1. Progress on the online tracking algorithm
Yutie Liang, Martin Galuska, Jifeng Hu, Wolfgang Kühn,
Jens Sören Lange, David Münchow, Björn Spruck, Hao Xu
II. Physikalisches Institut, JUSTUS-LIEBIG-UNIVERSITÄT GIESSEN
Dec

2. Drift circle with Legendre transformation
Outline
1. Introduction
Straw Tube, Conformal transformation, Hough transformation
Drift circle with Legendre transformation
Adaptive Hough transformation
Inclusion of MVD
5. Summary and outlook

3. Straw Tube Tracker(STT)
4636 Straw tubes
23-27 planar layers
15-19 axial layers(green) in beam direction
4 stereo double-layers for 3D reconstruction,
with ±2.89 skew angle(blue/red)
From STT : Wire position + drift time

4. Conformal transformation and Hough transformation
Real space
CF space
Conformal transformation:
Transform circles to straight lines
y(cm)
y’(cm-1)
x(cm)
x’(cm-1)
Hough transformation: Describing points in real space by parameters
For lines: y = mx + b (x1, y1), (x2, y2),… (m,b) or (r, θ)
R= x cos(θ) + y sin(θ)
Hough space
Use all possible angles
Save data in histogram
Peaks in histogram represent possible lines in point set

5. Method of Pt reconstruction with STT
Real space
CF space x y X’ Y’
Circle.
Approximation
Transform the drift circle to CF space.
Draw lines around the “circle” in CF space.
Fill the line parameter into the Hough space.
Legendre transformation

6. A look at one example event
Real Space
Conformal Space
y(cm)
y’(cm-1)
x(cm)
x’(cm-1) r
Legendre Space
Counts r θ θ

7. Momentum resolution σ: 0.2 GeV/c Truth: 3.45% Circle: 3.47%
Pt(GeV/c)
Pt(GeV/c)
MC truth position as input
Drift circles as input σ:
0.2 GeV/c
Truth: %
Circle: %
Hough space:
800 X 800
1 GeV/c
3.685%
3.694%

8. Adaptive Hough transformation
Initially, 16 X 16 Hough space, keep the best 16 bins for the next step.
For a peak(r, θ), locate the area of r-2 ~ r+1, θ-2 ~ θ+1
Divide the area into 16 X 16 Hough space. (equivalent to 64 X 64 )
Iteration
Hough space 16X16 64X X X X X16384

9. Adaptive Hough transformation
Iteration 4
Iteration 6
3.3%
2.6%
Pt(GeV/c)
Pt(GeV/c)
4096X4096
65536X65536
Iteration:
Truth position as input : % % % %
Drift circles as input: % % % %

10. Approximation as circle is not good in fine Hough space
Real Space
Conformal Space
y(cm) r
y’(cm-1)
Circle in real space,
center: (x0, y0)
radius: r
In CF space
When R >> r, still a circle
center: (x0’, y0’)
radius: r’ = r/(R2)
How good is the approximation?
|r1-r’|/r’ <= 3.2 % R
x(cm)
x’(cm-1)
y(cm)
y’(cm-1) r1
x(cm)
x’(cm-1)

11. Micro Vertex Detector(MVD)
4 barrel layers
6 forward disks
MC particle transport – Digitization – Noise emulation – Local reconstruction
– cluster finding
– hit reconstruction

12. Inclusion of MVD Y(cm) MVD Points Belong to MVD points
Pt 1GeV/c
Iteration:
STT: % % %
STT+MVD : 3.3% % %
No improvement on the pt resolution!
X(cm)

13. Inclusion of MVD Number of count under the expected peak
Red: STT alone
Blue: STT+MVD
Counts
Differences after the inclusion of MVD:
1: more counts under the expected peak 2: the Hough peak is narrow.
3: reconstruction efficiency improves (~90%  ~95%)
Next to study: MVD points used as reference for secondary particles.

14. T0 extraction Backup slides for more details
T0 scan + 220ns window size
If large burst:
ToT
Time based simulation of one burst with 50 events
Time (ns)

15. Summary
Use Legendre transformation (approximation) to handle the drift circle. A strict transformation is required for fine Hough space.
Adaptive Hough transformation makes the application of large Hough space possible, also saves large amount of computing time.
The inclusion of MVD points does not help with the pt resolution. But the Hough peak becomes narrow, and the reconstruction efficiency improves.
Next to do:
Need to use strict Legendre transformation.
Use MVD points as reference points.

16. Thank you

17. Remove MVD, Pipe. The pt resolution can reach 1.8%
No material at all!
0.8%

18. FairDetector *Mvd = new PndMvdDetector("MVD", kTRUE);
Mvd->SetGeometryFileName("Mvd-2.1_FullVersion.root"); // only sensors, update follows
Mvd->SetVerboseLevel(verboseLevel);
fRun->AddModule(Mvd);
PndMvdDigiTask* mvddigi = new PndMvdDigiTask();
mvddigi->SetVerbose(iVerbose);
fRun->AddTask(mvddigi);
PndMvdClusterTask* mvdmccls = new PndMvdClusterTask();
mvdmccls->SetVerbose(iVerbose);
fRun->AddTask(mvdmccls);

19. Inclusion of MVD --resolution
Stt +mvd

20. Conformal transformation
Status of the tracking algorithm in VHDL
Tracking Algorithm
Conformal transformation
Legendre
transformation
Fill
Legendre space
Find peak
x,y,z,r
Wire position
+drift distance
Done Done Done Done
For more realistic input, like id of tube or amplitude, lookup tables need to be added here.
128X128 Legendre space (r, θ) used.
Simulation with ISim
Pt(GeV/c)

21. Event Structure Marius talk 2 · 107 events/s 2000 ns revolution time
40 events per HESR revolution
200 ns gap
Time based simulation by T. Stockmanns
Darmstadt,
Marius C. Mertens

22. Can we extract the event start time?
Need fast detector to provide T0.
With T0, are the overlapped hits problem?
Y(cm)
T0=50
T0=80
T ns window size
T0=120
X(cm)
1: If T0 is known, the contamination hits from other events have wrong drift circle.
Wrong drift circle  bad peak in legendre space
2: Wrong T0  Wrong drift circle  bad peak
Can we use this to extract T0?

23. T0 extraction Time based simulation:
1) 4 events (single track) in one burst.
2) T0 (0, 90,160, 220) ns
T0 scan with tracking algorithm:
Step size: 2ns
C++
Hough space: 800 X 800
VHDL with ISim
Hough space: 128 X 128
Time (ns)
Time (ns)

24. T0 extraction — for large burst
T0 scan + 220ns window size
If large burst:
ToT
Time based simulation of one burst with 50 events
Time (ns)

25. T0 extraction — another idea
Tracking finding using wire position
y(cm)
Counts
smallest drift time of hits in one track
Drift time(ns)
Counts
x(cm)
Drift time(ns)
Not possible for too many hits in one burst!

26. Setup and test PC as data source and receiver. Ethernet.
Optical link (UDP by Grzegorz Korcyl )
(not integrated yet)
Ethernet via
Optical Link
FPGA
FIFO
Tracking algorithm
FIFO

27. Multiple tracks in one event
Counts θ θ r
Momentum resolution.
Reconstruction efficiency.
Fake track.
Pt(GeV/c)

28. If T0 is known:
For each T0, all hits within the 220ns window size need to be considered.
Assume, we have 100 hits in this window, 2 clock-cycle per hit.
100 hits * 2 = 200 clock-cycle per event
If T0 is unknown:
We need a T0 scan.
Assume a step size of 2ns, average time between two events of 50ns.
(200 clock-cycle per event) * 50ns/2ns = 5000 clock-cycle per event

29. Conformal transformation and Hough transformation
Transform circles to straight lines
Describing lines by parameters
y = mx + b  (m,b) or (r, θ)
Use all possible angles
Save data in histogram
Peaks in histogram represent possible lines in point set x y X’ Y’

30. Momentum resolution at 1GeVσ:
Truth: 3.685%
Old: 3.732%
New: 3.694%

31. Precision check One input hit. Generate two point candidates:……
(1.6619, , 2, ) Unit (0.1m)
Generate two point candidates:
…………………………………………………………….
x1, y1 ( , …) C++ calc: ( , )
…………………………………………………………….
x2, y2 ( , …) C++ calc: ( , )
Conformal transformation
…………………………………………………………….
x1’, y1’ ( , ) C++ calc: ( , )
…………………………………………………………….
x2’, y2’ ( , ) C++ calc: ( , )
Hough space, not listed here.

32. Summary table of the resource utilization
Hough Space
128*128 bins
Parallelized in θ
Serial in r
2 clock cycles per bin
252 clockcycles
2520 ns at 100 MHz
Estimation from
David’s previous design
pipeline => latency.

33. Behavior of transformations
P = 1GeV phi = 100 ~

34. Behavior of transformations
P = 1GeV phi = 190 ~

35. Behavior of transformations
P = 2GeV phi = 100 ~
Pt ~ 0.003/r

36. Real Space
Conformal Space
y(cm) r
y’(cm-1)
Circle in real space,
center: (x0, y0)
radius: r
In CF space
When R >> r, still a circle
center: (x0’, y0’)
radius: r’ = r/(R2)
How good is the approximation?
|r1-r’|/r’ <= 3.2 % R
x(cm)
x’(cm-1)
y(cm)
y’(cm-1) r1
x(cm)
x’(cm-1)
fb07:~/stt_online_tracking/points/calc_circle_cf.cpp

37. Can we extract the event start time?
T0(ns)
Need fast detector to provide T0.
With T0, are the overlapped hits problem?
220
160 90
T ns window size
ToT
ToT
1: If T0 is known, the contamination hits from other events have wrong drift circle.
Wrong drift circle  bad peak in legendre space
2: Wrong T0  Wrong drift circle  bad peak
Can we use this to extract T0?

38. Momentum resolution can reach this good, if no energy lost in the detector.