Utilization of a Combinatorial Hough Transform for Tracking in 3 Dimensions with a Drift Chamber

1. Utilization of a Combinatorial Hough Transform for Tracking in 3 Dimensions with a Drift Chamber Stephen C. Johnson, Federica Ceretto, Axel Drees, Thomas K. Hemmick, Barbara Jacak, John Noe

2. The PHENIX Detector • Specs: • Multi-subsystem (>10) experiment • Simultaneous measurements of e, m, g, hadrons. • Purpose: • Create nuclear matter at extreme T,r. • QGP, deconfined state • Chirally restored region • Quantify it’s properties. Stephen C. Johnson The University at Stony Brook

3. Current Progress • The PHENIX main facility hall, Brookhaven National Laboratory. • Detector/Collider Commissioning: • Spring 1999 • First Physics Run • Fall 1999 Stephen C. Johnson The University at Stony Brook

4. Unique Tracking Challenge • Multiplicity: • ~10,000 particles in the final state • Track Density: • 200-400 tracks in each arm (1 for every collaborator) Stephen C. Johnson The University at Stony Brook

5. Magnetic Field • To first order: • Axial Field Stephen C. Johnson The University at Stony Brook

6. Sample Trajectories • Primary bend plane: x-y • Focusing Spectrometer in the y-z plane Stephen C. Johnson The University at Stony Brook

7. The PHENIX Drift Chamber Stephen C. Johnson The University at Stony Brook

8. X and UV wire planes • X wires run parallel to the beam axis • Stereo (U,V) wires at ~50 relative to the x-wires uv1/uv2 x1/x2 Stephen C. Johnson The University at Stony Brook

9. ‘Normal’ Hough Transform Physical Space Feature Space y m Trajectory x b Stephen C. Johnson The University at Stony Brook

10. ‘Normal’ in PHENIX Space • The variables a and f are the natural coordinates for the PHENIX detector. • Unlike m and b, they are bounded • => f is point of intersection between track and reference radius. • => a is inclination angle at that point~ 1/p Stephen C. Johnson The University at Stony Brook

11. A first Hough Transform for PHENIX • Points in this space create a curved line. • When these lines overlap in space they create a peak corresponding to the a and f of our track. • Note long tail! f a Stephen C. Johnson The University at Stony Brook

12. Too many ghosts • This style of Hough transform creates long tails in our space • Leads to a large number of ghosts. • Calculationally intense! a f Stephen C. Johnson The University at Stony Brook

13. The Combinatorial Hough Transform Ben-Tzvi and Sandler, “A Combinatorial Hough Transform”,Pattern Recognition Lett, 11 (`90), 167-174. Physical Space Feature Space y m Trajectory x b Stephen C. Johnson The University at Stony Brook

14. Combinatorial Hough Transform in PHENIX • The smaller lever arm for combinations between x1 and x2 points coupled with a residual magnetic field bend in the drift chamber, couple to smear the resolution. • Therefore, only take combinations between x1 and x2 points. Stephen C. Johnson The University at Stony Brook

15. Sample space of Hough Transform Peaks are clearly distinguishable from the background in feature space Track finding algorithm e~97-99% Two track resolution given by bin size: df = 1 mrad da = 20 mrad Stephen C. Johnson The University at Stony Brook

16. Efficiencies with x-wires • As a function of the threshold on the Hough peak, the efficiency rises dramatically • The number of ghost tracks is <4% for all cuts Stephen C. Johnson The University at Stony Brook

17. y x UV wires • X hough transform constrains the reconstructed track to the x-plane. • UV wires intersect this plane to make points -> second Hough Stephen C. Johnson The University at Stony Brook

18. R’ b zed z In the UV plane • Second Hough Transform in this space: • combinations of all uv1/uv2 points • only one solution • Variables of UV Hough transform: • zed -- point where the trajectory intersect the mid point of the drift chamber in z. • b -- the polar angle at that point. uv2 uv1 Stephen C. Johnson The University at Stony Brook

19. Correlations in feature space b • Tracks from the vertex follow a very well defined line in b vs zed. • Note that this implies we can determine vertex from drift chamber. zed Stephen C. Johnson The University at Stony Brook

20. X Wire Algorithm UV Wire Algorithm • Fill X Hough Array • Find Maxima • Create plane associated with x soln • Intersect UV hits (lines) with plane • Fill UV Hough Array • Find Maximum Algorithm Flow Chart OO algorithm (C++) Data [list of lines] Solutions [list of DC lines] List of Candidates Associate Hits with Track Stephen C. Johnson The University at Stony Brook

21. Efficiencies • Efficiency is flat as a function of momentum. • ~92% for p > 200 MeV with the expected detector resolution Stephen C. Johnson The University at Stony Brook

22. Collisions at RHIC (high track density) provide an interesting test-bed for the study of robust tracking algorithms. An OO combinatorial Hough transform has been found to give very good performance for tracking through the PHENIX drift chamber Specs: High efficiency ~92% Low number of ghosts <2% Robust for high multiplicity Promising CPU studies two track resolution: df = 1 mrad da = 20 mrad db = 200 mrad dzed=1cm Postscript Stephen C. Johnson The University at Stony Brook