1. A Kalman Filter for HADES
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A Kalman Filter for HADES
Erik Krebs
17/11/2011
HADES Collaboration Meeting XXIII

2. Overview Theory of the Kalman filter.
Using the Kalman filter with the segment fitter.
Using the MDC wire information with the Deterministic Annealing Filter.
Comparison with Current Tracking.
Summary.

3. The Kalman Filter
Set of mathematical equations to estimate the state of system perturbed by process noise.
Least-squares estimator.
Optimal estimator for linear systems.
Works recursively on inaccurate measurements.
Advantages of the Kalman Filter
Recursive Approach: Measurements are processed as they arrive. Useful for real-time applications.
Takes multiple scattering and energy loss (ionization and radiation) into account.
Only small matrices need to be inverted.

4. Prediction (time update) Filter (measurement update)
Initialization
Initial estimate of state vector x0
and covariance matrix C0.
k = { 1, .., n }
Prediction (time update)
Transport state vector xkk-1
Propagate covariance Ckk-1.
Filter (measurement update)
Take measurement mk into account.
Compute state xk and covariance Ck.
Smooth Backwards
Update filtered state vectors xk and covariances Ck
using all available measurements.
The state vector x fully describes the internal state of the system.
Xkk-1 : estimate of state vector before processing the measurement at time k.
Xk: estimate of state vector after processing the measurement at time k.
Xkn: smoothed estimate of state at time k using all available measurements.

5. Example for Prediction and Filter Steps

6. HADES Tracking Now Intermediate Planned Candidate Search
Segment fitter
Segment fitter
MetaMatch
MetaMatch
MetaMatch
Spline task
Spline task
Spline task
Runge-Kutta task
Kalman filter
Kalman filter /
Det. Annealing Filter

7. Segment Fitting and Kalman Filter
Simulated Au+Au events at 1.25 AGeV with Geant.
Four segment hits as measurements.
Position resolution of segment hits Δx = 200 μm and Δy = 100 μm.
Smeared Geant momentum by 10%.
Cut tracks with 2 > 100.

8. Momentum Resolution
Protons

9. Segment Fitter and Kalman Filter
Positrons

10. Use Wire Information Challenge:
Two measurements from neighbouring cells for the same hit.
Fake hits.
Kalman filter can’t handle competing hits.
One Solution: Deterministic Annealing Filter.
Extension of the Kalman filter.
Measurements are assigned weights (probabilities).
Iterate these steps:
Run Kalman filter and smooth back using current weights.
Recalculate weights according to distance of the measurements to the Kalman filter estimates.
Lower annealing factor.
Annealing avoids local minima.

11. Comparison with Current Tracking
Protons
Current tracking.
Annealing filter with wire hits.

12. Comparison with Current Tracking
Protons
Current tracking.
Annealing filter with wire hits.

13. Comparison with Current Tracking
Positrons
Electrons
Current tracking.
Det. Annealing filter.

14. Summary Kalman filter for segment and wire hits has been implemented.
Includes multiple scattering and energy loss.
Kalman filter needs good initail values.
Det. Annealing Filter better than Runge-Kutta fit for high momenta and high theta tracks.
Open Issues:
Trace to META and vertex.
Particle hypothesis.
Initial momentum estimate (→ Spline?).
Test with real data.

15. Thank You.

16. Fit Quality
Current tracking.
Annealing filter with wire hits.

17. Ionization Loss Protons Energy loss in Kalman filter
Energy loss in Geant

18. Radiation and Ionization Loss for Positrons
Energy loss in Kalman filter
Energy loss in Geant

19. Protons Kalman filter Runge-Kutta with wire hits fit Det. Seg. fit
Annealing
Filter (DAF)
Seg. fit
and Kalman
filter
Encountered errors during reconstruction or 2 > 100

20. Protons
KF + wire
hits
RK Fit
Seg. Fit
+ KF
DAF

21. Protons
KF + wire
hits
RK Fit
Seg. Fit
+ KF
DAF

22. Positrons
RK Fit
Seg. Fit
+ KF
DAF

23. Positrons
RK Fit
Seg. Fit
+ KF
DAF

24. Electrons
RK Fit
Seg. Fit
+ KF
DAF

25. Electrons
RK Fit
Seg. Fit
+ KF
DAF