Fast Tracking of Strip and MAPS Detectors

transcript

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

x1 /P z

de te c to r Hough space

one track one po in t

x1 /P z

de te c to r Hough space

seven h its seven cu rves

x = z20.3 By

=0.3 By z2

=0.3 By (z cos + x sin)2

2 (z sin – x cos)

rotated by (Px/Pz):

Joachim Gläß, Univ. Mannheim, Institute of Computer Engineering

STS Tracking3-D Hough Transform

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)

de te c to r

STS TrackingHardware Implementation

hit coordinatesx, z

LUT shiftregisters

1 bit/row

Systolic processing of space points (1 hit/cycle)

de te c to r

hit coordinatesx, z

LUT shiftregisters

1 bit/row

de te c to r

hit coordinatesx, z

LUT shiftregisters

1 bit/row

de te c to r

hit coordinatesx, z

LUT shiftregisters

1 bit/row

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

de te c to r

STS Tracking of Strip DetectorsHardware Implementation

hit coordinatesx, z

LUT shiftregisters

1 bit/row

Processing of strip detector data

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

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

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

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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