Summary of Past Lectures. Target Tracking: Lecture 4 Multiple Target Tracking: Part I. Lecture Outline. What is a hypothesis?

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1 REGLERTEKNIK Summary of Past Lectures AUTOMATIC COROL Target Tracing: Lecture Multiple Target Tracing: Part I Emre Özan emre@isy.liu.se Division of Automatic Control Department of Electrical Engineering Linöping University Linöping, Sweden Lecture- Introduction to and overview of target tracing. Lecture- Trac initiation Single target data association Lecture- How to detect maneuvers. Detection and multiple model based maneuver compensation. November 5, 0 E. Özan Target Tracing November 5, 0 / 9 E. Özan Target Tracing November 5, 0 / 9 Lecture Outline What is a hypothesis? What is an hypothesis? What is Single Hypothesis Tracing (SHT)? Multiple Hypothesis Tracing (MHT)? Single Hypothesis Tracing Global nearest neighbor (GNN) Joint probabilistic data association (JPDA) Definition An (association) hypothesis is a partitioning of a set of measurements according to the their origin. At each time step, a single hypothesis tracing algorithm eeps only a single hypothesis about all of the measurements received in the past. Global nearest neighbor algorithm does this by selecting the best hypothesis according to a criterion. Joint probabilistic data association filter () combines all possible current hypotheses into a single one to form a single composite hypothesis. For this reason it can also be called as composite hypothesis tracing. A multiple hypothesis tracer, on the other hand, eeps multiple hypotheses about the origin of the received data and has much more computation and memory requirements. E. Özan Target Tracing November 5, 0 / 9 E. Özan Target Tracing November 5, 0 / 9

2 Basic Scenario Considered in the Lecture All the past is summarized by a trac hypothesis and possibly some tentative tracs. Tentative trac processing is the same as what we learned in Lecture-. Using single target tracing methods for each target gives only locally optimal results. The global picture must be taen into account for targets sharing measurements in their gates or possibly some other measurement-to-target association conflict. ŷ ŷ ŷ Single hypothesis tracing All the past is summarized by a single hypothesis. In this single hypothesis, we have n T tracs and n I initiators (or tentative tracs). Generally, the initiation procedure is separated from the main logic. When a set of new measurements arrives, one first gates the measurements with the existing (confirmed) targets. Using the gating results, association is carried out. Using association results, confirmed tracs are updated. Unprocessed remaining measurements are sent to the initiator logic. Y List of Tracs List of Initiators Gating Association Unprocessed Measurements Update Y Initiator Processing If Confirmed List of Tracs List of Initiators + time E. Özan Target Tracing November 5, 0 5 / 9 E. Özan Target Tracing November 5, 0 6 / 9 Gating Suppose there are n T = (confirmed) tracs in the hypothesis summarizing the past. Once we get the measurements Y = {,..., y }, using the gate criteria we can prepare the following matrix to facilitate hypothesis generation. T T Such a matrix is called as validation matrix. ŷ ŷ ŷ Association Hypotheses Iterate over measurements Hypotheses measurements T T T Repeat the procedure above for = and y =. ŷ ŷ ŷ Validation Matrix T T E. Özan Target Tracing November 5, 0 7 / 9 E. Özan Target Tracing November 5, 0 8 / 9

3 Association Hypotheses Probability of a Hypothesis We can define an association hypothesis θ formally as a mapping θ ( ) : {,,..., m } {,,,..., n T, } where m is the number of measurements in Y i.e., Y = {,..., ym } n T is the number of targets formed in the past. Example Hypotheses with m =, n T = ŷ ŷ ŷ Suppose we are at time at an intermediate stage of tracing. We have j =,..., n T targets established previously and have just received Y = {,..., ym } Suppose θ ( ) is an arbitrary hypothesis about the origin of Y. Number of false alarms m F A in S, the surveillance region is distributed as P F A (m F A ); False alarm spatial density is p F A (y) Number of new targets in S, the surveillance region is distributed as P (m ); New target spatial density is p (y); Detection probability of the jth target: P j D ; Gate probability of the jth target: P j G ; Predicted measurement density of jth target: p j (y). E. Özan Target Tracing November 5, 0 9 / 9 E. Özan Target Tracing November 5, 0 0 / 9 Standard Settings Fundamental Theorem of TT P F A (m F A ) = (β F AV S ) m F A exp(β F A V S ) m F A! p F A (y) = /V S when y V S. P (m ) = (β V S ) m exp(β V S ) m! p (y) = /V S when y V S. Theorem: Suppose θ is an association hypothesis about the current measurement set Y. Then the posterior probability of θ is given as P (θ Y 0: ) β mf A F A βm ( ) P j D pj y θ (j) ( P j D P j G ) j J D j J ND where J D is the set of indices of detected targets, i.e., indices of targets which were assigned a measurement by θ. J ND is the set of indices of non-detected targets i.e., indices of target that were not assigned a measurement by θ. θ (j) is the index of the measurements that is assigned to target when j J D. E. Özan Target Tracing November 5, 0 / 9 E. Özan Target Tracing November 5, 0 / 9

4 Fundamental Theorem of TT Fundamental Theorem of TT Since there is a single hypothesis for the past, the term n T ( P j D P j G ) = ( P j D P j G ) ( P j D P j G ) j= j J D j J ND is constant for all hypotheses. Then, we have P (θ Y 0: ) β mf A F A βm j J D P j D pj ( y θ (j) ( P j D P j G ) Taing the logarithm, we have ( log P (θ Y 0: ) = m F A log β F A +m log β + P j D pj y θ log j J D ( P j D P j G ) ) (j) ) We can write the summed elements in the log-probability in a matrix given as A T T l log β F A log β log β F A log β l l log β F A log β l log β F A where log β represents. l ij log P j D pj (y i ) ( P j D P j G ). This matrix is called assignment matrix. Finding the optimal association hypothesis then corresponds to finding the column indices {j, j, j, j }, j i j i for i i such that the sum i= A ij i is maximized. E. Özan Target Tracing November 5, 0 / 9 E. Özan Target Tracing November 5, 0 / 9 Assignment Problem Assignment Problem We can mae a formal definition of the problem as follows We are given the matrix A R m n with n m. Define the auxiliary matrix Z = [z ij ] where z ij {0, }. Problem Definition m Maximize z ij A ij n z ij = i= subject to m i and z ij j= i= j Considered first in economics. Possible equivalents are assigning personnel to jobs or assigning delivery trucs to locations. Earlier methods used linear programming techniques, lie Hungarian method which is computationally costly. Less computationally expensive methods appeared later when different applications were found. Munres algorithm JVC algorithm (by Joner and Volgenant) Auction algorithm (by Bertseas): Explained thoroughly in the boo. This problem is called as assignment problem in optimization literature. E. Özan Target Tracing November 5, 0 5 / 9 E. Özan Target Tracing November 5, 0 6 / 9

5 Global Nearest Neighbor (GNN) Algorithm GNN Algorithm A T T l log β F A log β log β F A log β l l log β F A log β l log β F A log β Choose the best (largest probability) association hypothesis. The measurements associated to targets in the best association hypothesis are processed by target KFs. The measurements associated to and are handled by the initiator logic. Therefore, we combine the and cases into the single category of external sources (EX). The external sources become Poisson with density β EX = β F A + β. A T T l log β F A log β log β F A log β l l log β F A log β l log β F A log β A T T EX EX EX EX l log β EX log β EX l l log β EX y l log β EX E. Özan Target Tracing November 5, 0 7 / 9 E. Özan Target Tracing November 5, 0 8 / 9 Global Nearest Neighbor (GNN) Algorithm PDA vs. JPDA GNN Time = 0: Send all measurements to initiation logic. Time > 0: Suppose we have m measurements and n T targets Form the assignment matrix A R m (n T +m ) auction(a) Process the targets with their associated measurements. Send the measurements associated to external sources (EX) to initiation logic. Process the initiators (tentative tracs with EX associated measurements). Add any confirmed tentative trac to the confirmed trac list. Figure taen from: R.J. Fitzgerald, Development of Practical PDA Logic for Multitarget Tracing by Microprocessor, American Control Conference, pp , Jun E. Özan Target Tracing November 5, 0 9 / 9 E. Özan Target Tracing November 5, 0 0 / 9

6 Joint Probabilistic Data Association (JPDA) Filter Soft decision equivalent of GNN in the way that PDA is a soft version of NN. y measurements All past is again summarized with a single hypothesis (n T confirmed targets n I tentative targets). The number of targets is assumed fixed in the association, hence no N T associations in the possible hypotheses. For each previously established target, we need to calculate P (T j y i ) and P (T j ) () for i that are in the gate of the target. The update is then made with PDA update formulas by using these probabilities instead. A separate trac initiation logic must run along with to detect and initiate new tracs. Hypotheses T l ij = P j Dp j (yi ) T T lj = PDP j j G p β F Al l l p βf Al l l p βf Al l l p βf Al l l p 5 βf Al l l p 6 βf Al l l p 7 βf Al l l p 8 βf Al l l p 9 βf Al l l p 0 βf A l l l ŷ ŷ ŷ E. Özan Target Tracing November 5, 0 / 9 E. Özan Target Tracing November 5, 0 / 9 Hypotheses y T T l ij = P j Dp j (yi ) y lj = P j DP j G measurements p β F A l l l p 6 β F Al l l p β F Al l l p 7 β F Al l l p β F Al l l p 8 β F Al l l p β F Al l l p 9 β F Al l l p 5 β F Al l l p 0 β F A l l l P (T y )= c P (T )= c P ( y )= c P ( )= c P ( )= c i {,6,,7} p i i {,,5,8,9,0} p i i {,8} p i i {,6,,9} p i i {,7,5,0} p i P (T )= 5 c i= p i P (T )= 0 c i=6 p i c= 0 i= p i Probability calculations show that the calculations can be done separately for the clusters of targets that does not share gated measurements. In other words our previous scenario can be grouped into two clusters T &, T and probability calculations can be done separately for the corresponding hypothesis trees. ŷ ŷ ŷ The clustering can be done using the T T 0 0 validation matrix E. Özan Target Tracing November 5, 0 / 9 E. Özan Target Tracing November 5, 0 / 9

7 Hypotheses T T measurements p 6 l p 7 β F A l p l l p β F A l l p β F A l l p β F A l l p 5 βf A l l l ij = PDp j j (yi ) lj = PDP j j G P (T y )= c (p +p ) P (T )= c (p +p +p 5 ) P ( y )= c p P ( )= c (p +p ) P ( )= c (p +p 5 ) P (T )= c p 6 P (T )= c p 7 c = 5 i= p i c =p 6 +p 7 Time = 0: Send all measurements to initiation logic. Time > 0: Suppose we have m measurements and n T targets Form the validation matrix. Group the targets into clusters in which targets share gated measurements. For each cluster, calculate PDA probabilities for each target in the cluster by using a hypothesis tree. Update targets with the weighted equivalent measurements as PDA. Send the unprocessed measurements and possibly gated extra measurements to initiation logic. Process the initiators. Add any confirmed tentative trac to the confirmed trac list. E. Özan Target Tracing November 5, 0 5 / 9 E. Özan Target Tracing November 5, 0 6 / 9 Trac Coalescence Problem of References R. J. Fitzgerald, Development of Practical PDA Logic for Multitarget Tracing by Microprocessor, American Control Conference, pp , Jun H. A. P. Blom, and E. A. Bloem, Probabilistic data association avoiding trac coalescence, IEEE Transactions on Automatic Control, vol.5, no., pp.7 59, Feb I. J. Cox and S. L. Hingorani, An efficient implementation of Reid s multiple hypothesis tracing algorithm and its evaluation for the purpose of visual tracing IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.8, no., pp.8 50, Feb R. Danchic and G. E. Newnam, Reformulating Reid s MHT method with generalised Murty K-best raned linear assignment algorithm, IEE Proceedings Radar, Sonar and Navigation, vol.5, no., pp., 6 Feb Figures taen from: R.J. Fitzgerald, Development of Practical PDA Logic for Multitarget Tracing by Microprocessor, American Control Conference, pp , Jun E. Özan Target Tracing November 5, 0 7 / 9 E. Özan Target Tracing November 5, 0 8 / 9

8 References S. Blacman and R. Popoli, Design and Analysis of Modern Tracing Systems. Norwood, MA: Artech House, 999. Y. Bar-Shalom and X. R. Li, Multitarget-Multisensor Tracing: Principles, Techniques. Storrs, CT: YBS Publishing, 995. E. Özan Target Tracing November 5, 0 9 / 9