1. 4/22/2017
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 University at Stony Brook
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. The University at Stony Brook
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:
tracks in each arm (1 for every collaborator)
Stephen C. Johnson
The University at Stony Brook

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

6. The University at Stony Brook
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. The University at Stony Brook
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. The University at Stony Brook
Too many ghosts a
This style of Hough transform creates long tails in our space
Leads to a large number of ghosts.
Calculationally intense! 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),
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. The University at Stony Brook
UV wires y x
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. The University at Stony Brook
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. b
zed z R’
uv2
uv1
Stephen C. Johnson
The University at Stony Brook

19. Correlations in feature spaceb
zed
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.
Stephen C. Johnson
The University at Stony Brook

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

21. The University at Stony Brook
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. The University at Stony Brook
Postscript
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
Stephen C. Johnson
The University at Stony Brook