Utilization of a Combinatorial Hough Transform for Tracking in 3 Dimensions with a Drift Chamber Stephen C. Johnson, Federica Ceretto, Axel Drees, Thomas.

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.


Current Progress

• The PHENIX main facility hall, Brookhaven National Laboratory.

– Detector/Collider Commissioning:

• Spring 1999

– First Physics Run• Fall 1999


Unique Tracking Challenge

• Multiplicity:– ~10,000 particles in the

final state

• Track Density:– 200-400 tracks in each arm

(1 for every collaborator)


Magnetic Field• To first order:

– Axial Field


Sample Trajectories

• Primary bend plane: x-y

• Focusing Spectrometer in the y-z plane


The PHENIX Drift Chamber


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


‘Normal’ Hough Transform

Physical Space Feature Space

y

x b

m

Trajectory


‘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


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!


Too many ghosts

• This style of Hough transform creates long tails in our space

• Leads to a large number of ghosts.

• Calculationally intense!


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.


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.


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


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


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


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


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


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]


Efficiencies

• Efficiency is flat as a function of momentum.

• ~92% for p > 200 MeV with the expected detector resolution


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

Utilization of a Combinatorial Hough Transform for Tracking in 3 Dimensions with a Drift Chamber Stephen C. Johnson, Federica Ceretto, Axel Drees, Thomas.

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