Pattern recognition

1. Pattern recognition in high energy physics

2. Measurement system

3. Measurement system

4. Measurement system • The forwardspectrometer is located downstream of the interaction region and measures charged particle tracksat small polar angles q below 5 degrees in the vertical and 10 degrees in the horizontal plane.

5. Measurement system • Measurement system is comprised of a total of six stations with four straw tube double layers each, two of which havevertical wires, the other two have wires inclined by 5 degrees. Chambers 1 and 2 will be placed in front,3 and 4 within, 5 and 6 behind the dipole magnet.

6. Measurement system Reconstructing coordinates from stereo views

7. Measurement system

8. Reconstruction of particles • Track finding (pattern recognition) • Global methods • Local methods • Track fitting • Usually least squares

9. Global methods • Template matching • Histogramming • Hough transform • Legendre transform • Neural networks • Elastic arms

10. Template matching/tree search

11. Hough transform

12. Legendre transform Alexopoulos, T., et al., 2008: Implementation of the Legendre Transform for track segment reconstruction in drift tube chambers Nucl. Instrum. Methods Phys.Res. A 592, 456.

13. Legendre transform • For a value p of the slope the Legendre transform of the function f is defined as follows: • The equationof a circle with center (x0, y0) and radius R is given by:

14. Legendre transform • The Legendre transform of thecircle is:

15. Neural networks • The basic idea is to associate each possible connectionbetween two hits with a neuron.Activation of such a neuron meansthat both hits are part of the same track.This desired behaviour can be obtained by an energy function:

16. Neural networks

17. Neural networks • Local minima • To overcome the problem some complicated cost function are necessary

18. Elastic arms • The basic idea can be described as follows: a set ofM deformable templates is created, whichcorrespond to valid parametrizationsof tracks with parameters {t1, ... tM}. • The algorithm should then move anddeform these templates such that they fit the pattern given by the positions ofN detector hits, which are represented by {ξ1 ... ξN}.

19. Elastic arms • We need an activation-likequantity Sia whose value is one if hit i is assigned to track a, and zero otherwise,and a function Mia(ξi, ta) describing a metric between track template and hit,typically the square of the spatial distance. The energy function can then bedefined as:

20. Elastic arms • The main challenge is to find the global minimum of the energy function • Simulated annealing is used • Method is time-consuming and requires very good initialization

21. Local methods • Track following • Kalman filter • (http://www-lc.kek.jp/subg/offl/kaltest/)

22. Track following • Seeds Creating seeds from drift chamber hit triplets. Seeding schemes with nearby layers

23. Track following • Naive track following • Starting from a seed, the trajectory isextrapolated to the detector part where the next hit is expected. • If a suitable hitis found, it is appended to the track candidate. • Where several hits are at disposal,naive track following selects the one closest to the extrapolated trajectory. • Thisprocedure is continued until the end of the tracking area is reached, or no furthersuitable hit can be found.

24. Track following • The main drawbacks of the naive scheme: • Some expected hits may be missing because of limited device efficiency. • Wrong hits may be closer to the presumed trajectory than the proper hitsand be picked up in their stead. • Left-right ambiguities in wire drift chambers double the number of choices.

25. Track following • Combinatorial Track Following • In each track following step,each continuation hit which is possible within a wide tolerance gives rise to anew branch of the procedure. • In general a whole tree of track candidatesemerges. • The final selection of the best candidate must be done in a subsequentstep, which may involve a full track fit on each candidate. • This kind of method ispotentially unbeatable in terms of track efficiency, but in general highly resourceconsuming and therefore only used in special cases with limited combinatorics.

26. Track following • Arbitration • In practical applications of track following, means are required to reduce its dependency on the starting point, and to decrease its vulnerability against stochastic influences.

27. Track following • Arbitration • It is mandatory not to depend on a single option of seeding tracks, which would lead to loss of a track if one of the seeding layer happens to be inefficient. • When an expected hit appears to be missing in a layer during propagation, it maybe advisable not to discard the candidate immediately, but to proceed furtheruntil a fault limit is exceeded. • In a case where more than one hit could present asuitable continuation for a track, one might want not to decide immediately forthe closest hit but create branches.

28. Arbitrated track following • R. Mankel, A Concurrent Track Evolution Algorithm for Pattern Recognitionin the HERA-B Main Tracking System, Nucl. Instr. and Meth. A395(1997) 169-184 • R. Mankel and A. Spiridonov, The Concurrent Track Evolution Algorithm:Extension for Track Finding in The Inhomogeneous Magnetic Field of TheHERA-B Spectrometer, Nucl. Instr. and Meth. A426 (1999) 268-282.

29. Kalman filter • Kalman filtering involves three steps: • Prediction – the estimation of the state vector at a future time • Filtering – the estimation of the present state vector based on past measurements • Smoothing – the estimation of the state vector in past based on all measurements taken to present Dynamic process Measurement

30. Kalman filter • R. Frühwirth:Application of Kalman filtering to track and vertex fitting, Nuclear Instruments and Methods in Physics Research Section A:Volume 262, Issues 2–3, 15 December 1987, Pages 444–450 • Bo Li, Keisuke Fujii,Yuanning Gao , Kalman-filter-based track fitting in non-uniform magnetic field with segment-wise helical track model, arXiv 1305.7300v2 [physics.ins-det] • C++library for Kalman filtering (linear and non-linear propagation): http://www-lc.kek.jp/subg/offl/kaltest

31. Metrics of algorithm quality • Assessment of track finding efficiency requires firstly a definition of a reference setof tracks that an ideally performing algorithm should find. Normally tracks willbe provided by a Monte Carlo simulation, and the selection of reference tracks willdepend on the physics motivation of the experiment.

32. Metrics of algorithm quality • Hit matching • This method analyzes the simulated origin of each hit in the reconstructedtrack using the Monte Carlo truth information. If the qualifiedmajority of hits, for example at least 70% originates from the same trueparticle, the track is said to reconstruct this particle.

33. Metrics of algorithm quality • Parameter matching • The reconstructed parameters of a track are comparedwith those of all true particles. If the parameter sets agree within certainlimits (which should be motivated by the physics goals of the experiment),the corresponding track is said to reconstruct this particle.

34. Metrics of algorithm quality • The reconstruction efficiency is defined as: • One should also control the abundance of non-reference tracks which are reconstructed:

35. Metrics of algorithm quality • Tracks produced by the pattern recognition algorithm that do not reconstruct any true particle within or without the reference set are called ghosts. A ghost rate can be defined as:

36. Metrics of algorithm quality • Redundant reconstructions of a single particle are called clones. For a given particle m with Nrecom tracks reconstructing it, the number of clones is: • Clone rate:

37. Metrics of algorithm quality • The quality of the estimate of a track parameter Xi is reflected in the parameter residual: • The estimate of the parameter covariance matrix can be used to define the normalized parameter residual • Ideally, the pull should follow a Gaussian distribution with a mean value of zero and a standard deviation of one.

38. Solution? • Totally we have 8 vertical layers of straws before the magnet • For 10 degrees for the scattered particles, we have to examine a region of about 1 m width, 3 meters from the target • For typical straw diameter (about 5 mm), about 200 straws will be examined in each of 8 layers

39. Solution? • Template based tree search can be used to find the track candidates (we need at most 8 levels of the template hierarchies) – it can be implemented in hardware.

40. Solution? • For each track matched to a template the Legendre transform can be used to find the single track parameters in xz plane – implemented in hardware (like Hough transform)

41. Solution? • We have 8 stereo layers to reconstruct tracks in yz plane

42. Solution? • „Drift” circles are mapped to elipses on S plane. • Stereo angles are constant • After rescaling vertical coordinate (multiply by sin(5O))we get circles on S plane. • Then we can apply Legendre transfrom to find the single track in S plane.

43. Magnetic field region • Can we assume Bx = Bz = 0, By = const? • If not – general methods for inhomogeneous field, based on measurements of B(x,y,z). • If yes: • Fx = -vz*By • Fy = 0 • Fz = vx*By, Fz << Fx => parabola

44. Magnetic field region • Independently on assumptions about B, track following methods can be used starting from seed tracks found in either the outer on the inner region. • If B vertical and homogeneous, problem can be split into two 2D problems. • If not – full 3D tracking must be implemented.

45. Magnetic field region • We can use track-following methods of: • R. Mankel, A Concurrent Track Evolution Algorithm for Pattern Recognitionin the HERA-B Main Tracking System, Nucl. Instr. and Meth. A395(1997) 169-184 • R. Mankel and A. Spiridonov, The Concurrent Track Evolution Algorithm:Extension for Track Finding in The Inhomogeneous Magnetic Field of TheHERA-B Spectrometer, Nucl. Instr. and Meth. A426 (1999) 268-282.