ellipsoid intersection other than testing ellipsoid pairs with intersecting d-ranges. (This difficulty arises from the fact that ellipsoids (L2 spheres) imply distances calculated with a Euclidean-like norm and d-ranges (L , spheres) imply distances calculated with an L, vector space norm).

of distributed states defined around a point, many tasks are defined for states distributed around such objects as a trajectory segment defined by a particular time period. One might define a suitable heuristic as follows: keeping in mind the minimal ellipsoidal volume of the distributed state at any particular point in time, the whole trajectory segment might either be circumscribed by a single box or represented by a chain of boxes. Once the objects are all decomposed into boxes gating as outlined above may proceed.

The efficiency of this method is twofold: the overall

For few dimensions, using box intersections appears

While we have defined minimal volumes for gating

bk = Xk

ACKNOWLEDCM ENT

The authors would like to thank David Book, Miguel Zuniga, and Mike Picone for their useful conversations.

X X j A i j X j = 1 and

APPENDIX

THEOREM. A box oriented along the rectangular coordinate ares that circumscribes a d-ellipsoid described by the expression C i , j x i A i j x j = 1 is determined by tangent hyperplanes b f ) defined by the equations

@ f ) ) i 6 i k = f d i k @. (27)

PROOF. The bounding hyperplanes are tangent to the

which reduces to

bk = Xk X X i A i j X j = 1 ; , j

C A l j X j = 0, V (1 # k) i

and consequently to

C A i j ~ j = 0, V (I # k) i

and finally to

and

and

and

Solving the second part,

C A l j x j = - A l k X k , V (1 # k ) j#k

is done in the usual way for an inhomogeneous equation using Kramer’s rule. The solution for the jth element of the vector x for j # k is given by the following: on the right-hand side the denominator is just the determinant of the matrix given by crossing out the kth row and column, which is just the cofactor of A k k ; the numerator is the product of - x k and the

CORRESPONDENCE 915

determinant of the matrix given by eliminating the j th column of A and substituting the kth column in its place, then crossing out the kth row, i.e.,

[61 Ch~e l l e , B. (1%) A functional approach to data structures and its use in multidimensional searching. SLAM Journal of Computing, 17, 3 (June 1988), 427-462.

where we make use of the definition det(M) = E. . MilMj2.. .Mrd ~; j . .= where M is of dimension d and the symbol ~ i , j , . . . ~ is the totally antisymmetric Levi-Civita alternating tensor of rank d . Accounting for the minus signs, this results in

J,J..-J

cofactor(Akj) x j = V (i # k ) (34) xk cofactor(Akk)

which is also obviously true for j = k. When substituted into

(35)

the result is

cofactor(& .) det(A) J = = 1

XkAkiXk cofactor(Akk) (xk)2 cofactor(Akk)

(36) reducing to the simple expression

(37) cofactor(Akk) 1

= A i k . ( x k ) 2 = det(A)

Q.E.D.

JOSEPH B. COLLINS JEFFREY K. UHLMA” Naval Research Laboratory Code 5571 WaShit#Ob DC 20375-5000

REFERENCES

[l] Bar-Shalom, Y., and Fortmann, T E. (1988) Eacking and Data Association. New York Academic Press, 1988.

In Gian Carlo Rota (Ed.), E r q c b p e d a of Mathematics and Irr Applications, Vol. 6 . Reading, MA: Addison-Wesley, 1978.

The complexity of computing the permanent. Theoretical Computer Science, 8 (1979), 189-201.

Multi-Target Tracking with Radnr Applicatwns. Dedham, MA: Artech House, 1986.

Bentley, J. L., and Wood, D. (1980) An optimal worst case algorithm for reporting intersections of rectangles. IEEE Transactions on Computers, C-29 (July 1980),

[2] Permanents, H. M. (1978)

[3] Valiant, L. G. (1979)

[4] Blackman, S. S. (1986)

[5]

571-577.

916

_ _ _ _

[7] Preparata, E, and Shamos, M. (1985) Computational Geometry: An Introduction. New York Springer-Vedag, 1985.

Optimal Data Fusion of Correlated Local Decisions in Multiple Sensor Detection Systems

Recently, Chair and Varshney have solved the data fusion

problem for fixed binary local detectors with statistically

independent decisions. We generalize their solution by using

the Bahadur-Lazarsfeld expansion of probability density

functions. The optimal data fusion rule is developed for correlated

local binary decisions, in terns of the conditional correlation

coefficients of all orders. We show that when all these coefficients

are zero, the rule coincides with the original Chair-Varshney

design

I. THE FUSION PROBLEM

We consider the following distributed-detection system: a bank of n sensors obtains observations from an environment on which two hypotheses are made: Ho (“no target”) and H1 (“target exists”). Each local detector collects an observation xi E R“ and transforms it, using a local mapping, to a local decision u i = g i ( x i ) , u i ~ ( O , l } , i = 1 , 2 ,..., n,where

(0. if fi is acceDted

U; = I 1; if H1 is accepted

The local decisions ( U $ ) are transmitted to a data fusion center (DFC) over noiseless communication channels. Using u1, u2,. . .,U,, the DFC makes a global decision U (U = 0 for accepting Ho, U = 1 for accepting HI) which minimizes the average Bayesian cost:

where the costs c j k are specified.

Manuscript received July 10, 1990; revised March 6, 1991.

IEEE Log No. 9107207.

’Ihis work was supported by the National Science Foundation under a PYI Award (ECS 9057587) and a Research Grant (ECS 8922142).

0018-9251/w%3.00 @ 1992 IEEE

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 28, NO. 3 JULY 1992