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Efficient data association with multivariate Gaussian distributed states
| Details |
Inventors: Collins, Joseph B.; Uhlmann, Jeffrey K.;
Assignee: The United States of America as represented by the Secretary of the Navy (Washington, DC)
Primary Examiner: Davis; George B.
Assistant Examiner:
Attorney, Agent or Firm: McDonnell; Thomas E., Miles; Edward
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-dimensional error ellipsoids for the state vectors which always overlap when a gating criterion is satisfied. Circumscribing boxes, or d-ranges, for the data ellipsoids are then found and whenever they overlap the association probability is computed. For efficiently determining the intersections of the d-ranges a multidimensional 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. Empirical testing for variously distributed data in both three and eight dimensions indicate that the scaling is significantly reduced from N.sup.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. |
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DETAILED DESCRIPTION Accordingly, an object of the invention is to track a plurality of objects, or fuse a plurality of data sets, in a computationally efficient manner.
Another object is to track a plurality of N objects, or fuse a plurality of data sets of N members, in a manner which requires a number of computations significantly less than N.
sup.
2.
In accordance with these and other objects made apparent hereinafter, the invention pertains to a method and apparatus of correlating a plurality of objects.
At two different times, sets of d dimensional data vectors {.
alpha.
} and {.
beta.
} are produced, each of whose elements a.
epsilon.
{.
alpha.
} and b.
epsilon.
{.
beta.
} corresponds to one of the plurality of objects.
The uncertainty of each element of each said data vector a or b is taken to be Gaussian, and thus has respective covariance matrices A and B.
A numerical value .
gamma.
.
sub.
AB is selected, which is used to determine which vector pairs a, b satisfy the gating criterion: .
gamma.
.
sub.
A.
gtoreq.
.
gamma.
.
sub.
AB, and .
gamma.
.
sub.
B.
gtoreq.
.
gamma.
.
sub.
AB, where .
gamma.
.
sub.
A =(r-a).
sup.
T A.
sup.
-1 (r-a) .
gamma.
.
sub.
B =(r-b).
sup.
T B.
sup.
-1 (r-b) .
gamma.
.
sub.
AB =(a-b).
sup.
T (A+B).
sup.
-1 (a-b) and where r is some d dimensional vector.
A relatedness criterion is selected for characterizing the degree of association between any such vector pair a,b, and the relatedness criterion applied to all vector pairs which satisfy the gating criterion.
Finally, one uses these results to fuse the data vectors of sets {.
alpha.
} and {.
beta.
} into a fused set {c} of data vectors each of whose members corresponds to one of the objects.
Because the gating criterion eliminated data pairs having small likelihood of correlation, the relatedness criterion need be applied only to those pairs not eliminated by the gating criterion, reducing the number of calculations necessary to correlate {.
alpha.
} and {.
beta.
}.
The invention is more fully understood from the following detailed description of preferred embodiments, presented in conjunction with the appended drawings, wherein:
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