Kalman Filter Techniques for Tracking

1. Kalman Filter Techniques for Tracking Rainer Mankel (DESY Hamburg) 28-Jul-2004

2. The HERMES Recoil Spectrometer Scenario (hopefully correct...) • /p in range 0.1-2 GeV • Field 1 T, 20% inhomogeneity • # tracks per event? R. Mankel, Kalman Filter Techniques

3. Layout Si beam pipe SciFi • 2 layers of double-sided Si give 4 (1D) measurements • or 2 space points with ghosts • 2 fibre super layers of 4 layers each give 8 (1D) measurements • or 2 space points R. Mankel, Kalman Filter Techniques

4. Challenges: Pattern recognition • Is pattern recognition an issue at all? • track density vs granularity  occupancy • “Tree-search” is fine if it can be applied • Concentrate on track fit R. Mankel, Kalman Filter Techniques

5. Challenges: Multiple Scattering • material budget O(4-5%)? • small in absolute size, but clearly non-negligible in relation to momentum range • some extended tracking volumes (SciFi) • thin scatterer approximation may be insufficient R. Mankel, Kalman Filter Techniques

6. Challenges: Energy Loss • Some thumb-based estimates... • mip ionization O(2 MeV/(g/cm2))? • pions (p=1 GeV) lose <1 % of their energy • pions (p=0.1 GeV) lose ~ 4% • protons (p=0.3 GeV) lose ~ 6% • protons (p=0.1 GeV) are stopped • very important to use appropriate correction • both mean value and variance ! R. Mankel, Kalman Filter Techniques

7. Challenges: Magnetic Field • 20% inhomogeneity cannot be neglected • deviation from axial direction? • 3D fit strongly recommended R. Mankel, Kalman Filter Techniques

8. Challenges: Robustness? • SciFi provide measurements with non-Gaussian error • Molière scattering introduces non-Gaussian angular distribution • Energy loss introduces non-Gaussian distribution • But this may not be a dominant effect R. Mankel, Kalman Filter Techniques

9. Least Squares Fitting Generally accepted solution: Kalman filter at Gaussian leveloptimal correction of multiple scattering (“process noise”) energy loss can be incorporated similarly with “smoother”, full information at every point of trajectory convenient for matching with other components R. Mankel, Kalman Filter Techniques

10. What the Kalman filter is • A progressive way of performing a least-squares fit • Mathematically equivalent to the latter What it is not: • a pattern recognition method (though it can be efficiently used within one) • a “robust” fitting method ( “ ) R. Mankel, Kalman Filter Techniques

11. An Example • The Kalman filter is used within the reconstruction code of the ZEUS MVD R. Mankel, Kalman Filter Techniques

12. Information Flow in the Track Fit Origin • Effects influencing the amount of information contained in the measurements • Information that the fit has to take into account Dilution of information Increase of information R. Mankel, Kalman Filter Techniques

13. How the Kalman Filter Works • Trajectory until point (k-1) point k-1 R. Mankel, Kalman Filter Techniques

14. How the Kalman Filter Works • Trajectory until point (k-1) • Prediction (without process noise) Prediction point k-1 R. Mankel, Kalman Filter Techniques

15. How the Kalman Filter Works Prediction • Trajectory until point (k-1) • Prediction (with process noise = mult. scattering) • Filter point k-1 Filtering of k-th point R. Mankel, Kalman Filter Techniques

16. How the Kalman Filter Works Multiple scattering Prediction • Trajectory until point (k-1) • Prediction (with process noise = mult. scattering) point k-1 R. Mankel, Kalman Filter Techniques

17. How the Kalman Filter Works Multiple scattering Prediction • Trajectory until point (k-1) • Prediction (with process noise = mult. scattering) • Filter point k-1 Filtering of k-th point R. Mankel, Kalman Filter Techniques

18. Some Math: Prediction Parameters & covariance matrix at (k-1) Prediction Prediction equations Process noise Transport matrix • Transports the information up to the (k-1)-th hit to the location of the k-th hit • Process noise takes random perturbations into account (e.g. multiple scattering, radiation) R. Mankel, Kalman Filter Techniques

19. Prediction vs. Measurement Measurement & covariance matrix at (k) Measurement equations Projection matrix Residual • Projection matrixHkconnects parameter vector (e.g. 5D) and the actual measurement (e.g. 1D) R. Mankel, Kalman Filter Techniques

20. In this formulation (“gain matrix formalism”), the matrix that needs to be inverted has only the dimension of the measurement (here: 1) Some Math: Filter “Gain matrix” Filter equations Filtered parameters & covariance matrix at (k) • In this formulation (“gain matrix formalism”), the matrix that needs to be inverted has only the dimension of the measurement (here: 1) R. Mankel, Kalman Filter Techniques

21. Along the Trajectory • Traditionally, the Kalman filter proceeds in the direction opposite to the particle’s flight • parameter estimate near point of origin contains information of all hits & is most precise production vertex direction of flight  production vertex  direction of filter R. Mankel, Kalman Filter Techniques

22. Along the Trajectory (cont’d) • If precise parameters at both ends are needed, two filters in opposite directions can be combined production vertex direction of filter 1  production vertex  direction of filter 2 R. Mankel, Kalman Filter Techniques

23. Along the Trajectory (cont’d) • The orthodox method of propagating the full information to all points of the trajectory is the “Kalman smoother” • Excellent, but computing intensive • one parameter vector size matrix to invert per step production vertex direction of flight  production vertex  direction of filter direction of smoother  R. Mankel, Kalman Filter Techniques

24. Process Noise & How to Calculate It • Important: multiple scattering model • evaluate contribution to covariance matrix • depends on track model (example is for tx = tan x, ty = tan y) • angular elements of Q (t = thickness in terms of radiation length) R. Mankel, Kalman Filter Techniques

25. Extended (“thick”) Scatterers • In this case, also the spatial components of the process noise matrix Q are non-zero (l = thickness in terms of radiation length, D=direction) R. Mankel, Kalman Filter Techniques

26. Nonlinear fit • With non-linear transport or measurement equation, generalizations are necessary • Optimal properties are retained if linear expansion is made in the right places • in general, this requires iteration R. Mankel, Kalman Filter Techniques

27. Nonlinear fit (cont’d) • Knowledge of derivatives important • for helical tracks, calculate analytically • for a parameterized inhomogeneous field, transport & calculation of derivatives are usually done numerically • e.g. embedded Runge-Kutta method (adaptive step size) • see T. Oest, HERA-B notes 97-165 and 98-001, and A. Spiridonov, HERA-B note 98-133 • In case derivatives depend on parameters, iteration may be needed R. Mankel, Kalman Filter Techniques

28. Example for Iteration • After an iteration, the trajectory has changed • transport derivatives have to be recalculated at new positions • #iterations needed depends on degree of non-linearity & targeted precision R. Mankel, Kalman Filter Techniques

29. Outlier Removal • In least squares fitting, outlier hits have bad influence on the parameter estimate • outliers should be removed • The traditional method of removing outliers is based on the 2 contribution of the hit to the fit • in Kalman filter language: smoothed 2s • Problems: • good hits can have a worse 2 than bad hits nearby that are causing the problem • “digital” decisions may result in bad convergence R. Mankel, Kalman Filter Techniques

30. The Deterministic Annealing Filter (DAF) • Invented in 1999 by Strandlie & Frühwirth (Comp.Phys.Comm. 120, 197) • Generalizes the Kalman filter with an additional weight for each measurement • weight depends on “distance” from trajectory • competition between several measurements in a layer possible • During iteration, the weight becomes a “soft” indicator of association to the track R. Mankel, Kalman Filter Techniques

31. DAF and Annealing • Even with “soft” associations, convergence to the global optimum of a track may be difficult • This problem can be solved with “annealing”, i.e. a simulated “temperature” effect that softens the potential, & is lowered stepwise. • The result is equivalent to the “elastic arms” method, but allows to incorporate material effects etc. R. Mankel, Kalman Filter Techniques

32. DAF and Elastic Arms elastic arms with inverse temperature =1/(2Vk) from Comp.Phys.Comm. 120, 197 • Weights of the DAF have an effect similar to the competitive potential of the elastic arms algorithm R. Mankel, Kalman Filter Techniques

33. Robust Estimation • Least squares fitting (& thereby Kalman filtering) reaches its limits when underlying statistics are far from Gaussian • typical example: 2distributions in presence of multiple scattering • This problem is more pressing in electron fitting with plenty of material  radiation • for general treatment, see Stampfer et al, Comp.Phys.Comm. 79, 157 R. Mankel, Kalman Filter Techniques

34. Robust Estimation (cont’d) • Extensions of the Kalman filter for treatment of non-Gaussian variables are possible • “Robust fitting”, Kitagawa Comp.Math.Appl. 18, 503; Frühwirth et al Comp.Phys.Comm.110, 80 • “Gaussian Sum Filter” • Perhaps not an urgent issue for HERMES recoil detector? R. Mankel, Kalman Filter Techniques

35. Kalman Filter & Pattern Recognition • Kalman filter can be used very efficiently at the core of track following methods • “Concurrent track evolution” • “Combinatorial Kalman filter” • within & without magnetic field see for example Nucl. Instr. Meth. A395, 169; Nucl. Instr. Meth. A426, 268 • will not be discussed in detail here R. Mankel, Kalman Filter Techniques

36. Further Reading • Many excellent papers exist, which unfortunately cannot be done justice by listing them all here • A review of tracking methods with many references to the original literature can be found inR. Mankel, Rep. Prog. Phys. 67 (2004) 553—622 (online at http://stacks.iop.org/RoPP/67/553) R. Mankel, Kalman Filter Techniques