2. Using Kalman Filter to Track Particles Saša Fratina advisor: Samo Korpar 2004-01-13

3. Overview Motivation Basic principles of Kalman Filter  example Application to particle tracking No big deal

4. R. E. Kalman Born 1930 in Hungary Studied at MIT / Columbia Developed filter in 1960/61

5. i i + 1 Illustration example Measuring parameters of a particle track in 2D particle track parameters: y, k measurement: m

6. System: our knowledge of the system Kalman filter – KF System state: unknown system parameters model measurement   noise ?KF When and where?

7. Tracking and navigation – Tracking missiles, aircrafts and spacecrafts – GPS technology – Visual reality Tracking in HEP experiments

8. KF assumptions Linear system – System parameters are linear function of parameters at some previous time – Measurements are linear function of parameters White Gaussian noise – White: uncorrelated in time – Gaussian: noise amplitude  KF is the optimal filter

9. KF description System: our knowledge of the system System state: unknown system parameters model measurement KF parameters v v i = A v i - 1 m i = H v i   noise + q i + r i using vectors and matrices estimation of parameters v ^

10. KF description: example System parameters: v System model: linear motion y = k x v i = A v i - 1 Measurement model: m i = H v i

11. Noise Noise: e Noise covariance matrix System noise: v i = A v i – 1 + q i  Q = E(qq T ) Measurement noise: m i = H v i + r i  R = E(rr T ) Gaussian  E(e 2 ) = σ 2

12. KF algorithm Prediction: v i - = A v i – 1 Correction: v i = v i - + K (m i – H v i - ) v i = A v i – 1 + q m i = H v i + r v ^ v KF Kalman gain matrix minimize the difference v - v ^ ^ ^^^ ^

13. Kalman gain matrix It is easy to show K = V - H T (H V - H T +R) -1, where V i - = AV i-1 A T + Q Minimize the expected error Limits: – system noise << measurement noise  v i = v i – – system noise >> measurement noise  v i = H –1 m i ^ ^ ^

14. Error on parameters Predictor: V i - = AV i-1 A T + Q – Q: system noise Corrector: V i = (I - KH) V i - – error reduced

15. Kalman filter Least squares Example Simulation y = k x Implemented KF – prediction – correction Compare with LS method x y

16. System noise x y Kalman filter Least squares

17. Matrix description of system state, model and measurement Progressive method Proper dealing with noise KF overview prediction correction

18. Application to particle tracking Detector: – Silicon vertex detector – Central drift chamber Description of track: 5 parameters

19. Advantages of using KF in particle tracking Progressive method – No large matrices has to be inverted Proper dealing with system noise Track finding and track fitting Detection of outliers Merging track from different segments

20. Modifications of KF (!) Non - linear system  extended Kalman filter Full precision only after the last step – Prediction – Correction – Smoothing Kalman filter Least squares x y

21. Conclusion We have demonstrated the principles predictor – corrector method combining model and measurement Very useful in tracking For given assumptions, KF is the optimal filter Extensions for non-linear systems Extensive application To sum up…

22. Tracking in BELLE detector Track finding Track fitting Track managing

23. Notation overview v: vector of parameters – v: our estimation – v - : predicted value m: vector of measurements A: matrix describing linear system  v i = A v i – 1 H: matrix describing measurements  m i = H v i V: error (on parameter) covariance matrix Q: system noise covariance matrix R: measurement noise covariance matrix K: Kalman gain matrix ^