Fast Tracking of Strip and MAPS Detectors

Fast Tracking of Strip and MAPS Detectors

Joachim GläßComputer Engineering, University of

Mannheim

Target application is trigger 1. do it fast 2. check precision

• Contents– STS Tracking (Strip Detectors)

• Hough Transform

– MAPS Tracking• Kalman Filter

October 7, 2004 CBM Collaboration Meeting

STS TrackingHough Transform of Parabola

z

x1 /P z

de te c to r Hough space

one track one po in t

z

x1 /P z

de te c to r Hough space

seven h its seven cu rves

x = z20.3 By

2 Pz

=0.3 By z2

2 x

Pz

1

=0.3 By (z cos + x sin)2

2 (z sin – x cos)

Pz

1

<=>

rotated by (Px/Pz):

Joachim Gläß, Univ. Mannheim, Institute of Computer Engineering

Z

X

Y

STS Tracking3-D Hough Transform

1/Pz

Px/PzPy/Pz


• 3-D according to the three parameters of a track– bending 1/Pz, angles and (Px/Pz, Py/Pz)

– Py/Pz detector slice corresponds to one 2-D Hough-histogram– 2-D Hough-histograms can be processed independently– Py/Pz planes are overlapping ( due to multiple scattering)

z

x

de te c to r

STS TrackingHardware Implementation

hit coordinatesx, z

LUT shiftregisters

1 bit/row

start

Systolic processing of space points (1 hit/cycle)

D Q

CNT


z

x

de te c to r


hit coordinatesx, z

LUT shiftregisters

1 bit/row

start

Systolic processing of space points (1 hit/cycle)

one hit -> one curve

Cell number of peakdetermines track parameters


STS TrackingSimulation Results

• Efficiency• e: found tracks/all tracks with P > 1GeV/c• g: ghost tracks/processed tracks• i: identified tracks/processed tracks

– 31 x 95 x 383 e: 95 %, g: 25 %, i: 45 %– 63 x 191 x 255 e: 93 %, g: 12 %, i: 65 %


STS TrackingSimulation Results

• Precision of the reconstructed momentum– 63 x 191 x 255



• Processing speed (rough estimations)• Real-time tracking (emphasis is on fast)

– 1 hit/cycle

– e.g. 10 Gb/s link with 64 bit/hit => 150 x 106 hits/s1 hit/cycle => 150 MHz

– 1500 to 10000 hits/event => 10µs to 100µs

– total number of processing unitsca. 200 x 10 Gb/s links needed for STS=> ca. 200 units


z

x

de te c to r

STS Tracking of Strip DetectorsHardware Implementation

hit coordinatesx, z

LUT shiftregisters

1 bit/row

start

Processing of strip detector data

one hit (x strip) -> one plane (horizontal)

stop


z

x

de te c to r

STS Tracking of Strip Detectors Hardware Implementation

hit coordinatesx, z

LUT shiftregisters

1 bit/row

startstop

Processing of strip detector data

one hit (y strip) -> one plane (vertical)

Logical AND gives same Hough Transform thanintersection point of strips(+ all fakes given by strip layout)

to do: angles other than 90°,especially small angles


• MAPS layer 1 and 2(monolithic active pixel sensors)– high resolution < 10 µm– slow readout > 10 µs

pile up of ca. 100 events

• Kalman Filter track following– track hits from L3 – L5 as

seed • later Hough transform

– emphasis is on fast:process 1 track/cycle

10

0 µ

m S

i

10

0 µ

m S

i 10

0 µ

m S

i

MAPS TrackingKalman Filter Track Following


• y-z plane (non-bending) => straight line– y = m * z + c

– start with m0 = y0/z0, c0=0

– predict position in previous layer yk = mk-1 * zk + ck-1

– measure position (distance predicted – real yk)

– update estimate with measurement• yk, mk, ck are simple function of mk-1, ck-1 and yk

• yk < 500 µm => needs few bits to code

• noise and error covariance are chosen to „believe“ the latest measurement

^

^



• x-z plane (magnetic field) => parabola– x = a z2 + b z + c

– start with a0, b0 from hits in layer 3, 4, 5 (or Hough-Transform), c0=0

– predict position in previous layer xk = ak-1 zk2 + bk-1 zk + ck-1

– measure position (distance predicted – real xk)

– update estimate with measurement• xk, ak, bk, ck are simple functions of ak-1, bk-1, ck-1, xk

• xk < 500 µm => needs few bits to code

• noise and error covariance are chosen to „believe“ the latest measurement

^


^



• no binning of data• max distance 0.5 mm

• nearest hit as function of PZ

• tracks with lower momentumare worse

• w/o pileup– 98% of nearest hits

from same track

• with pileup– no missing hits– less hits from same track

(ca. 10 %)

MAPS TrackingSimulation Results

• coefficients and parameters with 10 – 12 bit sufficient– no double precision floating point needed– old values -> LUTs -> adder -> LUT -> new value

• associative hit memory


hits from detector layer

predicted position x, y of nearest hit

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MAPS TrackingHardware Implementation

Summary

• Hough Transform– global algorithm– processing time ~ number of hits– possible implementation using FPGA and LUT– efficiency ca. 95% of tracks found– relatively high ghost rate– able to handle strip detectors

• Kalman Filter– MAPS pile up ca. 100 min. bias events– w/o pile up ca. 98% of nearest hits from same track– with pile up ca. 88% of nearest hits from same track

ca. 12 % of nearest hits from other events– possible implementation using FPGA and LUT

• simple calculation• associative hit memory


Fast Tracking of Strip and MAPS Detectors

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