1. Fast Tracking of Strip and MAPS Detectors Joachim Gläß Computer 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

2. STS Tracking Hough Transform of Parabola x = z 2 0.3 B y 2 P z = 0.3 B y z 2 2 x PzPz 1 = 0.3 B y (z cos  + x sin  ) 2 2 (z sin  – x cos  ) PzPz 1 rotated by  (P x /P z ): Joachim Gläß, Univ. Mannheim, Institute of Computer Engineering

3. STS Tracking 3-D Hough Transform 1/Pz Px/Pz Py/Pz Joachim Gläß, Univ. Mannheim, Institute of Computer Engineering 3-D according to the three parameters of a track –bending 1/P z, angles  and  (P x /P z, P y /P z ) –P y /P z detector slice corresponds to one 2-D Hough-histogram –2-D Hough-histograms can be processed independently –P y /P z planes are overlapping ( due to multiple scattering)

4. STS Tracking Hardware Implementation hit coordinates x, z LUT shift registers 1 bit/row start Systolic processing of space points (1 hit/cycle) DQ CNT Joachim Gläß, Univ. Mannheim, Institute of Computer Engineering

5. STS Tracking Hardware Implementation hit coordinates x, z LUT shift registers 1 bit/row start Systolic processing of space points (1 hit/cycle) DQ CNT Joachim Gläß, Univ. Mannheim, Institute of Computer Engineering

6. STS Tracking Hardware Implementation hit coordinates x, z LUT shift registers 1 bit/row start Systolic processing of space points (1 hit/cycle) DQ CNT Joachim Gläß, Univ. Mannheim, Institute of Computer Engineering

7. STS Tracking Hardware Implementation hit coordinates x, z LUT shift registers 1 bit/row start Systolic processing of space points (1 hit/cycle) one hit -> one curve Cell number of peak determines track parameters Joachim Gläß, Univ. Mannheim, Institute of Computer Engineering

8. STS Tracking Simulation 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 383e: 95 %,g: 25 %,i: 45 % –63 x 191 x 255e: 93 %,g: 12 %,i: 65 % Joachim Gläß, Univ. Mannheim, Institute of Computer Engineering

9. STS Tracking Simulation Results Precision of the reconstructed momentum –63 x 191 x 255 Joachim Gläß, Univ. Mannheim, Institute of Computer Engineering

10. STS Tracking Hardware Implementation 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 10 6 hits/s 1 hit/cycle => 150 MHz –1500 to 10000 hits/event => 10µs to 100µs –total number of processing units ca. 200 x 10 Gb/s links needed for STS => ca. 200 units Joachim Gläß, Univ. Mannheim, Institute of Computer Engineering

11. STS Tracking of Strip Detectors Hardware Implementation hit coordinates x, z LUT shift registers 1 bit/row start Processing of strip detector data one hit (x strip) -> one plane (horizontal) stop Joachim Gläß, Univ. Mannheim, Institute of Computer Engineering

12. STS Tracking of Strip Detectors Hardware Implementation hit coordinates x, z LUT shift registers 1 bit/row start stop Processing of strip detector data one hit (y strip) -> one plane (vertical) Logical AND gives same Hough Transform than intersection point of strips (+ all fakes given by strip layout) to do: angles other than 90°, especially small angles Joachim Gläß, Univ. Mannheim, Institute of Computer Engineering

13. 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 100 µm Si MAPS Tracking Kalman Filter Track Following Joachim Gläß, Univ. Mannheim, Institute of Computer Engineering

14. y-z plane (non-bending) => straight line –y = m * z + c –start with m 0 = y 0 /z 0, c 0 =0 –predict position in previous layer y k = m k-1 * z k + c k-1 –measure position (distance predicted – real  y k ) –update estimate with measurement y k, m k, c k are simple function of m k-1, c k-1 and  y k  y k needs few bits to code noise and error covariance are chosen to „believe“ the latest measurement ^ ^ Joachim Gläß, Univ. Mannheim, Institute of Computer Engineering MAPS Tracking Kalman Filter Track Following

15. x-z plane (magnetic field) => parabola –x = a z 2 + b z + c –start with a 0, b 0 from hits in layer 3, 4, 5 (or Hough-Transform), c 0 =0 –predict position in previous layer x k = a k-1 z k 2 + b k-1 z k + c k-1 –measure position (distance predicted – real  x k ) –update estimate with measurement x k, a k, b k, c k are simple functions of a k-1, b k-1, c k-1,  x k  x k needs few bits to code noise and error covariance are chosen to „believe“ the latest measurement ^ Joachim Gläß, Univ. Mannheim, Institute of Computer Engineering ^ MAPS Tracking Kalman Filter Track Following

16. Joachim Gläß, Univ. Mannheim, Institute of Computer Engineering no binning of data max distance 0.5 mm nearest hit as function of P Z tracks with lower momentum are worse w/o pileup –98% of nearest hits from same track with pileup –no missing hits –less hits from same track (ca. 10 %) MAPS Tracking Simulation Results

17. coefficients and parameters with 10 – 12 bit sufficient –no double precision floating point needed –old values -> LUTs -> adder -> LUT -> new value associative hit memory Joachim Gläß, Univ. Mannheim, Institute of Computer Engineering hits from detector layer predicted position  x,  y of nearest hit................................. MAPS Tracking Hardware Implementation

18. 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 Joachim Gläß, Univ. Mannheim, Institute of Computer Engineering