TimTrack is an algorithm for tracking charged particles using data from detectors. It estimates the parameters of a "SAETA", which defines a particle track using 6 parameters. TimTrack uses a least squares method to minimize the differences between measured and predicted values from a model. It can handle data from various detector geometries by defining appropriate configuration and data matrices. The method provides a matrix solution that is fast to compute and easy to implement.

1. Proyecto
timtrack
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timtrack timtrack
A Tracking Algorithm for
timtrack TRASGOS
timtrack
timtrack
timtrack
timtrack
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

2. About the TRASGO concept
A TRASGO
(TRAck reconStructinG mOdule)
is a detector able to work stand-alone offering full capabilities of timing and
tracking of charged particles
DAQ Electronics
Network
Power supplies
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

3. About SAETAS
A SAETA (SmAllest sEt of daTA) is the basic unit of information
in the timtrack algorithm and in the TRASGOs concept
A SAETA contains 6 parameters defining a charged particle track
In a cartesian coordinate system:
- X0 and Y0: 2 coordinates at a reference plane
- X’ and Y’ : 2 projected slopes in planes x-z and y-z
- T0 : The time at the reference plane respect a reference time
- V : The velocity
Saeta: s = (X0,X’,Y0,Y’,T0,V)
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

4. About SAETAS
From the mathematical point of view will be better to use:
Saeta: s = (X0,X’,Y0,Y’,T0,1/Vz)
where:
V = Vz · Sqrt(1+X’2+Y’2)
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

5. L
x
z
Saeta
X’ V
X0 T0
Vz
z=0 Y’
Y0 y
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

6. About timtrack
TimTrack is the algorithm developed to estimate SAETAS
1. It is based on a Least Squares Method (LSM)
2. It works directly with the primary data provided by detectors:
- Coordinates:
- Times: it is assumed that:
all times are refered to a common t=0
(all detector are WELL synchronized)
3. It lets free the six elements of a saeta:
(X0, X’, Y0, Y’, T0 and 1/Vz)
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

7. About timtrack
1st. Step
- To define the model, giving the cuantities that are measured
as function of the parameters of the saeta
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

8. x Times
Example Strip-like detector
X-type plane
z
T T’
z=zi
z=0
y
0
0
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

9. x Times
X-type plane
z
T
V T’
X’
X0 T0
z=zi
Y’
z=0
Y0 y
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

10. x Times
X-type plane
T z
T’
X’
V
X0 T0
z=zi
Vz Y’
z=0
Y0 y
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

11. x Times
X-type plane
Ti z
T’i
V
X’
X0 T0
z=zi
Y’
z=0
Y0 y
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

12. x Coordinates
X-type plane
z
Xi V
X’
X0 T0
z=zi
Y’
z=0
Y0 y
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

13. x
Ti
Y-type plane
z
V
X’
X0 T0
z=zi
z=0
Y’ T’i
Y0 Yi y
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

14. About timtrack
1st. Step
- To define the model giving the cuantities to be measured as
function of the parameters of the saeta
Either
or
3 equations (conditions) per plane!
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

15. About timtrack
2nd. Step
- To build the function S to be minimized
x
V
X’ n planes
X0 T0
Y’
Y0 y
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

16. About timtrack
2nd. Step
- S is a sum over n planes:
K = X or Y
K = Y or X
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

17. About timtrack
2nd. Step
- The expansion of the S function is:
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

18. About timtrack
2nd. Step
- That can be written in a more compact way:
where:
Saeta
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

19. About timtrack
K (configuration Matrix): depend on the detector layout
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

20. About timtrack
a (vector of reduced data): depend on the data
(They are just weighted sums and differences of the measurements)
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

21. About timtrack
3rd. Step
- To apply to LSM method.
From:
leads to:
As K is definite positive, K has an inverse and:
This equation provides the saeta directly from the data
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

22. About timtrack
3rd. Step
- Set of solutions (is just the Cramer rule):
where:
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

23. About timtrack
Error analysis
- The error matrix is
- Incertitudes can be easily calculated from the K matrix elements
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

24. About timtrack
Comments
- The method can be easily extended when there are correlations
between some of the measurements (e.G.: time readouts)
- Only two planes of strip-like detectors are enough to provide
unambiguously the 6 parameters of a saeta
- The solution has a matrix form: It’s very easy and fast of
implementing on computers
-There are many detector layouts with a K matrix having the
same structure (see next examples)
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

25. About timtrack
Other strip-like detector layouts
(with the same K-matrix structure)
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

26. About timtrack
Strip-like detectors with any shape:
x x
XBack
(X,Y) vs2
vs1
ymin YBack y
y
XFront
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

27. About timtrack
Strip-like detectors with any shape:
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

28. About timtrack
Strip-like detectors with any shape:
where:
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

29. About timtrack
Pads or pixel detectors :
∆Yi
X
∆Xi
Xi
X0 z
zi
z=0
Y0 Yi Y
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

30. About timtrack
Pads or pixel detectors :
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

31. About timtrack
Pads or pixel detectors :
where:
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

32. About timtrack
Other strip-like detector layouts
(with different K-matrix structure)
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

33. About timtrack
Other strip-like detector layouts
(with different K-matrix structure)
L
x
Ki
z
’ V
z=0
y
New transverse coordinates defined by an angle φ:
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

34. About timtrack
Other strip-like detector layouts (with different K-matrix structure)
K x
XBack x
XB Ti’
K
(Xp,Yp) -vs sinφ vs
Kim
X
vs cosφ
Kip
φ
φ
YBack + YFront YF - YB
y Y y
Ki
XF
XFront Ti
K=0 K=0
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

35. About timtrack
Other strip-like detector layouts
(with different K-matrix structure)
Remember:
ci = cos ϕi
si = sin ϕi
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

36. About timtrack
Other strip-like detector layouts
(with different K-matrix structure)
Again:
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

37. About timtrack
Other strip-like detector layouts
(with different K-matrix structure)
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

38. About timtrack
Other strip-like detector layouts
(with different K-matrix structure)
The solution of is (Cramer rules):
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

39. About timtrack
Comments
- The “problem” of the method is that there is an inversion of a
matrix. Sometimes it may give problems (when the matrix is not well
conditioned) but there are a lot of numerical methods to do it
(And it has to be done only once)
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

40. About timtrack
A typical example
2 parallel scintillators
T’2
T’1
y
➱
vs1
➱
➱
vs2
(Yo,Y’,V,T0)
➱
z1 z2 z
L1 T
T1
L2 τ=
T2 vs
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

41. About timtrack
A typical example: 2 parallel scintillators: different properties
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

42. About timtrack
A typical example: 2 parallel scintillators: identical properties
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

43. The END
Thanks!
Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010