METHOD AND SYSTEM FOR TRACKING MULTIPLE REGIONAL OBJECTS BY MULTI-DIMENSIONAL RELAXATION

[0001] This application is a continuation-in-part of U.S. patent appication Ser. No. 08/404,024, filed Mar. 14, 1995 and issued Jul. 16, 1996 as U.S. Pat. No. 5,537,119, which is a continuation-in-part of U.S. patent application Ser. No. 08/171,327 filed Dec. 21, 1993, now U.S. Pat. No. 5,406,289.

[0002] The invention relates generally to computerized techniques for processing data obtained from radiation reflections used to track multiple discrete object.

[0003] The invention relates generally to computerized techniques for processing data obtained from radar to track multiple discrete objects.

[0004] There are many situations where the courses of multiple objects in a region must be tracked. Typically, radar is used to scan the region and generate discrete images or snapshots based on sets of returns or observations. In some types of tracking systems, all the returns from any one object are represented in an image as a single point unrelated to the shape or size of the objects. Tracking is the process of identifying a sequence of points from a respective sequence of the images that represents the motion of an object. The tracking problem is difficult when there are multiple closely spaced objects because the objects can change their speed and direction rapidly and move into and out of the line of sight for other objects. The problem is exacerbated because each set of returns may result from noise as well as echoes from the actual objects. The returns resulting from the noise are also called false positives. Likewise, the radar will not detect all echoes from the actual objects and this phenomena is called a false negative or missed detect error. For tracking airborne objects, a large distance between the radar and the objects diminishes the signal to noise ratio so the number of false positives and false negatives can be high. For robotic applications, the power of the radar is low and as a result, the signal to noise ratio can also be low and the number of false positives and false negatives high.

[0005] In view of the proximity of the objects to one another, varied motion of the objects and false positives and false negatives, multiple sequential images should be analyzed collectively to obtain enough information to properly assign the points to the proper tracks. Naturally, the larger the number of images that are analyzed, the greater the amount of information that must be processed.

[0006] While identifying the track of an object, a kinematic model describing the object's location, velocity and acceleration may be generated. Such a model provides the means by which the object's future motion can be predicted. Based upon such a prediction, appropriate action may be initiated. For example, in a military application there is a need to track multiple enemy aircraft or missiles in a region to predict their objective, plan responses and intercept them. Alternatively, in a commercial air traffic control application there is a need to track multiple commercial aircraft around an airport to predict their future courses and avoid collision. Further, in these and other applications, such as robotic applications, may use radar, sonar, infrared or other object detecting radiation bandwidths for tracking objects. In particular, in robotic applications reflected radiation can be used to track a single object which moves relative to the robot (or vice versa) so the robot can work on the object.

[0007] Consider the very simple example of two objects being tracked and no false positives or false negatives. The radar, after scanning at time t₁, reports objects at two locations in a first observation set. That is, it returns a set of two observations o₁₁, o₁₂. At time t₂ it returns a similar set of two observations o₂₁, o₂₂) from a second observation set. Suppose from prior processing that track data for two tracks T₁ and T₂ includes the locations at to of two objects. Track T₁ may be extended through the points in the two sets of observations in any of four ways, as may track T₂. The possible extensions of T₁ can be described as: T₁, o₁₁, o₂₁, T₁, o₁₂, o₂₂, T₁, o₁₂, o₂₁ and T₁, o₁₂, o₂₂. Tracks can likewise be extended from T₂ in four possible ways including, T₂, o₁₂, o₂₁. FIG. 1 illustrates these five (out of eight) possible tracks (to simplify the problem for purposes of explanation) . The five track extensions are labeled h₁₁, h₁₂, h₁₃, h₁₄, and h₂₁ wherein h₁₁ is derived from T₁, o₁₁, o₂₁, h₁₂ is derived from T₁, o₁₁, o₂₂, h₁₃ is derived from T₁, o₁₂, o₂₁, h₁₄ is derived from T₁, o₁₂, o₂₂, and h₂₁ is derived from (T₂, o₁₁, o₂₁. The problem of determining which such track extensions are the most likely or optimal is hereinafter known as the assignment problem.

[0008] It is known from prior art to determine a figure of merit or cost for assigning each of the points in the images to a track. The figure of merit or cost is based on the likelihood that the point is actually part of the track. For example, the figure of merit or cost may be based on the distance from the point to an extrapolation of the track. FIG. 1 illustrates costs _{21 22 21} modeled target characteristics. The function to calculate the cost will normally incorporate detailed characteristics of the sensor, such as probability of measurement error, and track characteristics, such as likelihood of track maneuver.

[0009] FIG. 2 illustrates a two by two by two matrix, c, that contains the costs for each point in relation to each possible track. The cost matrix is indexed along one axis by the track number, along another axis by the image number and along the third axis by a point number. Thus, each position in the cost matrix lists the cost for a unique combination of points and a track, one point from each image. FIG. 2 also illustrates a 0, 1 assignment matrix, z, which is defined with the same dimensions as the cost matrix. Setting a position in the assignment matrix to one means that the equivalent position in the cost matrix is selected into the solution. The illustrated solution matrix selects the h₁₄, h₂₁ solution previously described. Note that for the above example of two tracks and two snapshots, the resulting cost and assignment matrices are three dimensional. As used in this patent application, the term dimension means the number of axes in the cost or assignment matrix while size refers to the number of elements along a typical axis. The costs and assignments have been grouped in matrices to facilitate computation.

[0010] A solution to the assignment problem satisfies two constraintsfirst, the sum of the associated costs for assigning points to a track extension is minimized and, second, if no false positives or false negatives exist, then each point is assigned to one and only one track.

[0011] When false positives exist, however, additional hypothetical track extensions incorporating the false positives will be generated. Further note that the random locations of false positives will, in general, not fit well with true data and such additional hypothetical track extensions will result in higher costs. Also note that when false negative errors exist, then the size of the cost matrix must grow to include hypothetical track extensions formulated with gaps (i.e., data omissions where there should be legitimate observation data) for the false negatives. Thus, the second criteria must be weakened to reflect false positives not being as signed and also to permit the gap filler to be multiply assigned. With hypothetical cost calculated in this manner then the foregoing criteria for minimization will tend to materialize the false negatives and avoid the false positives.

[0012] For a 3-dimensional problem, as is illustrated in FIG. 1, but with N₁ (initial) tracks, N₂ observations in scan 1, N₃ observations in scan 2, false positives and negatives assumed, the assignment problem can be formulated as: $\begin{array}{cc}\left(a\right)\mathrm{Minimize}\text{:}\underset{i_1=0}{\overset{N_1}{}}\underset{i_2=0}{\overset{N_2}{}}\underset{i_3=0}{\overset{N_3}{}}{c}_{i_1{i}_2{i}_3}{z}_{i_1{i}_2{i}_3}\left(b\right)\mathrm{Subject}\mathrm{to}\text{:}\underset{i_2=1}{\overset{N_2}{}}\underset{i_3=1}{\overset{N_3}{}}{z}_{i_1{i}_2{i}_3}=1,i_1=1,,N_1\left(c\right)\underset{i_1=1}{\overset{N_1}{}}\underset{i_3=1}{\overset{N_3}{}}{z}_{i_1{i}_2{i}_3}1,i_2=1,N_2\left(d\right)\underset{i_1=1}{\overset{N_1}{}}\underset{i_2=1}{\overset{N_2}{}}{z}_{i_1{i}_2{i}_3}1,i_3=1,N_3\left(e\right)z_{i_1{i}_2{i}_3}\left\{0,1\right\}{z}_{i_1{i}_2{i}_3}& \left[1.0\right]\end{array}$

[0013] where c is the cost and z is a point or observation assignment, as in FIG. 2.

[0014] The minimization equation or equivalently objective function 1.0 (a) specifies the sum of the element by element product of the c and z matrices. The summation includes hypothesis representations z_i1i2i3 with observation number zero being the gap filler observation. Equation 1.0(b) requires that each of the tracks T₁, . . . T_N1 be extended by one and only one hypothesis. Equation 1.0 (c) relates to each point or observation in the first observation set and requires that each such observation, except the gap filler, can only associate with one track but because of the less than condition it might not associate with any track. Equation 1.0 (d) is like 1.0 (c) except that it is applicable to the second observation set. Equation 1.0 (e) requires that elements of the solution matrix z be limited to the zero and one values.

[0015] The only known method to solve Problem Formulation 1.0 exactly is a method called Branch and Bound. This method provides a systematic ordering of the potential solutions so that solutions with a same partial solution are accessible via a branch of a tree describing all possible solutions whereby the cost of unexamined solutions on a branch are incrementally developed as the cost for other solutions on the branch are determined. When the developing cost grows to exceed the previously known minimal cost (i.e., the bound) then enumeration of the tree branch terminates. Evaluation continues with a new branch. If evaluation of the cost of a particular branch completes, then that branch has lower cost than the previous bound so the new cost replaces the old bound. When all possible branches are evaluated or eliminated then the branch that had resulted in the last used bound is the solution. If we assume that N₁N₂N₃n and that branches typically evaluate to half there full length, then workload associated with branch and bound is proportional to ${\left(n!|\frac{n}{2}!\right)}^2.$

[0016] This workload is unsuited to real time evaluation.

[0017] The Branch and Bound algorithm has been used in past research on the Traveling Salesman Problem. Messrs. Held and Karp showed that if the set of constraints was relaxed by a method of Lagrangian Multipliers (described in more detail below) then tight lower bounds could be developed in advance of enumerating any branch of the potential solution. By initiating the branch and bound algorithm with such a tight lower bound, significant performance improvements result in that branches will typically evaluate to less than half their full length.

[0018] Messrs. Frieze and Yadagar in dealing with a problem related to scheduling combinations of resources, as in job, worker and work site, showed that Problem Formulation 1.0 applied. They further described a solution method based upon an extension of the Lagrangian Relaxation previously mentioned. The two critical extensions provided by Messrs. Frieze and Yadagar were: (1) an iterative procedure that permitted the lower bound on the solution to be improved (by hill climbing described below) and (2) the recognition that when the lower bound of the relaxed problem was maximized, then there existed a method to recover the solution of the non-relaxed problem in most cases using parameters determined at the maximum. The procedures attributed to Messrs. Frieze and Yadagar are only applicable to the 3-dimensional problem posed by Problem Formulation 1.0 and where the cost matrix is fully populated. However, tracking multiple airborne objects usually requires solution of a much higher dimensional problem.

[0019] FIGS. 1 and 2 illustrate an example where look ahead data from the second image improved the assignment accuracy for the first image. Without the look ahead, and based only upon a simple nearest neighbor approach, the assignments in the first set would have been reversed. Problem Formulation 1.0 and the prior art only permit looking ahead one image. In the prior art it was known that the accuracy of assignments will improve if the process looks further ahead, however no practical method to optimally incorporate look ahead data existed. Many real radar tracking problems involve hundreds of tracks, thousands of observations per observation set and matrices with dimensions in the range of 3 to 25 including many images of look ahead.

[0020] It was also known that the data assignment problem may be simplified (without reducing the dimension of the assignment problem) by eliminating from consideration for each track those points which, after considering estimated limits of speed and turning ability of the objects, could not physically be part of the track. One such technique, denoted hereinafter the cone method, defines a cone as a continuation of each previously determined track with the apex of the cone at the end of the previously defined track. The length of the cone is based on the estimated maximum speed of the object and the size of the arc of the cone is based on the estimated maximum turning ability of the object. Thus, the cone defines a region outside of which no point could physically be part of the respective track. For any such points outside of the cones, an infinite number could be put in the cost matrix and a zero could be preassigned in the assignment matrix. It was known for the tracking problem that these elements will be very common in the cost and selection matrices (so these matrices are sparse).

[0021] It was also known in the prior art that one or more tracks which are substantially separated geographically from the other tracks can be separated also in the assignment problem. This is done by examining the distances from each point to the various possible tracks. If the distances from one set of points are reasonably short only in relation to one track, then they are assigned to that track and not further considered with the remainder of the points. Similarly, if a larger group of points can only be assigned to a few tracks, then the group is considered a different assignment problem. Because the complexity of assignment problems increases dramatically with the number of possible tracks and the total number of points in each matrix, this partitioning of the group of points into a separate assignment problem and removal of these points from the matrices for the remaining points, substantially reduces the complexity of the overall assignment problem.

[0022] A previously known Multiple Hypothesis Testing (MHT) algorithm (see Blackman, Multiple-Target Tracking with Radar Applications, Chapter 10, Artech House Norwood Mass., 1986) related to formulation and scoring. The MHT procedure describes how to formulate the sparse set of all reasonable extension hypothesis (for FIG. 1 the set h₁₁. . . h₂₄) and how to calculate a cost of the hypothesis T_i, o_1j, o_2k based upon the previously calculated cost for hypothesis T₁, o_1j. The experience with the MHT algorithm, known in the prior art, is the basis for the assertion that look ahead through k sets of observations results in improved assignment of observations from the first set to the track.

[0023] In theory, the MHT procedure uses the extendable costing procedure to defer assignment decision until the accumulated evidence supporting the assignment becomes overwhelming. When it makes the assignment decision it then eliminates all potential assignments invalidated by the decision in a process called pruning the tree. In practice, the MHT hypothesis tree is limited to a fixed number of generations and the overwhelming evidence rule is replaced by a most likely and feasible rule. This rule considers each track independently of others and is therefore a local decision rule.

[0024] A general object of the present invention is to provide an efficient and accurate process for assigning each point object in a region from multiple images to a proper track and then taking an action based upon the assignments.

[0025] A more specific object of the present invention is to provide a technique of the foregoing type which determines the solution of a k-dimensional assignment problem where k is greater than or equal to three.

[0026] The present invention relates to a method and apparatus for tracking objects. In particular, the present invention tracks movement or trajectories of objects by analyzing radiation reflected from the objects, the invention being especially useful for real-time tracking in noisy environments.

[0027] In providing such a tracking capability, a region containing the objects is repeatedly scanned to generate a multiplicity of sequential images or data observation sets of the region. One or more points (or equivalently observations), in each of the images or observation sets are detected wherein each such observation either corresponds to an actual location of an object or is an erroneous data point due to noise. Subsequently, for each observation detected, figures of merit or costs are determined for assigning the observation to each of a plurality of previously determined tracks. *Afterwards, a first optimization problem is specified which includes:

[0028] (a) a first objective function for relating the above mentioned costs to potential track extensions through the detected observations (or simply observations); and

[0029] (b) a first collection of constraint sets wherein each constraint set includes constraints to be satisfied by the observations in a particular scan to which the constraint set is related. In general, there is a constraint for each observation of the scan, wherein the constraint indicates the number of track extensions to which the observation may belong.

[0030] In particular, the first optimization problem is formulated, generated or defined as an M-dimensional assignment problem wherein there are M constraint sets in the first collection of constraint sets (i.e., there are M scans being examined) and the first objective function minimizes a total cost for assigning observations to various track extensions wherein terms are included in the cost, such that the terms have the figures of merit or costs for hypothesized combinations of assignments of the observations to the tracks. Subsequently, the formulated M-dimensional assignment problem is solved by reducing the complexity of the problem by generating one or more optimization problems each having a lower dimension and then solving each lower dimension optimization problem. That is, the M-dimensional assignment problem is solved by solving a plurality of optimization problems each having a lower number of constraint sets.

[0031] The reduction of the M-dimensional assignment problem to a lower dimensioned problems is accomplished by relaxing the constraints on the points of one or more scans thereby permitting these points to be assigned to more than one track extension. In relaxing the constraints, terms having penalty factors are added into the objective function thereby increasing the total cost of an assignment when one or more points are assigned to more than one track. Thus, the reduction in complexity by this relaxation process is iteratively repeated until a sufficiently low dimension is attained such that the lower dimensional problem may be solved directly by known techniques.

[0032] In one embodiment of the invention, each k-dimensional assignment problem 2<kM, is iteratively reduced to a k1 dimensional problem until a 2-dimensional problem is specified or formulated. Subsequently, the 2-dimensional problem formulated is solved directly and a recovery technique is used to iteratively recover an optimal or near-optimal solution to each k-dimensional problem from a derived (k1) dimensional problem k2,3,4, . . .M.

[0033] In performing each recovery step (of obtaining a solution to a k-dimensional problem using a solution to a (k1)-dimensional problem) an auxiliary function, is utilized. In particular, to recover an optimal or near-optimal solution to a k-dimensional problem, an auxiliary function, _k-1, k4,5, . . . , M, is specified and a region or domain is determined wherein this function is maximized, whereby values of the region determine the penalty factors of the (k1)-dimensional problem such that another 2-dimensional problem can be formulated which determines a solution to the k-dimensional problem using the penalty factors of the (k1)-dimensional problem.

[0034] Each Auxiliary function _k of both lower dimensional problem penalty factors and a solution at the dimension k at which the penalized cost function is solved directly (typically a 2-dimensional problem). Further, in determining, for auxiliary function ₄ determined, and utilized to identify the peak region. Thus, gradients are used for each of the approximation functions _{3 4} determining penalty factors penalty factors until tm1 is used in determining the penalty factors for the M1 dimensional problem. Subsequently, once the M-dimensional problem is solved (using a 2-dimensional problem to go from an (M1) dimensional solution to an M-dimensional solution), one or more of the following actions are taken based on the track assignments: sending a warning to aircraft or a ground or sea facility, controlling air traffic, controlling anti-aircraft or anti-missile equipment, taking evasive action, working on one of the objects.

[0035] According to one feature of this first embodiment of the present invention, the following steps are also performed before the step of defining the auxiliary function. A preliminary auxiliary function is defined for each of the lower dimensional problems having a dimension equal or one greater than the dimension at which the penalized cost function is solved directly. The preliminary auxiliary function is a function of lower order penalty values and a solution at the dimension at which the penalized cost function was solved directly. In determining a gradient of the preliminary auxiliary function, step in the direction of the gradient to identify a peak region of the preliminary auxiliary function and determine penalty factors at the peak region. Iteratively repeat the defining, gradient determining, stepping and peak determining steps to define auxiliary functions at successively higher dimensions until the auxiliary function at 6-18 (k1) dimension is determined. In an alternative second embodiment of the present invention, instead of reducing the dimentiality of the M-dimensional assignment problem by a single dimension at a time, a plurality of dimensions are relaxed simultaneously. This new strategy has the advantage that when the M-dimensional problem is relaxed directly to a 2-dimensional assignment problem, then all computations may be performed precisely without utilizing an auxiliary function such as _k embodiment. More particularly, the second embodiment solves the first optimization problem (i.e., the M- dimensional assignment problem) by specifying (i.e., creating, generating, formulating and/or defining) a second optimization problem. The second optimization problem includes a second objective function and a second collection of constraint sets wherein:

[0036] a) the second objective function is a combination of the first objective function and penalty factors or terms determined for incorporating M-m constraint sets of the first optimization problem into the second objective function;

[0037] b) the constraint sets of the second collection include only m of the constraint sets of the first collection of constraints,

[0038] wherein 2<mM1. Note that, once the second optimization problem has been specified or formulated, an optimal or near-optimal solution is determined and that solution is used in specifying (i.e, creating, generating, formulating and/or defining) a third optimization problem of M-m dimensions (or equivalently constraint sets) . The third optimization problem is subsequently solved by decomposing it using the same procedure of this second embodiment as was used to decompose the first optimization problem above. Thus, a plurality of instantiations of the third optimization problem are specified, each successive instantiation having a lower number of dimensions, until an instance of the third optimization problem is a two dimensional assignment problem which can be solved directly. Subsequently, whenever an instance of the third optimization problem is solved, the solution is used to recover a solution to the instance of the first optimization problem from which this instance of the third optimization was derived. Thus, an optimal or near-optimal solution to the original first optimization problem may be recovered through iteration of the above steps.

[0039] As mentioned above, the second embodiment of the present invention is especially advantageous when m2, since in this case all computations may be performed precisely and without utilizing auxiliary functions.

[0040] Other features and benefits of the present invention will become apparent from the detailed description with the accompanying drawings contained hereinafter.

[0041] FIG. 1 is a graph of images or data sets generated by a scan of a region and possible tracks within the images or data sets according to the prior art.

[0042] FIG. 2 illustrates cost and assignment matrices for the data sets of FIG. 1 according to the prior art.

[0043] FIG. 3 is a block diagram of the present invention.

[0044] FIG. 4 is a flow chart of a process according to the prior art for solving a 3-dimensional assignment problem.

[0045] FIG. 5 Is a flow chart of a process according to the present invention for solving a k-dimensional assignment problem where k is greater than or equal to 3.

[0046] FIG. 6 is a graph of various functions used to explain the present invention.

[0047] FIG. 7 is another block diagram of the present invention for solving the k-dimensional assignment problem where k is greater than or equal to 3.

[0048] FIG. 8 is a flowchart describing the procedure for solving a n-dimensional assignment problem according to the second embodiment of the invention.

[0049] Referring now to the other figures in detail wherein like reference numerals indicate like elements throughout the several views, FIG. 3 illustrates a system generally designated 100 for implementing the present invention. System 100 comprises, for example, a radar station 102 (note sonar, microwave, infrared and other radiation bandwidths are also contemplated) for scanning a region which may be, for example, an aerial region (in aerial surveillance applications) or a work region (in robotic applications) and generating signals indicating locations of objects within the region. The signals are input to a converter 104 which converts the signals to data points or observations in which each object (or false positive) is represented by a single point. The output of the converter is input to and readable by a computer 106. As described in more detail below, the computer 106 assigns the points to respective tracks, and then displays the tracks and extrapolations of the tracks on a monitor 110. Also, the computer 106 determines an appropriate action to take based on the tracks and track extensions. For example, in a commercial application at an airport, the computer can determine if two aircraft being tracked are on a collision course and if so, signal a transmitter 112 to warn each aircraft, or if a scheduled take-off will pose the risk of collision, delay the take-off. For a military application on a ship or base, the computer can determine subsequent coordinates of enemy aircraft and send the coordinates to an antiaircraft gun or missile 120 via a communication channel 122. In a robotic application, the computer controls the robot to work on the proper object or portion of the object.

[0050] The invention generates k-dimensional matrices where k is the number of images or sets of observation data in the look ahead window plus one. Then, the invention formulates a k-dimensional assignment problem as in 1.0 above.

[0051] The k-dimensional assignment problem is subsequently relaxed to a (k1)-dimensional problem by incorporating one set of constraints into the objective function using a Lagrangian relaxation of this set. Given a solution of the (k1)-dimensional problem, a feasible solution of the k-dimensional problem is then reconstructed. The (k1)-dimensional problem is solved in a similar manner, and the process is repeated until it reaches the two-dimensional problem that can be solved exactly. The ideas behind the Lagrangian relaxation scheme are outlined next.

[0052] Consider the integer programming problem $\begin{array}{cc}\mathrm{Minimize}v\left(z\right)=c^{T}z\mathrm{Subject}\mathrm{to}\text{:}\mathrm{Az}b\mathrm{Bz}d{z}_i\mathrm{is}\mathrm{an}\mathrm{integer}\mathrm{for}iI& \text{[1.0.1]}\end{array}$

[0053] where the partitioning of the constraints is natural in some sense. Given a multiplier vector u0, the Lagrangian relaxation of 1.0.1 relative to the constraints Bzd is defined to be $\begin{array}{cc}{}^{TT}\mathrm{Subject}\mathrm{to}\text{:}\mathrm{Az}b{z}_i\mathrm{is}\mathrm{an}\mathrm{integer}\mathrm{for}iI& \text{[1.0.2]}\end{array}$

[0054] If the constraint set Bzd is replaced by Bzd, the nonnegativity constraint on u is removed. c^Tzu^T(Bzd) is the Lagrangian relative to the constraints Bzd, and hence the name Lagrangian relaxation. The following fact gives the relationship between the objective functions of the original and relaxed problems.

[0055] FACT A.1. If {overscore (z)} is an optimal solution to 1.0.1, then v({overscore (z)}) for all u0. If an optimal solution {circumflex over (z)} of 1.0.2 is feasible for 1.0.2, then {circumflex over (z)} is an optimal solution for 1.0.1 and

[0056] Algorithm k _k²⁸ 0 converging to the solution {overscore (u)} of Maximize 0 and a corresponding sequence of feasible solutions z_kk0 of 1.0.1 as follows:

[0057] 1. Generate in initial approximation u₀.

[0058] 2. Given u_k, choose a search direction s_k and a search distance _{k k k k k} estimate u_k by u_k1u_{kk k}

[0059] 3. Given u_k1 and a feasible solution {circumflex over (z)}_k1(u_k1) of 1.0.2, recover a feasible solution z_k1(u_k1) of the integer programming problem 1.0.1.

[0060] 4. Check the termination criteria. If the algorithm is not finished, set kk1 and return to Step 2. Otherwise, terminate the algorithm.

[0061] If {circumflex over (z)} is an optimal solution of 1.0.1, then

[0062] Since the optimal solution {circumflex over (z)}{circumflex over (z)}(u) of 1.0.2 is usually not a feasible solution of 1.0.1 for any choice of the multipliers,

[0063]

[0064]

[0065]

[0066] _ppk in z_{1^{i . . . i}k}^k in problem 1.1 below indicates that the track extension represented by z_i1^k. . . ⁱ_k includes a false positive in the p^th observation set. Note that this implies that a hypothesis be formed incorporating an observation with k1 gap fillers, e.g. , z_{0 . . . 0ip0 . . . 0}, i_p0. Thus, the resulting generalization of Problem Formulation 1.0 without the less than complication within the constraints is the following k-Dimensional Assignment Problems in which k3: $\begin{array}{cc}\mathrm{Minimize}\underset{i_1=0}{\overset{N_1}{}}\underset{i_k=0}{\overset{N_k}{}}{c}_{i_1{i}_k}{z}_{i_1{i}_k}\mathrm{Subject}\mathrm{to}\text{:}\underset{i_2=0}{\overset{N_2}{}}\underset{i_k=0}{\overset{N_k}{}}{z}_{i_1{i}_k}=1,i_1=1,N_1\underset{i_1=0}{\overset{N_1}{}}\underset{i_{j-1}=0}{\overset{M_{j-1}}{}}\underset{i_{j+1}=0}{\overset{M_{j+1}}{}}\underset{i_k=0}{\overset{N_k}{}}{z}_{i_1{i}_k}=1,\mathrm{for}{i}_j=1,,N_j\mathrm{and}j=2,,k-1,\underset{i_1=0}{\overset{N_1}{}}\underset{i_{k-1}=0}{\overset{N_{k-1}}{}}{z}_{i_1{i}_k}=1,i_k=1,N_k{z}_{l_1{i}_k}\left\{0,1\right\}{i}_{n}n=1,,k& \left(1.1\right)\end{array}$

[0067] where c and z are similarly dimensioned matrices representing costs and hypothetical assignments. Note, in general, for tracking problems these matrices are sparse.

[0068] After formulating the k-dimensional assignment problem as in 1.1, the present invention solves the resulting problem so as to generate the outputs required by devices 110, 112, 122 and 130. For each observation set O_i received from converter 104 at time t_i where i1. . . , N₁, the computer processes O_i in a batch together with the other observation sets O_ik1, . . . , O_i and the track T_ik, i.e., T_j is the set of all tracks that have been defined up to but not including O_j. (Note, bold type designations refer to the vector of elements for the indicated time, i.e., the set of all observations in the scan or tracks existing at the time, etc.) The result of this processing is the new set of tracks T_ik1 and a set of cost weighted possible solutions indicating how the tracks might extend to the current time t_i. At time t_i1 the batch process is repeated using the newest observation set and deleting the oldest. Thus, there is a moving window of observation sets which is shifted forward to always incorporate the most recent observation set. The effect is that input observation sets are reused for k1 cycles and then on the observation set's k-th reuse each observation within the observation set is integrated into a track.

[0069] FIG. 7 illustrates various processes implemented upon receipt of each observation set. Except for the addition of the k-dimensional assignment solving process 300 and the modification to scoring process 154 to build data structures suitable for process 300, all processes in FIG. 7 are based upon prior art. The following processes 150 and 152 extend previously defined tracks h_i1 based on new observations. Gate formulation and output process 156 determines, for each of the previously defined tracks, a zone wherein the track may potentially extend based on limits of velocity, maneuverability and radar precision. One such technique to accomplish this is the cone method described previously. The definition of the zone is passed to gating process 150. When a new observation set O_i is received, the gating process 150 will match each member observation with the zone for each member of the hypothetical set h_i1. After all input observations from O_i are processed the new hypothesis set h_i is generated by extending each track of the prior set of hypothetical tracks h_i1 either with missed detect gap fillers or with all new observation elements satisfying the track's zone. This is a many to many matching in that each hypothesis member can be extended to many new observations and each new observation can be used to extend many hypotheses. It, however, is not a full matching in that any hypothesis will neither be matched to all observations nor vice versa. It is this matching characteristic that leads to the sparse matrices involved in the tracking process. Subsequently, gating 150 forwards the new hypothesis set h_i to filtering process 152. Filtering process 152 determines a smooth curve for each member of h_i. Such a smooth curve is more likely than a sharp turn from each point straight to the next point. Further, the filtering process 152 removes small errors that may occur in generating observations. Note that in performing these tasks, the filtering process 152 preferably utilizes a minimization of a least squares test of the points in a track hypothesis or a Kalman Filtering approach.

[0070] As noted above, the foregoing track extension process requires knowledge of a previous track. For the initial observations, the following gating process 158 and filtering process 160 determine the previous track based on initial observations. In determining the initial tracks, the points from the first observation set form the beginning points of all possible tracks. After observation data from the next observation set is received, sets of simple two point straight line tracks are defined. Then, promotion, gate formulation, and output step 162 determines a zone in which future extensions are possible. Note that filtering step 160 uses curve fitting techniques to smooth the track extensions depending upon the number of prior observations that have accumulated in each hypothesis. Further note that promotion, gate formulation and output process 162 also determines when sufficient observations have accumulated to form a basis for promoting the track to processes 150 and 152 as described above.

[0071] The output of each of the filtering processes 152 and 160 is a set of hypothetical track extensions. Each such extension contains the hypothetical initial conditions (from the previous track), the list of observations incorporated in the extension, and distance between each observation and the filtered track curve. Scoring process 154 determines the figure of merit or cost of an observation being an extension of a track. In one embodiment, the cost is based on the above-mentioned distance although the particular formula for determining the cost is not critical to the present invention. A preferred formula for determining the cost utilizes a negative log likelihood function in which the cost is the negative of the sum of: (a) the logs of the distances normalized by sensor standard deviation parameters, and (b) the log likelihoods for events related to: track initiation, track termination, track maneuver, false negatives and false positives. Note that track maneuvers are detected by comparing the previous track curve with the current extension. Further note that some of the other events related to, for example, false negatives and false positives are detected by analyzing the relative relationship of gap fillers in the hypothesis. Thus, after determining that one of these events occurred, a cost for it can be determined based upon suitable statistics tables and system input parameters. The negative log likelihood function is desirable because it permits effective integration of the useful components. Copies of the set of hypothetical track extensions which are scored are subsequently passed directly to one of the gate formulation and output steps 156 and 162. Note that the scoring process 154 also arranges the actual scores in a sparse matrix based upon observation identifiers, and passes them to k-dimensional assignment problem solving process 300.

[0072] The assignment solving process 300 is described below. Its output is simply the list of assignments which constitute the most likely solution of the problem described by Equation 1.1. Note that both gate formulation and output processes 156 and 162 use (at different times) the list of assignments to generate the updated track history T_i to eliminate or prune alternative previous hypotheses that are prohibited by the actual assignments in the list, and subsequently to output any required data. Also note that when one of the gate formulation and output processes 156 and 162 accomplish these tasks, the process will subsequently generate and forward the new set of gates for each remaining hypothesis and the processes will then be prepared to receive the next set of observations. In one embodiment, the loop described here will generate zones for a delayed set of observations rather than the subsequent set. This permits processes 156 and 162 to operate on even observation sets while the scoring step 154 and k-dimensional solving process operate on odd sets of observations, or vice versa.

[0073] The assignment solving process 300 permits the present invention to operate with window size of k1 for k3. The upper limit on k depends only upon the computational power of the computer 106 and the response time constraints of system 100. The k1 observation sets within the processing window plus the prior track history result in a k-dimensional Assignment Problem as described by Problem Formulation 1.1. The present invention solves this generalized problem including the processes required to consider false positives and negatives, and also the processes required to consider sparse matrix problem formulations.

[0074] I. A FIRST EMBODIMENT OF THE k-DIMENSIONAL ASSIGNMENT SOLVER 300

[0075] In describing a first embodiment of the k-dimensional assignment solver 300, it is worthwhile to also discuss the process of FIG. 4 which is used by the solver 300. FIG. 4 illustrates use of the Frieze and Yadagar process as shown in prior art for transforming a 3-dimensional assignment problem into a 2-dimensional assignment problem and then use a hill climbing algorithm to solve the 3-dimensional assignment problem. The solver 300 uses a Lagrangian Relaxation technique (well known in the art) to reduce the dimension of an original k-dimensional assignment problem (k>3) down to a 3-dimensional problem and then use the process of FIG. 4 to solve the 3-dimensional problem. Further note that the Lagrangian Relaxation technique is also utilized by the process of FIG. 4 and that in using this technique the requirement that each point is assigned to one and only one track is relaxed. Instead, an additional cost, which is equal to a respective Lagrangian Coefficient u, is added to the cost or objective function 1.0(a) whenever a point is assigned to more than one track. This additional cost can be picked to weight the significance of each constraint violation differently, so this additional cost is represented as a vector of coefficients u which are correlated with respective observation points. Hill climbing will then develop a sequence of Lagrangian Coefficients sets designated (u₀,. . . u_j, u_j1, . . . , u_p). That correspond to an optimum solution of the 2-dimensional assignment problem. The assignments at this optimum solution are then used to recover the assignment solution of the 3-dimensional assignment problem.

[0076] In step 200 of FIG. 4, initial values are selected for the u₀ coefficients. Because the Lagrangian Relaxation process is iterative, the initial values are not critical and are all initially selected as zero. In step 202, these additional costs are applied to the objective function 1.0(a). With the addition of the costs u, the goal is still to assign the points which minimize the total cost. This transforms Equation 1.0(a) , written for k3 and altered to exclude mechanisms related to false positives and negatives, into objective function 2.1(a). In the first iteration it is not necessary to consider the u matrix because all u values are set to zero. To relax the requirement that each point be assigned to one and only one track, the constraint Equation 1.0 (d) is deleted, thereby permitting points from the last image to be assigned to more than one track. Note that while any axis can be chosen for relaxation, observation constraints are preferably relaxed. The effect of this relaxation is to create a new problem which must have the same solution in the first two axes but which can have a differing solution in the third axis. The result is constraints 2.1(bd). $\begin{array}{cc}\left(a\right)\mathrm{Minimize}\text{:}\underset{i_1=0}{\overset{N_1}{}}\underset{i_2=0}{\overset{N_2}{}}\left\{\left(c_{i_1{i}_2{i}_3}-u_{j_{i_3}}\right)\right\}{z}_{i_1{i}_2{i}_3}\left(b\right)\mathrm{Subject}\mathrm{to}\text{:}\underset{i_2=1}{\overset{N_2}{}}\underset{i_3=1}{\overset{N_3}{}}{z}_{i_1{i}_2{i}_3}=1,i=1,,N_1\left(c\right)\underset{i_1=1}{\overset{N_1}{}}\underset{i_3=1}{\overset{N_3}{}}{z}_{i_1{i}_2{i}_3}1,i_2=1,,N_2\left(d\right)z_{i_1{i}_2{i}_3}\left\{0,1\right\}{z}_{i_1{i}_2{i}_3}& \left[2.1\right]\end{array}$

[0077] Step 204 then generates from the 3-dimensional problem described by Problem Formulation 1.0 a new 2-dimensional problem formulation which will have the same solution for the first two indices. As Problem Formulation 2.1 has no constraints on the 3^rd axis, any value within a particular 3^rd axis can be used in a solution, but using anything other than the minimum value from any 3-rd axis has the effect of increasing solution cost. Conceptually, the effect of step 204 is to change the 3-dimensional arrays in Problem Formulation 2.1 into 2-dimensional arrays as shown in Problem Formulation 2.2 and to generate the new 2-dimensional matrix m_i1i2 defined as shown in Equation 2.3. $\begin{array}{cc}\left(a\right)\mathrm{Minimize}\text{:}\underset{i_1=0}{\overset{N_1}{}}\underset{i_2=0}{\overset{N_2}{}}\left\{\begin{array}{cc}\mathrm{Min}\text{:}& \\ & \left(c_{i_1{i}_2{i}_3}-u_{j_{i_3}}\right)\\ i_3& \end{array}\right\}{z}_{i_1{i}_2}\left(b\right)\mathrm{Subject}\mathrm{to}\text{:}\underset{i_2=1}{\overset{N_2}{}}\underset{i_3=1}{\overset{N_3}{}}{z}_{i_2{i}_3}=1,i_1=1,,N_1\left(c\right)\underset{i_1=1}{\overset{N_1}{}}\underset{i_3=1}{\overset{N_3}{}}{z}_{i_1{i}_3}1,i_2=1,,N_2\left(d\right)z_{i_1{i}_2}\left\{0,1\right\}{z}_{i_1{i}_2}& \left[2.2\right]\\ m_{i_1{i}_2}=\mathrm{Min}\text{:}\arg \mathrm{minimize}\left\{c_{i_1{i}_2}-u_{j_i}|i=1,,N_k\right\}& \left[2.3\right]\end{array}$

[0078] The cost or objective function for the reduced problem as defined by 2.2(a), if evaluated at all possible values of u is a surface over the domain of U³. This surface is referred to as u

[0079] u

[0080] u_j

[0081] coefficients u_j1 whose corresponding Problem Formulation 2.2 problem solution is a cost value closer to the peak of 0 u

[0082] u_p

[0083] dimensional assignment problem requires solving the Equation 2.2 problem corresponding to u_p.

[0084] In step 206, the two dimensional problem is solved directly using a technique known to those skilled in the art such as Reverse Auction for the corresponding cost and solution values. This is the evaluation of one point on the surface or for the first iteration u₀

[0085] Thus, after this first iteration, the points have been assigned based on all u values being arbitrarily set to zero. Because the u values have been arbitrarily assigned, it is unlikely that these assignments are correct and it is likely that further iterations are required to properly assign the points. Step 208 determines whether the points have been properly assigned after the first iteration by determining if for this set of assignments whether a different set of u values could result in a higher total cost. Thus, step 208 is implemented by determining the gradient of objective function 2.2(a) with respect to u_j. If the gradient is substantially non-zero (greater than a predetermined limit) then the assignments are not at or near enough to the peak of the u

[0086] u_j1

[0087] Hill climbing Step 212 determines the u_j1 values based upon the u_j values, the direction resulting from protecting the previous gradient into the U³ domain, and a step size. The solution value of the 2-dimensional problem is the set of coefficients that minimize the 2-dimensional problem and the actual cost at the minimum. Those coefficients augmented by the coefficients stored in m_i1i2 permit the evaluation (but not the minimization) of the cost term in Problem Formulation 2.1. These two cost terms are lower and upper bounds on the actual minimized cost of the 3-dimensional problem, and the difference between them in combination with the gradient is used to compute the step size.

[0088] With this new set of u values, steps 202-210 are repeated as a second iteration. Steps 212 and 202-210 are repeated until the gradient as a function of u determined in step 208 is less than the predetermined limit. This indicates that the u_p values which locate the peak area of the u surface are determined and that the corresponding Problem Formulation 2.2 has been solved. Step 214 will attempt to use the assignments that resulted from this particular 2-dimensional assignment problem to recover the solution of the 3-dimensional assignment problem as described below. If the limit was chosen properly so that the u values are close enough to the peak, this recovery will yield the proper set of assignments that rigidly satisfies the constraint that each point be assigned to one and only one track. However, if the u values are not close enough to the peak, then the limit value for decision 210 is reduced and the repetition of steps 212 and 202-210 is continued.

[0089] Step 214 recovers the 3-dimensional assignment solution by using the assignment values determined on the last iteration through step 208. Consider the 2-dimensional z assignment matrix to have 1's in the locations specified by the list L₁(a_i, b_i)_i1^N. If the 3-dimensional z matrix is specified by placing 1's at the location indicated by the list L₂(a_i, b_i, m_aib1)_{{haeck over (N)}}1 then the result is a solution of Problem Formulation 2.1. Let L₃(m_aib1){haeck over (N)}1 be the list formed by the third index. If each member of L₃ is unique then the L₂ solution satisfies the third constraint so it is a solution to Problem Formulation 1.0. When this is not the case, recovery determines the minimal substitutions required within list L₃ so that it plus L₁ will be a feasible solution, i.e., a solution which satisfies the constraints of a problem formulation, but which may not optimize the objective function of the problem formulation. This stage of the recovery process is formulated as a 2-dimensional assignment problem: Form a new cost matrix c_i,ji,j1^N where c_i,jc_aibij, for j1. . . N_i and the N_i term is the total number of cost elements in the selected row of the 3-dimensional cost matrix. Attempt to solve this 2-dimensional problem for the minimum using two constraints sets. If a feasible solution is found then the result will have the same form as list L₁. Replace the first set of indexes by the indicated (a₁, b_i) pairs taken from list L₁ and the result will be a feasible solution of Problem Formulation 1.1. If no feasible solution to the new 2-dimensional problem exists then further effort to locate the peak of u

[0090]

[0091]

[0092]

[0093]

[0094] by the set of observations o_{{overscore (y)}}i1, . . . , M1, is not valid. Because the matrix is sparse the list of cost elements is stored as a packed list, and then for each dimension of the matrix, a vector of a variable length list of pointers to the cost elements is generated and stored. This organization means that for a particular observation o_{{overscore (y)}} the j^th list in the i^th vector will be a list of pointers to all hypotheses in which o_{{overscore (y)}} participates. This structure is further explained in the following section dealing with problem partitioning.

[0095] The objective of the assignment solving process is to select from the set of all possible combinations of track extensions a subset that satisfies two criteria. First, each point in the subset of combinations should be assigned to one and only one track and therefore, included in one and only one combination of the subset, and second, the total of the scoring sums for the combinations of the subset should be minimized. This yields the following M-dimensional equations where kM: $\begin{array}{cc}\left(a\right)\mathrm{Minimize}\text{:}{v}_k\left(z^k\right)=\underset{i_1=0}{\overset{N_1}{}}\underset{i_k=0}{\overset{N_k}{}}{c}_{i_1{i}_k}^k{z}_{i_1{i}_k}^k\left(b\right)\mathrm{Subject}\mathrm{to}\text{:}\underset{i_2=0}{\overset{N_2}{}}\underset{i_k=0}{\overset{N_k}{}}{z}_{i_1{i}_k}^k=1i_1=1,N_1\left(c\right)\underset{i_1=0}{\overset{N_1}{}}\underset{i_{j-1}=0}{\overset{N_{j-1}}{}}\underset{i_{j+1}=0}{\overset{N_{j+1}}{}}\underset{i_k=0}{\overset{N_k}{}}{z}_{i_1{i}_k}^k=1\mathrm{for}{i}_j=1,N_j\mathrm{and}j=2,k-1\left(d\right)\underset{i_1=0}{\overset{N_1}{}}\underset{i_{k-1}}{\overset{N_{k-1}}{}}{z}_{i_1{i}_k}^k=1i_k=1,N_k\left(e\right)z_{i_1{i}_k}^k\left\{0,1\right\}\mathrm{for}\mathrm{all}{i}_1{l}_k& \left[3.1\right]\end{array}$

[0096] and where c^k is the cost matrix c_il^k. . . ⁱ_k which is a function of the distance between the observed point z^k and the smoothed track determined by the filtering step, and v_k is the cost function. This set of equations is similar to the set presented in Problem Formulation 1.1 except that it includes the subscript and superscript k notation. Thus, in solving the M-dimensional Assignment Problem the invention reduces this problem to an M-1 dimensional Assignment Problem and then to an N-2 dimensional Assignment Problem, etc. Further, the symbol k3, . . . ,M customizes Problem Formulation 3.1 to a particular relaxation level. That is, the notation is used to reference data from levels relatively removed as in c^k1 are the cost coefficients which existed prior to this level of relaxed coefficients c^k. Note that actual observations are numbered from 1 to N_i where N_i is the number of observations in observation set i. Further note that the added 0 observation in each set of observations is the unconstrained gap filler. This element serves as a filler in substituting for missed detects, and a sequence of these elements including only one true observation represents the possibility that the observation is a false positive. Also note that by being unconstrained a gap filler may be used in as many hypotheses as required.

[0097] While direct solution to 3.1 would give the precise assignment, the solution of k-dimensional equations directly for large k is too complex and time consuming for practice. Thus, the present invention solves this problem indirectly.

[0098] The following is a short description of many aspects of the present invention and includes some steps according to the prior art. The first step in solving the problem indirectly is to reduce the complexity of the problem by the previously known and discussed Lagrangian Relaxation technique. According to the Lagrangian Relaxation technique, the absolute requirement that each point is assigned to one and only one track is relaxed such that for some one image, points can be assigned to more than one track. However, a penalty based on a respective Lagrangian Coefficient u^k is added to the cost function when a point in the image is assigned to more than one track. The Lagrangian Relaxation technique reduces the complexity or dimension of the formulation of the assignment problem because constraints on one observation set are relaxed. Thus, the Lagrangian Relaxation is performed iteratively to repeatedly reduce the dimension until a 2-dimensional penalized cost function problem results as in Problem Formulation 2.1. This 2-dimensional problem is solved then directly by a previously known technique such as Reverse Auction. The penalized cost function for the 2-dimensional problem defines a valley or convex shaped surface which is a function of various sets of u^kk3, . . . ,M penalty values and one set of assignments for the points in two dimensions. That is, for each particular u³ there is a corresponding 2-dimensional penalized cost function problem and its solution. Note that the solution of the 2-dimensional penalized cost function problem identifies the set of assignments for the particular u³ values that minimize the penalized cost function. However, these assignments are not likely to be optimum for any higher dimensional problem because they were based on an initial arbitrary set of u_k values. Therefore, the next step is to determine the optimum assignments for the related 3-dimensional penalized cost function problem. There exists a 2-dimensional hill shaped function $u^3{u}^k|k>3\}$

[0099] values originally assigned. Then, the gradient of the hill-shaped

[0100] u³

[0101] previously selected for the one point on the hill (corresponding to the minimum of the penalized cost

[0102] which the corresponding problem will result in the peak of the characteristic. The next task is to recover the solution of the proper u⁴ values for the 4-dimensional problem. The foregoing hill climbing process will not work again because the foregoing hill climbing process when required to locate the 4-dimensional u⁴ values for the peak of ³

[0103] exact definition of the ³

[0104] case of ^{2 3}

[0105] the iteration can result in a greater than approximation of ³

[0106] function

[0107] u^k

[0108] gradient of the

[0109] u⁴ u⁴

[0110] results in a new 3-dimensional problem and requires the 2-dimensional hill climbing based upon new 2-dimensional problems. At the peak of the 3-dimensional to a complete solution. This process is repeated iteratively until the u^k values that result in a corresponding solution at the peak of the highest order

[0111]

[0112] Coefficient u^M penalty values initially equal to zero. The Lagrangian Coefficients associated with the M^th constraint set are a N_M1 element vector. The reduction of this M-dimensional problem in step 322 to a M1 dimensional problem uses the two step process described above. First, a penalty based on the value of the respective u^M coefficient is added to the cost function when a point is assigned to more than one track and then the resultant cost function is minimized. However, during the first iteration, the penalty is zero because all u^M values are initially set to zero. Second, the requirement that no point from any image can be assigned to more than one track is relaxed for one of the images. In the extreme this would allow a point from the relaxed image to associate with every track. However, the effect of the previous penalty would probably mean that such an association would not minimize the cost. The effect of the two steps in combination is to remove a hard constraint while adding the penalty to the cost function so that it operates like a soft constraint. For step 322 this two step process results in the following penalized cost function problem with kM: $\begin{array}{cc}\begin{array}{c}\left(a\right){}_k^k\mathrm{Minimize}\text{:}{}_k^{kk}\\ =\underset{i_1=0}{\overset{N_1}{}}\underset{i_k=0}{\overset{N_k}{}}{c}_{i_1{i}_k}^k{z}_{i_1{i}_k}^k-\\ \underset{i_k=0}{\overset{N_k}{}}{u}_{i_k}^k\underset{i_1=0}{\overset{N_1}{}}\underset{i_{k-1}}{\overset{N_{k-1}}{}}{z}_{i_1{i}_k}^k-1\\ =\underset{i_1=0}{\overset{N_1}{}}\underset{i_k=0}{\overset{N_k}{}}\left(c_{i_1{i}_k}^k-u_{i_k}^k\right)z_{i_1{i}_k}^k+\\ \underset{i_k=0}{\overset{N_k}{}}{u}_{i_k}^k\end{array}& \left[3.2\right]\\ \left(b\right)\mathrm{Subject}\mathrm{to}\text{:}\underset{i_2=0}{\overset{N_2}{}}\underset{i_k=0}{\overset{N_k}{}}{z}_{i_1{i}_k}^k=1i_1=1,N_1\left(c\right)\underset{i_1=0}{\overset{N_1}{}}\underset{i_{j-1}=0}{\overset{N_{j-1}}{}}\underset{i_{j+1}=0}{\overset{N_{j+1}}{}}\underset{i_k=0}{\overset{N_k}{}}{z}_{i_1{i}_k}^k=1\mathrm{for}{i}_j=1,N_j\mathrm{and}j=2,k-1\left(d\right)z_{i_1{i}_k}^k\left\{0,1\right\}\mathrm{for}\mathrm{all}{i}_1{l}_k& \end{array}$

[0113] Because the constraint on single assignment of elements from the last image has been eliminated, a M1 dimensional problem can be developed by eliminating some of the possible assignments. As shown in Equations 3.3, this is done by selecting the smallest cost clement from each of the M^th axis vectors of the cost matrix. Reduction in this manner yields a new, lower order penalized cost function defined by Equations 3.3 which has the same minimum cost as does the objective function defined by 3.2(a) above.

[0114] The cost vectors are selected as follows. Define an index array m_{i1 . . . ik}^1k and new cost array c_i1^k1. . . i_k1 by: $\begin{array}{cc}{m}_{i_1{i}_k}^k=\mathrm{Min}\text{:}\arg \mathrm{minimize}\left\{c_{i_1{i}_k}^k-u_{i_k}^k|i_k=0,1,,N_k\right\}{c}_{i_1{i}_{k-1}}^{k-1}=c_{i_1{i}_{k-1}{m}_{i_1{i}_k}^k}^k\mathrm{for}\left(i_1,,i_{k-1}\right)\left(0,,0\right)c_{0,,0}^{k-1}=\underset{i_k=0}{\overset{N_k}{}}\min \left\{0,c_{00}^k-u_{i_k}^k\right\}& \left[3.3\right]\end{array}$

[0115] The resulting M1 dimensional problem is (where kM): $\begin{array}{cc}{}_k^k{}_{k-1}^{k-1}\underset{i_1=0}{\overset{N_1}{}}\underset{i_{k-1}=0}{\overset{N_{k-1}}{}}{c}_{i_1{i}_{k-1}}^{k-1}{z}_{i_1{i}_{k-1}}^{k-1}\mathrm{Subject}\mathrm{to}\text{:}\underset{i_2=0}{\overset{N_2}{}}\underset{i_k=0}{\overset{N_k}{}}{z}_{i_1{i}_k}^{k-1}=1i_1=1,N_1\underset{i_1=0}{\overset{N_1}{}}\underset{i_{j-1}=0}{\overset{N_{j-1}}{}}\underset{i_{j+1}=0}{\overset{N_{j+1}}{}}\underset{i_k=0}{\overset{N_k}{}}{z}_{i_1{i}_k}^{k-1}=1\mathrm{for}{i}_j=1,,N_j\mathrm{and}j=2,k-1\underset{i_1=0}{\overset{N_1}{}}\underset{i_{k-2}=0}{\overset{N_{k-2}}{}}{z}_{i_1{i}_{k-1}}^{k-1}=1i_{k-1}=1,N_{k-1}{z}_{i_1{i}_k\left\{0,1\right\}}^k\left\{0,1\right\}\mathrm{for}\mathrm{all}{i}_1{l}_{k-1}& \left[3.4\right]\end{array}$

[0116] Assignment Problem 3.1 and Problem Formulation 3.4 differ only in the dimension M vs. M1, respectively. An optimal solution to 3.4 is also an optimal solution to equation 3.2. This relationship is the basis for an iterative sequence of reductions indicated by steps 320-332 through 330-332 and 200-204 in which the penalized cost function problem is reduced to a two dimensional problem. As these formula will be used to describe the processing at all levels, the lowercase k is used except where specific reference to the top level is needed. In step 206, the 2-dimensional penalized cost function is solved directly by the prior art Reverse Auction technique. Each execution of 206 produces two results, the set of z² values that minimize the problem and the cost that results v² when these z values are substituted into the objective function 3.4(a).

[0117] In step 208, according to the prior art, solution z² values are substituted into the 2-dimensional derivative of the ₂ u³

[0118] should be adjusted so as to perform the hill climbing function. As was previously described the objective is to produce a sequence of u₁³ values which ends when the u_p³ value in the domain of the peak of the

[0119]

[0120] u_i^k

[0121] to the peak of ₂

[0122] flow moves to step 214 which will attempt to recover the 3-dimensional solution as previously described. When further adjustment is required then the flow progresses to step 212 and the new values of u^k are computed. At the 2-dimensional level the method of the prior art could be used for the hill climbing procedure. However, it is not practical to use this prior art hill climbing technique to determine the updated Lagrangian Coefficients u^k or the Max on the next (or any) higher order

[0123]

[0124] u^k

[0125] values which result in z values that are closer to the proper z values for the highest k-dimension. This hill climbing process (which is different than that used in the prior art of FIG. 4 for recovering only the three dimensional solution) is used iteratively at all lower dimensions k of the M-dimensional problem (including the 3-dimensional level where it replaces prior art) even when k is much larger than three. FIG. 6 helps to explain this new hill climbing technique and illustrates the original k-dimensional cost function v_k of Problem Formulation 3.1. However, the actual k-dimensional cost surface v_k defined by 3.1(a) comprises scalar values at each point described by k-dimensional vectors and as such can not be drawn. Nevertheless, for illustration purposes only, FIG. 6 ignores the reality of ordering vectors and illustrates a concave function v_k (z^k) to represent Equation 3.1. The v_k surface is illustrated as being smooth to simplify the explanation although actually it can be imagined to be terraced. The goal of the assignment problem is to find the values of {overscore (z)}^k; these values minimize the k-dimensional cost function v_k.

[0126] For purposes of explanation, assume that in FIG. 6, k4 (the procedure is used for all k3). This problem is reduced by two iterations of Lagrangian Relaxation to a 2-dimensional penalized cost function _j^(k1)_i. This cost function, and all other cost functions described below are also non-smooth and continuous but are illustrated in FIG. 6 as smooth for explanation purposes. Solving the _j^(k1)_i problem results in one set of z² assignments and the value of _(k1)i at the point u_ij². A series of functions _j^(k1)_i, . . . , _l^(k1)_i each generated from a different u³ are shown. The particular series illustrates the process of locating the peak of _(k1)ⁱ. The 2-dimensional penalized cost functions _j^(k1)_i, . . . ,_l^(k1)_i can be solved directly. Each such solution provides the information required to calculate the next u³ value. Each iteration of the hill climbing improves the selection of u³ values, i.e., yields ² solution is closer to those at the solution of the ³ The result of solving _l^(k1)_i is values that are on both

[0127] _(k1)1 and _{k k}

[0128] _k

[0129] and 3.4 were solved at all possible values of u^k the function _{k k k}

[0130] than the v_k surface except at the peak as described in Equation 3.5 and its maximum occurs where the v_k surface is minimum. Because the _k

[0131] the _{k k}

[0132] v_k surface. The _k

[0133] minimization problem described by 3.1(a). Let {overscore (z)}^k be the unknown solution to Problem Formulation 3.1 and note that:

[0134] Consequently, the z^k values at the peak of _k greatest lower bound on the cost of the relaxed problem), can be substituted into the k-dimensional penalized cost function to determine the proper assignments. Consequently, the present invention attempts to find the maximum on the _k surface. However, it is not possible to use the prior art hill climbing to hill climb to the peak of _k definition of _k functions. As the solution of _l^k in that higher order u values are not yet optimized, its value can be larger than the true peak of _k lower bound on v_k and it can not be used to recover the required solution.

[0135] Instead, the present invention defines all auxiliary function _k

[0136] dimensional penalized cost function problem, lower order z^k values and u^k values determined previously by the reduction process. The _{k k} and its gradient is used for hill climbing to its peak. The z^k values at the peak of the _k into Problem Formulation 3.2 to determine the proper assignments. To define the _k

[0137] explicitly makes the function _k u^k

[0138] order sets of Lagrangian Coefficients with the expanded notation: _k u^k u^k1 u^k is defined recursively, using the _k $\begin{array}{cc}{}_3\left(u^3\right)=v_2+\underset{i_3=0}{\overset{N_3}{}}{u}_{i_3}^3={}_3\left(u^3;u^4,,u^K\right)& \text{[3.6] (a)}\end{array}$

[0139] where v₂ is the solution value for the most recent 2-dimensional penalized cost function problem. For k>3 $\begin{array}{cc}{}_k\left(u^3,,u^{k-1};u^k\right)=\{\begin{array}{cc}{}_k\left(u^k;u^{k+1},,u^k\right)& \mathrm{If}\mathrm{known}\\ {}_{k-1}\left(u^3,,u^{k-2};u^{k-1}\right)+\underset{i_k=0}{\overset{N_k}{}}{u}_{i_k}^k& \mathrm{otherwise}\end{array}& \text{[3.6] (b)}\end{array}$

[0140] From the definition of _k

[0141] compared with Problem Formulation 3.2): ${}_k\left(u^k;u^{k+1},,u^M\right)=v_{k-1}\left(z^{k-1}\right)+\underset{i_k=0}{\overset{N_k}{}}{u}_{i_k}^k$

[0142] it follows that: ${}_3\left(u^3\right)=v_2+\underset{i_3=0}{\overset{N_3}{}}{u}_{i_3}^3={}_3\left(u^3;u^4,,u^M\right)v_3^3\left(z^3\right)$

[0143] and with that Equation 3.5 is extended to:

[0144] This relationship means that either _{k k} hill climbing to update the Lagrangian Coefficients u. _k the preferred choice, however it is only available when the solution to Problem Formulation 3.2 is a feasible solution to Problem Formulation 3.1 (as in hill climbing from the second to third dimension which is why prior art worked). For simplicity in implementation, the function _k

[0145] that it equals _k

[0146] therefore always used, even for hill climbing from the second dimension. The use of the

[0147] u

[0148] u

[0149] _k

[0150] climbing/peak detection are to determine the gradient of the _k

[0151] and then move up the _k

[0152] increasing portion of the gradient. As shown in FIG. 6, any upward step on the _k

[0153] the _kj u^k

[0154] is closer to the ideal set of u^k values which correspond to the minimum of the v_k function. While it is possible to iteratively step up this _k

[0155] and then determine if the peak has been reached, the present invention optimizes this process by determining the single step size from the starting point at the minimum of _ki that will jump to the peak and then calculating the u^k values at the peak. Once the f. u s u at the peak at _k determined, then the u^k values can be substituted into Problem Formulation 3.2 to determine the proper assignment. (However, in a more realistic example, where k is much greater than three, then the u^k values at the peak of the along with the lower order u^k values and those assigned and yielded by the reduction steps are used to define the next higher level

[0156] _{k k} u^k

[0157] ^k

[0158] determine the gradient of each

[0159]

[0160] bundle) are determined at several points on the _k in the region of the starting point (i.e., minimum of _ki) and then averaged. Statistically, the averaged result should point toward the peak of the _k Subgradient Algorithm (Wolfe75, Wolfe79) for minimization was previously known in another environment to determine a gradient of a non-smooth surface using multiple subgradients and can be used with modification in the present invention. Wolfe's algorithm is further described in A Method of Conjugate Subgradients for Minimizing Nondifferentiable Functions page 147-173 published by Mathematical Programming Study 3 in 1975 and Finding the Nearest Point in a Polytope page 128-149 published by Mathematical Programming Study 11 in 1976. The modification to Wolfe's algorithm uses the information generated for _k u^k3 u^kk2 u^kk1 for calculating the subgradients. The definition of a subgradient v of _k u^k3 u^kk

[0161] subdifferential set defined as:

[0162] (where v^T is the transpose of v)

[0163] Next, a subgradient vector is determined from this function. If z^k is the solution of Problem Formulation 3.2, then differentiating _k u³ u^k1 u^k u_ik^k evaluating the result with respect to the current selection matrix z^k yields a subgradient vector: $g=\left(0,\left(1-\underset{i_1=0}{\overset{N_1}{}}\underset{i_{k-1}=0}{\overset{N_k}{}}{z}_{i_1}^k,i_k{i}_k=1,,N_k\right)\right)$

[0164] The iterative nature of the solution process at each dimension yields a set of such subgradients. Except for a situation described below where the resultant averaged gradient does not point toward the peak, the most recent set of such subgradients are saved and used as the bundle for the peak finding process for this dimension. For example, at the k level there is a bundle of subgradients of the _k the minimum of the _ki surface determined as a result of solving Problem Formulation 3.1 at all lower levels. This bundle can be averaged to approximate the gradient. Alternately, the previous bundle can be discarded so as to use the new value to initiate a new bundle. This choice provides a way to adjust the process to differing classes of problems, i.e., when data is being derived from two sensors and the relaxation proceeds from data derived from one sensor to the other then the prior relaxation data for the first sensor could be detrimental to performance on the second sensors data.

[0165] I.2.3. Step Size and Termination Criterion

[0166] After the average gradient of the _k determined, the next step is to determine a single step that will jump to the peak of the _k

[0167] is to first specify an arbitrary step size, and then calculate the value of _k

[0168] gradient. If the _k

[0169] this probably means that the step has not yet reached the peak. Consequently, the step size is doubled and a new _k value determined and compared to the previous one. This process is repeated until the new _k

[0170] previous one. At that time, the step has gone too far and crossed over the peak. Consequently, the last doubling is rolled-back and that step size is used for the iteration. If the initial estimated _k

[0171] step size is decreased by 50%. If this still results in a smaller _k

[0172] previous step size is decreased by 25%. The following is a more detailed description of this process.

[0173] With a suitable bundle of subgradients determined as just described. Wolfe's algorithm can be used to determine the effective subgradient (d) and the Upgraded value u_u1^k. From the previous iteration, or from an initial condition, there exists a step length value (t). The value,

[0174] is calculated as an estimate of u_j1^k. To determine if the current step size is valid the we evaluate _k u³ u^k2 u If the result represents an improvement then we double the step size. Otherwise we halve the step size. In either case a new u is calculated. The doubling or halving continues until the step becomes too large to improve the result, or until it becomes small enough to not degrade the result. The resulting suitable step size is saved with d as part of the subgradient bundle. The last acceptable u is assigned to u_j1^k.

[0175] Three distinct criterion are used to determine when u_j^k is close enough to {overscore (u)}^k:

[0176] 1. The Wolfe's algorithm criterion of d0 given that the test has been repeated with the bundle containing only the most recent subgradient.

[0177] 2. The difference between the lower bound _k u^k upper bound v_k (z^k, u^k) being less than a preset relative threshold. (Six percent was found to be an effective threshold for radar tracking problems.)

[0178] 3. An iteration count being exceeded.

[0179] The use of limits on the iteration are particularly effective for iterations at the level 3 through n-1 levels, as these iterations will be repeated so as to resolve higher order coefficient sets. With limited iterations the process is in general robust enough to improve the estimate of upper order Lagrangian Coefficients. By limiting the iteration counts then the total processing time for the algorithm becomes deterministic. That characteristic means the process can be effective for real time problems such as radar, where the results of the last scan of the environment must be processed prior to the next scan being received.

[0180] I.2.4. Solution Recovery

[0181] The following process determines if the u^k values at what is believed to be the peak of the _k

[0182] approximate the u^k values at the peak of the corresponding _k function. This is done by determining if the corresponding penalized cost function is minimized. Thus, the u^k values at the peak of the _k

[0183] Formulation 3.2 to determine a set of z assignments for k1 points. During the foregoing reduction process, Problem Formulation 3.4 yielded a tentative list of k selected z points that can be described by their indices as: ${\left\{\left(i_l^j{i}_{k=1}^j\right)\right\}}_{j=1}^{N_0}$

[0184] where N₀ is the number of cost elements selected into the solution. One possible solution of Problem Formulation 3.1 is the solution of Problem Formulation 3.2 which is described as ${\left\{\left(i_l^j{i}_{k-1}^j{m}_{i_1{i}_{k-1}}^k\right)\right\}}_{j=1}^{N_0}$

[0185] with m_i1^k. . . i_k1 as it was defined in Problem Formulation 3.3. If this solution satisfies the k-th constraint set then it is the optimal solution.

[0186] However, if the solution for Problem Formulation 3.2 is not feasible (decision 355), then the following adjustment process determines if a solution exists which satisfies the k-th constraint while retaining the assignments made in solving Problem Formulation 3.4. To do this, a 2-dimensional cost matrix is defined based upon all observations from the k-th set which could be used to extend the relaxed solution.

[0187] and j0, . . . , N₀3.8

[0188] If the resulting 2-dimensional assignment problem, $\begin{array}{cc}\text{Minimize:}\underset{j=0}{\overset{N_0}{}}\underset{l=0}{\overset{N_1}{}}{h}_{\mathrm{j1}}{w}_{\mathrm{j1}}\text{Subject to:}\underset{l=0}{\overset{N_k}{}}{w}_{\mathrm{j1}}=1j=1,,N_0\underset{l=0}{\overset{N_k}{}}{w}_{\mathrm{j1}}=1j=1,,N_0{w}_{\mathrm{j1}}\left\{0,1\right\}j=0,,N_0,l=0,,N_k& \left[3.9\right]\end{array}$

[0189] has a feasible solution then the indices of that solution map to the solution of Problem Formulation 3.1 for the k-dimensional problem. The first index in each resultant solution entry is the pointer back to an element of the ${\left\{\left(i_1^j{i}_{k-1}^j{m}_{i_1{i}_{k-1}}^k\right)\right\}}_{j=1}^{N_0}$

[0190] list. That element supplies the first k-1 indices of the solution. The second index of the solution to the recovery problem is the k^th index of the solution. Together these indexes specify the values of z^k that solve Problem Formulation 3.1 at the k^th level.

[0191] If Problem Formulation 3.9 does not have a feasible solution then the value of u_i^k which was thought to represent u_P^k is not representative of the actual peak and further iteration at the k^th level is required. This decision represent the branch path from step 214 and equivalent steps.

[0192] I.3. Partitioning

[0193] A partitioning process is used to divide the cost matrix that results from the Scoring step 154 into as many independent problems as possible prior to beginning the relaxation solution. The partitioning process is included with the Problem Formulation Step 310. The result of partitioning is a set of problems to be solved, i.e., there will be p₁ problems that consist of a single hypothesis, p₂ problems that consist of two hypothesis, etc. Each such problem is a group in that one or more observations or tracks are shared between members of the group.

[0194] In partitioning to groups no consideration is given to the actual cost values. The analysis depends strictly on the basis of two or more cost elements sharing the same specific axis of the cost matrix. In a 2-dimensional case two cost elements must be in the same group if they share a row or a column. If the two elements are in the same row, then each other cost clement that is also in the same row, as well as any cost elements that are in columns occupied by members of the row must be included in the group. The process continues recursively. In literature it is referred to as Constructing a Spanning Forest. The k-dimensional case is analogous to the 2-dimensional case. The specific method we have incorporated is a depth first search, presented by Aho, Hopecraft, and Ullman, in Design and Analysis of Computer Algorithms, section 5.2, published by Addison-Westley, Mass.

[0195] The result of partitioning at level M is the set of problems described as P_iji1, p_j and j1, . . . ,N, where N is the total number of hypothesis. The problems labeled P_ili1, . . . , p₁ are the cases where their is only one choice for the next observation at each scan and that observation could be used for no other track, i.e., it is a single isolated track. The hypothesis must be in included in the solution set and no further processing is required.

[0196] As hypothesis are constructed the first clement is used to refer to the track id. Any problem in the partitioned set which does not have shared costs in the first scan represent a case where a track could be extended in several ways but none of the ways share an observation with any other track. The required solution hypothesis for this case is the particular hypothesis with the maximum likelihood. For this case all likelihoods are determined as was described in Scoring and the maximum is selected.

[0197] In addition to partitioning at the M level, partitioning is applied to each subsequent level M1, . . . , 2. For each problem that was not eliminated by either by the prior selection partitioning is repeated, ignoring observations that are shared in the last set of observations. Partitioning recognizes that relaxation will eliminate the last constraint set and thus partitioning is feasible for the lower level problems that will result from relaxation. This process is repeated for all levels down to k3. The full set of partitionings can be performed in the Problem Formulation Step 310, prior to initiating the actual relaxation steps. The actual 2-dimensional solver used in step 206 includes an equivalent process so no advantage would be gained by partitioning at the k2 level.

[0198] There are two possible solution methods for the remaining problems. Branch and Bound as was previously described, or assigned to both tracks. The fruitfulness of partitioning increases for lower and lower levels of relaxation.

[0199] The following is a more detailed description of the partitioning method. Its application at all but the M level depends upon the relaxation process described in this invention. The recursive partitioning is therefore a distinct part of this invention. The advantage of this method is greatly enhanced by the sparse cost matrix resulting from tracking problems. However the sparse nature of the problem requires special storage and search techniques.

[0200] A hypothesis is the combination of the cost element c_n the selection variable z_n and all observations that made up the potential track extension. It can be written as, h_nc_n, z_n, o_nko_kik1, . . . , Mi1, . . . , N_k, i.e., cost, selection variable and observations in the hypothesis. n1, . . . , N where N is the total number of hypotheses in the problem. While the cost and assignment matrices were previously referenced, these matrices are not effective storage mechanisms for tracking applications. Instead the list of all hypothesis and sets of lists for each dimension that reference the relaxation method that this invention describes. If any of the partitions have 5-10 possible tracks and less than 50 to 20 hypotheses, then the prior art Branch and Bound algorithm generally executes faster than does the relaxation due to its reduced level of startup overhead. The Branch and Bound algorithm is executed against all remaining M level problems that satisfy the size constraint. For the remainder the Relaxation algorithm is used. The scheduling done in Problem Formulation allows each Problem Formulation 3.2 cost matrix resulting from the first step of relaxation to be partitioned. The resulting partitions can be solved by any of the four methods, isolated track direct inclusion, isolated track tree evaluation, small group Branch and Bound or an additional stage of relaxation as has been fully described.

[0201] The partitioning after each level of Lagrangian Relaxation is effective because when a problem is relaxed, the constraint that requires each point to be assigned to only one track is eliminated (for one image at a time). Therefore, two tracks previously linked by contending for one point will be unlinked by the relaxation which permits the point to be the hypothesis set are stored. The hypothetical set in list form is: ${\left\{h_n\right\}}_{n=1}^{n=N}$

[0202] For each dimension k1 . . . M there exists a set of lists, with each list element being a pointer to a particular hypothesis: $L_{k_i}={\left\{p_{k_j}|k_j=1,,N_{k_i}\right\}}_{i=1,k=1}^{i=N_k,k=M}$

[0203] where N_ki is number of hypothesis containing the i-th observation from scan k. This structure is a multiply linked list in that any observation is associated with a set of pointer to all hypothesis it participates in, and any hypothesis has a set of pointers to all observations that formed it. (These pointers can be implemented as true pointers or indices depending upon the particular programming language utilized.)

[0204] Given this description of the matrix storage technique then the partitioning technique is as follows: Mark the first list L_ki and follow out all pointers in that list to the indicated hypothesis h_Pki for i1, . . . , N_ki. Mark all located hypothesis, and for each follow pointers back the particular L_ok for k1, . . . M. Those L's if not previously marked get marked and also followed out to hypothesis elements and back to L's. When all such L's or h's being located are marked, then an isolated sub-problem has been identified. All marked elements can be removed from the lists and stored as a unique problem to be solved. The partitioning problem then continues by again starting at the first residual L set. When none remain, the original problem is completely partitioned.

[0205] Isolated problems can result from one source track having multiple possible extensions or from a set of source tracks contending for some observations. Because one of the indices of k (in our implementation it is k1) indicate the source track then it is possible to categorize the two problem types by observing if the isolated problem includes more than one L-list from the index level associated with tracks.

[0206] I.4. Subsequent Track Prediction

[0207] As noted above, after the points are assigned to the respective tracks, some action is taken such as controlling take-offs and landings to avoid collision, advising airborne aircraft to change course, warning airborne aircraft of an adjacent aircraft in a commercial environment, or aiming and firing an anti-aircraft (or anti-missile) missile, rocket or projectile, or taking evasive action in a military environment. Also, the tracking can be used to position a robot to work on an object. For some or all of these applications, usually the tracks which have just been identified are extended or extrapolated to predict a subsequent position of the aircraft or other object. The extrapolation can be done in a variety of prior art ways; for example, straight line extensions of the most likely track extension hypothesis, parametric quadratic extension of the x versus time, y versus time, etc. functions that are the result of the filtering process described earlier, least square path fitting or Kalman filtering of just the selected hypothesis.

[0208] The process of gating, filtering, and gate generation as they were previously described require that a curve be fit through observations such that fit of observations to the likely path can be scored and further so that the hypothetical target future location can be calculated for the next stage of gating. In the implementation existing, quadratic functions have been fit through the measurements that occur within the window. A set of quadratics is used, one per sensor measurement. To calculate intercept locations these quadratics can be converted directly into path functions. Intercept times are then calculated by prior methods based upon simultaneous solution of path equations.

[0209] The use of the fitted quadratics is not as precise as more conventional filters like the Kalman Filter. They are however much faster and sufficiently accurate for the relative scores required for the Assignment problem. When better location prediction is required then the assignment problem is executed to select the solution hypothesis and based upon the observations in those hypothesis the more extensive Kalman filter is executed. The result is tremendous computation savings when compared with the Kalman Filter being run on all hypothetical tracks.

[0210] Based on the foregoing, apparatus and methods have been disclosed for tracking objects. However, numerous modifications and substitutions can be made without deviating from the scope of the present invention. For example, the foregoing

[0211] u^k values are used to eliminate the higher than approximation characteristic of the _k

[0212]

[0213] _k

[0214] be as efficient it seems likely that the method would converge. Therefore, the invention has been disclosed by way of example and not limitation, and reference should be made to the following claims to determine the scope of the present invention.

[0215] II. AN ALTERNATIVE EMBODIMENT OF THE MULTI-DIMENSIONAL ASSIGNMENT SOLVING PROCESS

[0216] The following description provides an alternative second embodiment of the multi-dimensional assignment solving process of section I.1. However, in order to clearly describe the present alternative embodiment, further discussion is first presented regarding the data assignment problem of partitioning measurements into tracks and false alarms and, in particular, the representation of multi-dimensional assignment problems.

[0217] II.1. Formulation of the Assignment Problem

[0218] The goal of this section is to explain the formulation of the data association problems, and more particularly multi-dimensional assignment problems, that govern large classes of problems in centralized or hybrid centralized-sensor level multisensor/multitarget tracking. The presentation is brief; technical details are presented for both track initiation and maintenance in A. B. Poore, Multi-dimensional assignment formulation of data association problems arising from multitarget tracking and multisensor data fusion, Computational Optimization and Applications, 3 (1994), pp. 27-57, for nonmaneuvering targets and in Ingemar J. Cox, Pierre Hansen, and Beta Julesz, eds., Partitioning Data Sets, DIMACS Series in Discrete Mathematics and Theoretical Computer Science. American Mathematical Society, Providence, R.I., v. 19, Feb. 1995, pp. 169-198 for maneuvering targets. Note that the present formulation can also be modified to include target features (e.g., size and type) into the scoring process 154.

[0219] The data assignment problems for multisensor and multitarget tracking considered here are generally posed as that of maximizing the posterior probability of the surveillance region (given the data) according to $\begin{array}{cc}\mathrm{Maximize}\left\{\frac{P\left(=|Z^M\right)}{P\left(={}^0|Z^M\right)}|{}^{*}\right\}& \left[5.1\right]\end{array}$

[0220] where Z^M represents M data sets, of the data (and thus induces a partition of the data), * the finite collection of all such partitions,

[0221] P(Z^M) is the posterior probability of a partition true given the data Z^M. The term partition is defined below.

[0222] Consider M observation sets Z(k) (k1, . . . ,N) each of N_k observations ${\left\{z_{i_k}^k\right\}}_{i_k=1}^{N_k}$

[0223] and let Z^M denote the cumulative data set defined by $\begin{array}{cc}Z\left(k\right)={\left\{z_{i_k}^k\right\}}_{i_k=1}^{N_k}\mathrm{and}{Z}^M=\left\{Z\left(1\right),,Z\left(M\right)\right\},& \left[5.2\right]\end{array}$

[0224] respectively. In multisensor data fusion and multitarget tracking the data sets Z(k) may represent different classes of objects, and each data set can arise from different sensors. For track initiation the objects are measurements that must be partitioned into tracks and false alarms. In the formulation of track extensions a moving window over time of observations sets is used. The observation sets will be measurements which are: (a) assigned to existing tracks, (b) designated as false measurements, or (c) used for initiating tracks. However, note that alternative data objects instead of observation sets may also be fused such as in sensor level tracking wherein each sensor forms tracks from its own measurements and then the tracks from the sensors are fused in a central location. Note that, as one skilled in the art will appreciate, both embodiments of the present invention may be used for this type of data fusion.

[0225] The data assignment problem considered presently is represented as a case of set partitioning defined in the following way. First, for notational convenience in representing tracks, a zero index is added to each of the index sets in 5.2, a gap filler z₀^k is added to each of the data sets Z(k) in 5.2, track of data is defined as (z_i1¹, . . . , z_iN^N) where i_k and z_ik^k can now assume the values of 0 and z₀^k, respectively. A partition of the data will refer to a collection of tracks of data or track extensions wherein each observation occurs exactly once in one of the tracks of data and such that all data is used up. Note that the occurrence of the gap filler is unrestricted. The gap filler z₀^k serves several purposes in the representation of missing data, false observations, initiating tracks, and terminating tracks. Note that a reference partition may be defined which is a partition in which all observations are declared to be false.

[0226] Next under appropriate independence assumptions it can be shown that: $\begin{array}{cc}\frac{P\left(=|Z^M\right)}{P\left(={}^0|Z^M\right)}{L}_{}\underset{\left(i_1{i}_N\right)}{}{L}_{i_1{i}_M},& \left[5.3\right]\end{array}$

[0227] where L_il^k. . . ⁱ_M is a likelihood ratio containing probabilities for detection, maneuvers, and termination as well as probability density functions for measurement errors, track initiation and termination. Then if c_i1^k. . . ⁱ_NlnL_i1^k, . . . iM, $\begin{array}{cc}-\ln \frac{P\left(|Z^M\right)}{P\left({}^0|Z^M\right)}=\underset{\left(i_1,,i_M\right)}{}{c}_{i_1{i}_N}.& \left[5.4\right]\end{array}$

[0228] Using 5.3 and the zero-one variable z_i1^k . . . ⁱ_N1 if (i₁, . . . , i_M) $\begin{array}{cc}\underset{i_1=0}{\overset{N_1}{}}\underset{i_M=0}{\overset{N_M}{}}{c}_{i_1{i}_M}{z}_{i_1{i}_M}\underset{i_2=0}{\overset{N_2}{}}\underset{i_M=0}{\overset{N_M}{}}{z}_{i_1{i}_M}=1,l{}_1\left(c\right)\underset{i_1=0}{\overset{N_1}{}}\underset{i_{k-1}=0}{\overset{N_{k-1}}{}}\underset{i_{k+1}=0}{\overset{N_{k+1}}{}}\underset{i_M=0}{\overset{N_M}{}}{z}_{i_1{i}_M}=1,\mathrm{for}{i}_k=1,,N_k\mathrm{and}k=2,,M-1,\left(d\right)\underset{i_1=0}{\overset{N_1}{}}\underset{i_{N-1}=0}{\overset{N_{M-1}}{}}{z}_{i_1{i}_M}=1,i_M=1,,N_M\left(e\right)z_{i_1{i}_M}\left\{0,1\right\}\mathrm{for}\mathrm{all}{i}_1,,i_M& \left[5.5\right]\end{array}$

[0229] where c_{0 . . . 0} is arbitrarily defined to be zero. Here, each group of sums in the constraints 5.5 (b)-5.5(e) represents the fact that each non-gap filler observation occurs exactly once in a track of data. One can modify this formulation to include multiassignments of one, some, or all the actual observations. The assignment problem 5.5 is changed accordingly. For example, if z_ik^k is to be assigned no more than, exactly, or no less than n_lk ^k times, then the 1 in the appropriate one of the constraints 5.5 (b)-5.5(d) is changed to , , n_ik^k, respectively.

[0230] Expressions for the likelihood ratios L_i1^k. . . ⁱ_M such as 5.5 are well known. In particular, discussions of these ratios may be found in: A. B. Poore, Multi-dimensional assignment formulation of data association problems arising from multitarget tracking and multisensor data fusion,

[0231] Computational Optimization and Applications, 3 (1994), pp. 27-57; and Ingemar J. Cox, Pierre Hansen, and Beta Julesz, eds., Partitioning Data Sets, DIMACS Series in Discrete Mathematics and Theoretical Computer Science. American Mathematical Society, Providence, R.I., v. 19, 1995, pp. 169-198. Furthermore, the likelihood ratios are easily modified to include target features and to account for different sensor types. Also note that in track initiation, the M observation sets provide observations from M sensors, possibly all the same. Additionally note that for track maintenance, a sliding window of M observation sets and one data set containing established tracks may be used. In this latter case, the formulation is the same as above except that the dimension of the assignment problem is now M1.

[0232] II.2. Overview of the New Lagrangian Relaxation Algorithms

[0233] Having formulated an M-dimensional assignment problem 5.5, we now turn to a description of the Lagrangian relaxation algorithm. Subsection II.2.1 below presents many of the relaxation properties associated with the relaxation of an n-dimensional assignment problem to an m-dimensional one via a Lagrangian relaxation of n-m sets of constraints wherein m<nM and preferably in the present invention embodiment nm>1. Although any nm constraint sets can be relaxed, the description here is based on relaxing the last n-m sets of constraints and keeping the first m sets. Given either an optimal or suboptimal solution of the relaxed problem, a technique for recovering a feasible solution of the n-dimensional problem is presented in subsection II.2.2 below and an overview of the Lagrangian relaxation algorithm is given in subsection II.2.3.

[0234] The following notation will be used throughout the remainder of the work. Let M be an integer such that M3 and let n3, . . . ,M. The n-dimensional assignment problem is $\begin{array}{cc}\left(a\right)\mathrm{Minimize}\text{:}{v}_n\left(z\right)=\underset{i_1=0}{\overset{N_1}{}}\underset{i_n=0}{\overset{N_m}{}}{c}_{i_1{i}_n}^n{z}_{i_1{i}_n}^n\left(b\right)\mathrm{Subject}\mathrm{to}\text{:}\underset{i_2=0}{\overset{N_2}{}}\underset{i_n=0}{\overset{N_m}{}}{z}_{i_1{i}_n}^n=1,i_1=1,,N_1,\left(c\right)\underset{i_1=0}{\overset{N_1}{}}\underset{i_{k-1}=0}{\overset{N_{k-1}}{}}\underset{i_{k+1}=0}{\overset{N_{k-1}}{}}\underset{i_n=0}{\overset{N_m}{}}{z}_{i_1{i}_n}^n=1,\mathrm{for}{i}_k=1,,N_k\mathrm{and}k=2,,n-1,\left(d\right)\underset{i_1=0}{\overset{N_1}{}}\underset{i_{n-1}=0}{\overset{N_{m-1}}{}}{z}_{i_1{i}_n}^n=1,i_n=1,,N_m,\left(e\right)z_{i_1{i}_n}^n\left\{0,1\right\}\mathrm{for}\mathrm{all}{i}_1,,i_n& \left[6.1\right]\end{array}$

[0235] To ensure that a feasible solution of 6.1 always exists, all variables with exactly one nonzero index (i.e., variables of the form z_{0 . . . 0ik0 . . . 0}ⁿ for i_k0) are assumed free to be assigned and the corresponding cost coefficients are well-defined.

[0236] II.2.1. The Lagrangian Relaxed Assignment Problem

[0237] The n-dimensional assignment problem 6.1 has n sets of constraints. A (N_k1)-dimensional multiplier vector associated with the k-th constraint set will be denoted by u^k(u₀^k, u₁^k, . . . , u_Mk^k) with u₀^k0 and k1, . . . , n. The n-dimensional assignment problem 6.1 is relaxed to an m-dimensional assignment problem by incorporating n-m of the n sets of constraints into the objective function 6.1 (a). Although any constraint set can be relaxed, the description of the relaxation procedure for 6.1 will be based on the relaxation of the last n-m sets of constraints. The relaxed problem is $\begin{array}{cc}\begin{array}{c}{}_{mn}{}^{m+1n}\mathrm{Minimize}{}_{mn}{}^{nM+1n}\\ \underset{i_1=0}{\overset{N_1}{}}\underset{i_n=0}{\overset{N_m}{}}{c}_{i_1{i}_n}^n{z}_{i_1{i}_n}^n+\\ \underset{k=m+1}{\overset{n}{}}\underset{i_k=0}{\overset{N_k}{}}{u}_{i_k}^k\underset{i_1=0}{\overset{N_1}{}}\underset{i_{k-1}=0}{\overset{N_{k-1}}{}}\underset{i_{k+1}=0}{\overset{N_{k+1}}{}}\underset{i_n=0}{\overset{N_m}{}}{z}_{i_1{i}_n}^n-1\\ \underset{i_1=0}{\overset{N_1}{}}\underset{i_n=0}{\overset{N_m}{}}\left[c_{i_1{i}_n}^n+\underset{k=m+1}{\overset{n}{}}{u}_{i_k}^k\right]z_{i_1{i}_n}^n-\\ \underset{k=m+1}{\overset{n}{}}\underset{i_k=0}{\overset{N_k}{}}{u}_{i_k}^k\end{array}& \left[6.2\right]\\ \mathrm{Subject}\mathrm{to}\text{:}\underset{i_2=0}{\overset{N_2}{}}\underset{i_n=0}{\overset{N_n}{}}{z}_{i_1}^n{i}_n=1,i_1=1,,N_1,\underset{i_1=0}{\overset{N_1}{}}\underset{i_{k-1}=0}{\overset{N_{k-1}}{}}\underset{i_{k+1}=0}{\overset{N_{k+1}}{}}\underset{i_n=0}{\overset{N_m}{}}{z}_{i_1{i}_n}^n=1,\mathrm{for}{i}_k=1,,N_k\mathrm{and}k=2,,m,z_{i_1{i}_n}^n\left\{0,1\right\}\mathrm{for}\mathrm{all}{i}_1,,i_n.& \end{array}$