A Kalman Filter for
HADES
Erik Krebs
17/11/2011
HADES Collaboration Meeting XXIII
1
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.
2
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.
3
Initialization
Initial estimate of state vector x0
and covariance matrix C0.
k = { 1, .., n }
Prediction (time update)
k-1
Transport state vector xk
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.4
Example for Prediction and Filter
Steps
5
HADES Tracking
Now
Intermediate
Planned
Candidate Search
Candidate Search
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
6
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.
7
Momentum Resolution
Protons
8
Segment Fitter and Kalman Filter
Positrons
9
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.
10
Comparison with Current Tracking
Protons
Current tracking.
Annealing filter with wire hits.
11
Comparison with Current Tracking
Protons
Current tracking.
Annealing filter with wire hits.
12
Comparison with Current Tracking
Positrons
Electrons
Current tracking.
Det. Annealing filter.
13
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.
14
Thank You.
15
Fit Quality
Current tracking.
Annealing filter with wire hits.
16
Ionization Loss
Protons
Energy loss in Kalman filter
Energy loss in Geant
E  E final  Estart
17
Radiation and Ionization Loss for
Positrons
Positrons.
Energy loss in Kalman filter
Energy loss in Geant
18
Protons
Runge-Kutta
fit
Seg. fit
and Kalman
filter
Kalman filter
with wire hits
Det.
Annealing
Filter (DAF)
Encountered errors during reconstruction or 2 > 100
19
Protons
RK Fit
KF + wire
hits
Seg. Fit
+ KF
DAF
20
Protons
RK Fit
KF + wire
hits
Seg. Fit
+ KF
DAF
21
Positrons
RK Fit
Seg. Fit
+ KF
DAF
22
Positrons
RK Fit
Seg. Fit
+ KF
DAF
23
Electrons
RK Fit
Seg. Fit
+ KF
DAF
24
Electrons
RK Fit
Seg. Fit
+ KF
DAF
25