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