Lecture Outline. Target Tracking: Lecture 7 Multiple Sensor Tracking Issues. Multi Sensor Architectures. Multi Sensor Architectures

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1 Lecture Outline Target Tracing: Lecture 7 Multiple Sensor Tracing Issues Umut Orguner umut@metu.edu.tr room: EZ-12 tel: 4425 Department of Electrical & Electronics Engineering Middle East Technical University Anara, Turey Multiple Sensor Tracing Architectures Multiple Sensor Tracing Problems Trac Association Trac Fusion 1 / 37 2 / 37 Multi Sensor Architectures Multi Sensor Architectures 3 / 37 4 / 37

2 Multi Sensor Architectures: Centralized Multi Sensor Architectures: Hierarchical without Memory 5 / 37 6 / 37 Multi Sensor Architectures: Hierarchical with Memory Multi Sensor Architectures: Hierarchical with Feedbac without Memory 7 / 37 8 / 37

3 Multi Sensor Architectures: Hierarchical with Feedbac with Memory Multi Sensor Architectures: Decentralized without Memory 9 / / 37 Multi Sensor Architectures: Decentralized with Memory Multi Sensor Architectures: Pros & Cons The traditional centralized architecture gives optimal performance but Requires high bandwidth communications. Requires powerful processing resources at the fusion center. There is a single point of failure and hence reliability is low. For distributed architectures Communications can be reduced significantly by communicating tracs less often. Computational resources can be distributed to different nodes Higher survivability. It is a necessity for legacy systems e.g. radars sometimes might not supply raw data. 11 / / 37

4 Problems in Multi Sensor TT Correlation Registration: Coordinates (both time and space) of different sensors or fusion agents must be aligned. Bias: Even if the coordinate axes are aligned, due to the transformations, biases can result. These have to be compensated. Correlation: Even if the sensors are independently collecting data, processed information to be fused can be correlated. Rumor propagation: The same information can travel in loops in the fusion networ to produce fae information maing the overall system overconfident. This is actually a special case of correlation. Out of sequence measurements: Due to delayed communications between local agents, sometimes measurements belonging to a target whose more recent measurement has already been processed, might arrive to a fusion center. Centralized Case y 1 Fusion Center y 2 Decentralized Case y 1 y 2 Tracer-1 Tracer-2 ˆx 1 ˆx Fusion Center ˆx 2 ˆx Suppose the fusion center have the prediction ˆx 1 in both cases. Centralized Case [ ] y 1 = y 2 [ C1 C 2 ] x + [ e 1 e 2 where e 1 and e2 are independent. Decentralized Case [ ] ˆx 1 [ I = I ˆx 2 ] x [ x 1 x 2 ] ] 13 / / 37 Correlation Correlation Suppose the target follows the dynamics x = Ax 1 + w and the ith sensor measurement is given as y i = C ix + e i Then with the KF equations ˆx i =Aˆxi Ki (yi C iaˆx i 1 1 ) =Aˆx i Ki C i(x Aˆx i 1 1 ) + Ki ei =Aˆx i Ki C ia(x 1 ˆx i 1 1 ) + Ki C iw + K i ei Define x i x ˆx i, then x i =x Aˆx i 1 1 Ki C ia(x 1 ˆx i 1 1 ) Ki C iw K i ei =Ax 1 + w Aˆx i 1 1 Ki C ia(x 1 ˆx i 1 1 ) K i C iw K i ei =(I K i C i)a x i 1 + (I Ki C i)w K i ei Hence x i =(I Ki C i)a x i 1 + (I Ki C i)w K i ei x j =(I Kj C j)a x j 1 + (I Kj C j)w K j ej We can calculate the correlation matrix Σ ij E( xi xjt ) as Σ ij = (I Ki C i)aσ ij 1 AT (I K j C j) T + (I K i C i)q(i K j C j) T 15 / / 37

5 Correlation Assuming that Σ ij 0 = 0, we can calculate the correlation between the estimation errors of the local tracers recursively as Correlation Illustration Maneuvers mae this problem more dominant and visible. 120 Σ ij = (I Ki C i)aσ ij 1 AT (I K j C j) T + (I K i C i)q(i K j C j) T This necessitates that the fusion center nows the individual Kalman gains K i and Kj of the local tracers which is not very practical. Assuming that the errors are independent is not a good idea either. Neglecting the correlation maes the resulting estimates overconfident i.e., very small covariances meaning that too small gates and smaller Kalman gains. When Q = 0, Σ ij = 0, i.e., no correlation when no process noise. y (m) y (m) σa =0.3m/s x (m) σa =0.3m/s x (m) Position Errors (m) Position Errors (m) time (s) time (s) 17 / / 37 Rumor Propagation Hierarchical Case: Rumor always flows to the fusion center Rumor Propagation Decentralized case: Rumor propagates everywhere. 19 / / 37

6 Trac Association: Testing Test for Trac Association [Bar-Shalom (1995)]: Two estimates ˆx i, ˆxj and the covariances Σi, Σj are given from ith and jth local systems. We calculate the difference vector ij ij ˆxi ˆxj Then we calculate covariance Γ ij Then test statistics D ij E( ij ijt Γ ij = Σi + Σj Σij ΣijT D ij calculated as = ijt (Γ ij ) 1 ij γij can be used for checing trac association. ) as Trac Association What about the cross covariance Σ ij? Simple method is to set it Σ ij = 0. It can be calculated using Kalman gains if they are transmitted to the fusion center. Approximation for cross covariance from [Bar-Shalom (1995)]: The following cross-covariance approximation was proposed: Σ ij ρ (Σ i. Σj ). 1 2 where multiplication and power operations are to be done element-wise. For negative numbers, square root must be taen on the absolute value and sign must be ept. The value of ρ must be adjusted experimentally. ρ = 0.4 was suggested for 2D tracing. 21 / / 37 Trac Association Method proposed by [Blacman (1999)] for two local agents Suppose local agent i and j have N i T and N j T tracs respectively. A trac association hypothesis θ between local agents i and j can be represented as a NT i N j T -size binary matrix Z = [z mn {0, 1}] such that If trac m of local agent i 1, is associated with trac n of local agent j z mn = 0, otherwise Note that the constraints N i T m=1 z mn 1 n and N j T z mn 1 n=1 m must be satisfied for a valid trac association hypothesis. Trac Association Method proposed by [Blacman (1999)] for two local agents Define the quantities β T : target density (number of targets/state-space volume) P i j : probability that local agent i has a trac in the common field of view with local agent j given that there is a target there. βf i T : False trac density of the tracer of local agent i (same unit as β T ). Then the probability of a trac association hypothesis is given by P (θ ) (β i NA) N i NA (β j NA )N j NA {m,n z mn=1} β T P i j P j i N (ˆx m ˆxn ; 0, Γmn ) where βna i = β T P i j (1 P j i ) + βf i T and βj NA = β T P j i (1 P i j ) + β j F T NNA i N T i N i T N j T m=1 n=1 z mn and N j NA N j T N i T N j T m=1 n=1 z mn 23 / / 37

7 Trac Association Trac Association: Assignment Problem Method proposed by [Blacman (1999)] for two local agents P (θ ) (β i NA) N i NA (β j NA )N j NA Divide by the constant (β j NA )N j T P (θ ) (β i NA) N i NA {m,n z mn=1} {m,n z mn=1} β T P i j P j i N (ˆx m ˆxn ; 0, Γmn ) β T P i j P j i N (ˆx m ˆxn ; 0, Γmn ) β j NA Maximizing this probability is equivalent to maximizing log of it. log P (θ ) = N i NA log β i NA + {m,n z mn=1} log β T P i j P j i N (ˆx m ˆxn ; 0, Γmn ) β j NA + C Method proposed by [Blacman (1999)] for two local agents log P (θ ) = N i NA log β i NA + Form the assignment matrix: {m,n z mn=1} log β T P i j P j i N (ˆx m ˆxn ; 0, Γmn ) β j NA A ij T j 1 T j 2 T j 3 T j 4 NA 1 NA 2 NA 3 T i 1 l 11 l 12 l 13 l 14 log β i NA T i 2 l 21 l 22 l 23 l 24 log β i NA T i 3 l 31 l 32 l 33 l 34 log β i NA where l mn log β T P i j P j i N (ˆx m ˆxn ;0,Γmn β j NA ) Then, use auction(a ij ) to get trac association decisions.. + C 25 / / 37 Trac Association Trac Fusion: Independence Assumption Trac association for more than two local agents. One way is to solve multi dimensional assignment problem. The simpler way is to do the so-called sequential pairwise trac association. Suppose we have N L local agents whose tracs need to be fused. Then, we order the local agents according to some criteria e.g. accuracy, priority, etc. Tracs of Local Agent 1 Tracs of Local Agent 2 Trac Association & Trac Fusion Tracs of Local Agent 3 Trac Association & Trac Fusion Tracs of Local Agent NL Trac Association & Trac Fusion Final Fused Tracs Once we associate two tracs, we have to fuse them to obtain a fused trac. This is called as trac fusion. Consider the trac fusion at point A assuming t CR = t CT = t. Independence assumption gives (Σ A t ) 1 =(Σ B t ) 1 + (Σ C t ) 1 (Σ A t ) 1ˆx A t =(Σ B t ) 1ˆx B t + (Σ C t ) 1ˆx C t This is simplistic and expected to give very bad results here. This is also called as naive fusion. 27 / / 37

8 Trac Fusion: Optimal Solution Consider the trac fusion at point A assuming t CR = t CT = t. Optimal solution (Σ A t ) 1 = ( I (Σ C t ) 1 Σ CB ) t 1 (Σ A t ) 1ˆx A t = ( I (Σ C t ) 1 Σ CB t where B ) 1 B ˆxB t + ( I (Σ B t ) 1 Σ BC ) t 1 + ( I (Σ B t ) 1 Σ BC t C ) 1 C ˆxC t Trac Fusion: Channel Filter Consider the trac fusion at point A assuming t CR = t CT = t. Channel filter, equivalent measurements, traclets etc. (Σ A t ) 1 =(Σ B t ) 1 + (Σ C t ) 1 (Σ D t t t d ) 1 (Σ A t ) 1ˆx A t =(Σ B t ) 1ˆx B t + (Σ C t ) 1ˆx C t (Σ D t t t d ) 1ˆx D t t t d One can define ẑt C and Zt C, which are called equivalent measurements or traclets in the literature, as B Σ B t Σ BC t (Σ C t ) 1 Σ CB t C Σ C t Σ CB t (Σ B t ) 1 Σ BC t (Z C t ) 1 (Σ C t ) 1 (Σ D t t t d ) 1 This is very difficult to compute in a scalable way for variable networs (no fixed-structure). (Z C t ) 1 ẑ C t (Σ C t ) 1ˆx C t (Σ D t t t d ) 1ˆx D t t t d Transmitting these quantities instead from a local agent, one can use the independent trac fusion formulas. 29 / / 37 Trac Fusion Illustration of Correlation Independent Schemes. z T (Σ B t ) 1 z = 1 z T (Σ C t ) 1 z = 1 Trac Fusion: LEA Algorithm Largest ellipsoid algorithm or safe fusion: Suppose we have local estimates ˆx B t, Σ B t and ˆx C t, Σ C t. Find SVD of Σ B t = U 1 Λ 1 U1 T Define the transformation T 1 = Λ 1/2 1 U1 T Transform Σ C t with T 1 and define P C = T 1 Σ C t T1 T. Find SVD of P C = U 2 Λ 2 U2 T. Define the transformation T 2 = U T 2 T 1 Transform ˆx B t, Σ B t and ˆx C t, Σ C t with T 2. ẑ B t = T 2ˆx B t and ẑ C t = T 2ˆx C t z T (Σ A t ) 1 z = 1 Largest Ellipsoid Algorithm Covariance Intersection... Z B t = T 2 Σ B t T T 2 = I nx and Z C t = T 2 Σ C t T T 2 = Λ 2 31 / / 37

9 Trac Fusion: LEA Algorithm Trac Fusion: CI Largest ellipsoid algorithm continued:... Define set of indices I = {i 1 i n x, [Λ 2 ] ii < 1} Find vector ẑ A t and covariance Z A t as { [ẑt A [ẑt B ] ] i i i I [Zt B ] ii i = j, i I [ẑt C ] i i I, [ZA t ] ij [Zt C ] ii i = j, i I 0 i j Find fused estimate and covariance ˆx A t =T 1 2 ẑ A t Σ A t =T 1 2 Z A t T T 2. Covariance intersection Define Find using optimization. (Σ A t (w)) 1 = w(σ B t ) 1 + (1 w)(σ C t ) 1 w = arg min w [0,1] ΣA t (w) Then the fused estimate and covariance are given as (Σ A t ) 1 =w (Σ B t ) 1 + (1 w )(Σ C t ) 1 (Σ A t ) 1ˆx A t =w (Σ B t ) 1ˆx B t + (1 w )(Σ C t ) 1ˆx C T 33 / / 37 Trac Fusion References According to the recent wor (recommended) K. C. Chang, Chee-Yee Chong and S. Mori, On scalable distributed sensor fusion, Proceedings of 11th International Conference on Information Fusion, Jul channel filter seems to be the best algorithm for trac fusion in terms of scalability; estimation errors; and memory. M. E. Liggins II, Chee-Yee Chong, I. Kadar, M. G. Alford, V. Vannicola and V. Thomopoulos, Distributed fusion architectures and algorithms for target tracing, Proceedings of the IEEE, vol.85, no.1, pp , Jan M. E. Liggins and Kuo-Chu Chang, Distributed fusion architectures, algorithms, and performance within a networ-centric architecture, Ch.17, Handboo of Multisensor Data Fusion: Theory and Practice, Taylor & Francis, Second Edition, K. C. Chang, Chee-Yee Chong and S. Mori, On scalable distributed sensor fusion, Proceedings of 11th International Conference on Information Fusion, Jul / / 37

10 References S. Blacman and R. Popoli, Design and Analysis of Modern Tracing Systems. Norwood, MA: Artech House, Y. Bar-Shalom and X. R. Li, Multitarget-Multisensor Tracing: Principles, Techniques. Storrs, CT: YBS Publishing, / 37