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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
Stephen C. JohnsonThe University at Stony Brook
The PHENIX Detector
• Specs:– Multi-subsystem (>10)
experiment
– Simultaneous measurements of e, , , hadrons.
• Purpose:– Create nuclear matter at
extreme T,.• QGP, deconfined state
• Chirally restored region
– Quantify it’s properties.
Stephen C. JohnsonThe 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. JohnsonThe University at Stony Brook
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. JohnsonThe University at Stony Brook
Magnetic Field• To first order:
– Axial Field
Stephen C. JohnsonThe University at Stony Brook
Sample Trajectories
• Primary bend plane: x-y
• Focusing Spectrometer in the y-z plane
Stephen C. JohnsonThe University at Stony Brook
The PHENIX Drift Chamber
Stephen C. JohnsonThe 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
x1/x2uv1/uv2
Stephen C. JohnsonThe University at Stony Brook
‘Normal’ Hough Transform
Physical Space Feature Space
y
x b
m
Trajectory
Stephen C. JohnsonThe University at Stony Brook
‘Normal’ in PHENIX Space
• The variables and are the natural coordinates for the PHENIX detector.– Unlike m and b, they are
bounded
• => is point of intersection between track and reference radius.
• => is inclination angle at that point~ 1/p
Stephen C. JohnsonThe University at Stony Brook
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 and of our track.
• Note long tail!
Stephen C. JohnsonThe University at Stony Brook
Too many ghosts
• This style of Hough transform creates long tails in our space
• Leads to a large number of ghosts.
• Calculationally intense!
Stephen C. JohnsonThe University at Stony Brook
The Combinatorial Hough Transform
Physical Space Feature Space
y
x b
m
Trajectory
Ben-Tzvi and Sandler, “A Combinatorial Hough Transform”,Pattern Recognition Lett, 11 (`90), 167-174.
Stephen C. JohnsonThe University at Stony Brook
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. JohnsonThe University at Stony Brook
Sample space of Hough Transform
Peaks are clearly distinguishable from the background in feature spaceTrack finding algorithm ~97-99%
Two track resolution given by bin size: = 1 mrad = 20 mrad
Stephen C. JohnsonThe University at Stony Brook
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. JohnsonThe University at Stony Brook
UV wires
•X hough transform constrains the reconstructed track to the x-plane.•UV wires intersect this plane to make points -> second Hough
y
x
Stephen C. JohnsonThe 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.
-- the polar angle at that point.
zed
z
R’
uv1
uv2
Stephen C. JohnsonThe University at Stony Brook
Correlations in feature space
• Tracks from the vertex follow a very well defined line in vs zed.
• Note that this implies we can determine vertex from drift chamber.
zed
Stephen C. JohnsonThe University at Stony Brook
Algorithm Flow ChartOO algorithm (C++)
Data[list of lines]
Associate Hits with Track
X Wire Algorithm
•Fill X Hough Array•Find Maxima
List of Candidates
UV Wire Algorithm
•Create plane associated with x soln•Intersect UV hits (lines) with plane
•Fill UV Hough Array•Find Maximum
Solutions[list of DC lines]
Stephen C. JohnsonThe 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. JohnsonThe 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: = 1 mrad = 20 mrad = 200 mrad zed=1cm
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