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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
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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.
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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
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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)
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Stephen C. JohnsonThe University at Stony Brook
Magnetic Field• To first order:
– Axial Field
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Stephen C. JohnsonThe University at Stony Brook
Sample Trajectories
• Primary bend plane: x-y
• Focusing Spectrometer in the y-z plane
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Stephen C. JohnsonThe University at Stony Brook
The PHENIX Drift Chamber
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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
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Stephen C. JohnsonThe University at Stony Brook
‘Normal’ Hough Transform
Physical Space Feature Space
y
x b
m
Trajectory
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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
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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!
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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!
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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.
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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.
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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
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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
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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
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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
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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
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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]
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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
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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