Particle Filter Track Before Detect Algorithms

Transcription

1 Particle Filter Track Before Detect Algorithms Theory and Applications Y. Boers and J.N. Driessen JRS-PE-FAA THALES NEDERLAND Hengelo The Netherlands

2 Outline Introduction Filtering Detection Examples Overview Conclusions 2

3 Classical vs. TBD Introduction Raw Data Detect Cluster Extract Track Tracks TBD TBD integrates the information over time. Detection is based on power/energy that has been integrated over time (multiple scans). Classical tracking : single scan based detection. * TBD provides higher probability of detection (P d ) at the same level of probability of false alarm (P fa ) * TBD circumvents the data association problem. 3

4 Introduction Twofold problem The TBD problem is twofold:. Filtering 2. Detection 4

5 Filtering The System s k+ = f(t k, s k, d k, w k ), k N Prob{d k+ = j d k = i} = [Π(t k )] ij z k = h(t k, s k, d k, v k ), k N Filtering Problem: Determine p(s k, d k Z k ) 5

6 Filtering Basic idea of the particle filter Describe the a posteriori pdf p(s k, d k Z k ) by a cloud of N particles that propagates in time such that the cloud approximately equals an N-sample drawn from p(s k, d k Z k ) NOTE: This is more than just a (point) estimate!!!! 6

7 Filtering Kalman vs. PF representation

8 Filtering Using a (proper) particle filter on the system: The following holds N i= N δ(s si k ) a.s. p(s k Z k ) i.e. almost sure convergence... Popular (point) estimators obtained from particle cloud: ŝ MV k = R n s kp(s k Z k )ds k N i= N si k ŝ MAP k = arg max p(s s k R n k Z k ) s i k where i = arg max i q i k 8

9 Detection Deciding upon presence of target: Hypothesis testing: Given two hypotheses H : no signal present z(j) = v(j), j =,..., k H : signal present z(j) = h(s(j), v(j)), j =,..., k where s(k) evolves according to dynamical system 9

10 Detection Using particle filter output for detection Every optimal detector can be expressed in terms of a Likelihood Ratio Test: THEOREM: L(Z(k, l)) τ L(Z(k, l)) = p(z(k l + ),..., z(k) H ) p(z(k l + ),..., z(k) H ) kj=k l+ ( N i= q i (j)) N l k j=k l+ p v (z(j))

11 Detection Using particle filter output for detection Elements of the Proof: p(z(l),..., z(m) H ) = m j=l p v (z j ) p(z(l),..., z(m) H ) = where p(z(j) Z(j )) = = S S m j=l p(z(j) Z(j )) p(z(j), s Z(j ))ds p(z(j) s, Z(j ))p(s Z(j ))ds = E p(s Z(j )) p(z(j) s, Z(j )) N N i= p(z(j) s i (j)) = N N i= q i (j)

12 Example - Detection Linear Gaussian scalar system: s(k + ) = s(k) + w(k) z(k) = s(k) + v(k) w(k) N(, ), v(k) N(, ) and s() N(, ) Data has been generated according to the above model. Particle filter solution (2 particles) and the exact (Kalman) solution have been calculated. 2

13 22 Example - Detection True states and estimates Ratio exact and p.f. likelihood

14 Example - MTT TBD The Fighter-Missile Example: Multi target track before detect application for small to very small closely spaced targets. Early detection is crucial. Modelling details in: Y. Boers, J.N. Driessen, F. Verschure, W.P.M.H. Heemels and A. Juloski. A Multi Target Track Before Detect Application. Workshop on Multi Object Tracking, Madison, WI, June 23. 4

15 Example - MTT TBD System: s k+ = f(t k, s k, d k ) + g(t k, s k, d k )w k where f(t k, s k, d k ) = T T s k The process noise input model is given by g(t k, s k, d k ) = 2 ( 3 a x,max)t 2 3 a x,max T 2 ( 3 a y,max)t 2 3 a y,maxt 5

16 Example- MTT TBD System: The discrete mode d k represents one of three hypotheses (each have a different measurement equation!!) d k = : There is no target present. d k = : The prime target is present. d k = 2: There are two targets present. Markov process: Π(t k ) =

17 Example - MTT TBD Simulations Initially, there is no target present. The first target (fighter: SNR=3dB) appears after 5 seconds (T = s) and moves at a constant velocity of 2ms towards the sensor. After 2 seconds, a second target (missile: SNR 3dB) spawns from the first and accelerates to a velocity of 3ms in 3 scans. particles have been used in a plain vanilla particle filter implementation 7

18 Example- MTT TBD Simulations Matlab movies 8

19 Example - MTT TBD Estimation of the mode.8 Probability Time.8 True mode no target present target present 2 targets present Probability Time 9

20 Overview Related work (co)authored by presenter: Y. Boers and J.N. Driessen. A Particle Filter Based Detection Scheme. IEEE Signal Processing Letters, October, 23. Y. Boers, J.N. Driessen, F. Verschure, W.P.M.H. Heemels and A. Juloski. A Multi Target Track Before Detect Application. Workshop on Multi Object Tracking, Madison, WI, June 23. Y. Boers and J.N. Driessen. Hybrid State Estimation: A Target Tracking Application. Automatica, vol. 38, no.2, 22. Y. Boers and J.N. Driessen. An Interacting Multiple Model Particle Filter. To appear in IEE Proceedings - Radar, Sonar and Navigation, 24. 2

21 Overview Some Other Related work: D.J. Ballantyne, H. Y. Chan and M.A. Kouritzin. A novel branching particle method for tracking. SPIE Aerosense 2 proceedings, volume 448, pp , Orlando FL, April 2. D.J. Salmond and H. Birch, A Particle Filter for Track- Before-Detect, In Proc. of the American Control Conference, June 25-27, 2, Arlington, VA. C. Kreucher et al., Multi Target Tracking Using A Particle Representation of The Joint Multi Target Density. Submitted to IEEE Transactions AES / In Proceedings of SPIE Small Targets Conference, 23 2

22 Overview General Particle Filter literature: As an excellent general book on Particle Filtering with a lot of theory, applications and references: A. Doucet, N.J. Gordon and N. de Freitas eds. Sequential Monte Carlo Methods in Practice, Springer Verlag, New York, 2. 22

23 Conclusions Specific Conclusions Every optimal detector can be expressed in terms of PF weights...very important result both from a theoretical and practical point of view. A multi target particle filter for closely spaced targets has been presented for a TBD application. The algorithm can be applied in real time. Questions/Remarks/Discussions 23