1. Status of the Pattern Recognition with
Julich, September 7th, 2009
Status of the Pattern Recognition with
the STT system alone.
Gianluigi Boca

2. Y Z X  Parametrization of the Helix trajectory  = Kz + 0
x – x0 = Rcos(Kz + 0)
y – y0 = Rsin(Kz + 0) Z X Y
 = Kz + 0
P(x ,y, z)
5 parameters : R
x0  abscissa of center of
cylinder
(x0 ,y0) 
y0  ordinate of center of
cylinder
0  azimuthal angle at z = 0
R  radius of cylinder
K  rate of increase of 

3. ..at last Torino meeting, the good news
Total P (Gev/c)
# generated tracks
Cputime/event (sec)
0.5 1
0.11 2
0.12 3
0.15
1.0
0.10
0.13
0.14
2.0
3.0

4. ..at last Torino meeting, the bad news
Total P
(GeV/c)
Generated
tracks
Found R, D,  (%)
Found K, 0 (%)
Hits belonging to Helix found in parallel straws (%)
Hits belonging to Helix found in skew straws (%)
Spurious tracks found
(%)
0.5 1 89 80
100
Bad 2 44 13 11 3 86 85
112
1.0 92 83 94 93
154
2.0 84 82 12 81
110
3.0 56 49 22 63 54 76 66 69

5. Conclusions at the Torino meeting
Algorithm is very fast now thanks to the use of the Z  space
and semplifying assumptions of the mathematics behind it.
The peak finding in the Hough plots is now the key issue and the
toughest. One one hand that’s not surprise since finding shape
patterns with a computer algorithm is generally very difficult.
In this case the problem is even worse since the algorithm
cannot be to slow.
To deal with the multitrack case I will try to find suitable ‘roads’
such that I will reduce the problem in some sense to the single
track case.

6. Progress
So I started all over again.
I ABANDONED the Hough transform method. Impossible to
find the right peaks when there are many tracks.
I used the more traditional approach of using ‘roads’ of hits
and then doing a (fast) partial fit to get tracklet parameters.
The remaining hits are then found extrapolating the tracklet and
collecting the hits near to it.

7. Progress
Track finding performed in two steps.
First step : find the circular trajectory of the helix projected in the
XY plane. At this stage use only the hits of the straws parallel
to the Z axis.
Second step : find the remaining two parameters of the helix
using the skew straw hits.
WARNING : I am assuming the tracks come from a vertex at 0,0
with a possible spread of the order of 2 mm.
For Vee’s pattern finding this code must be modified.

8. Y Z X  Parametrization of the Helix trajectory  = Kz + 0
x – x0 = Rcos(Kz + 0)
y – y0 = Rsin(Kz + 0) Z X Y
 = Kz + 0
P(x ,y, z)
5 parameters : R
x0  abscissa of center of
cylinder
(x0 ,y0) 
y0  ordinate of center of
cylinder
0  azimuthal angle at z = 0
R  radius of cylinder
K  rate of increase of 

9. Y X An average 10 track event, the most difficult case, P = 0.3 GeV/c
(P┴ < 0.3 GeV/c) in the XY projection Y
green lines = MC
truth, notice that they
don’t go through the
hits exactly X

10. First step : finding circle i XY projection
First I perform a conformal transformation from X and
Y of the hits to U and V :
U  X/ (X2 + Y2)
V  Y/ (X2 + Y2)
This mapping has the virtue of transforming a circle trajectory
passing through the origin into a straight line. This is a very
useful semplification when one tries to find clusters of hits
belonging to the same track.

11. V U The same event viewed in the conformal UV plane
Since trajectory are approximately straight lines, one can find clusters
of hits of the same track in a easier way. V
green lines = MC
truth.
blue lines =
subdivision of
the conformal
space in cells
used to build
the roads. U

12. V U Example First a cluster finding algorithm
selects hits in neighboring cells …

13. V U Example …. then a fast fitter (see later) find straight
line of the tracklet.

14. V U Example The straight line is then extrapolated and more hits
added to the track candidate

15. Fitting the tracklet straight
line in a FAST way

16. The problem in fitting with the usual 2 minimization is that
normally it is quite slow
also in the case of the straws detector there is a left-right ambiguity
finally ‘wrong’ hits in the track candidate tend to spoil the fit
result since there is no way to ‘undeweight’ them
Solution : linearize the problem by minimizing
m, q = fit parameters
iNhits |Vi – di – m Ui – q | / i
Vi , Ui = straw position in conformal space
di = drift radius
i = error on the hit
instead of the usual 2 which is a quadratic function (the absolute
value is also a non linear function but it can be transformed in a
a number of linear functions subject to linear constraints, see below)

17. The minimization of linear equation subject to linear costraints
is a tool used in various fields (economics, traffic flow
regulation, even Sudoku solving…) and on the market there is a lot
of software already.
It is called Linear Programming, an instance of which is the
SIMPLEX algorithm used by Minuit.
Usually this programs are MUCH FASTER that the traditional
2 minimization.
In the case of the straws there is also a right-left ambiguity to
solve.
This can be achieved by using an INTEGER variable in the equations
connected to each straw hit, which can be 0 or 1.
Consequently the minimizer must deal with integer and continuous
variables at the same time.
This is also a situation arising often in other fields, and it is called
Mixed Integer Linear Programming (MILP)

18. Finally the ‘wrong’ or ‘noise’ hits in the track candidate that would
tend to spoil the fit result with the traditional 2 minimization can be
UNDERWEIGHTED with the ‘Big M’ technique.
As a MILP minimizing program I use the GNU C++ library
GLPK which is a stand alone pakage for the time being, and that
can be integrated, I hope, later in the Pandaroot framework.

19. So, bottom line, the linear equations and constraints are :

20. Linear fit parameters (i labels the ith hit in the track candidate):
m1 , m2 , q1, q2 , s+i , s-i : real variables  0
l+ + l- : integer variables equal to 0 or 1 (0 the hit
used, 1 the hit is not used)
Fixed numbers depending on the hit : Ui , Vi , Ri , Di
M  the ‘Big M’ , a fixed large (compared to the straw radius in
the conformal space) number
Function to minimize : iNhits i / i
Linear constraints :
(m1-m2)Ui + q1-q2 - l M - si  Vi  Ri
(-m1+m2)Ui - q1 + q2 - l M - si  -Vi -( Ri)
- l M + si  Di
(l -1)M - si  M - Di
1  l+ + l-  2
si  0

21. Result of the fit for the previous event, in the conformal plane
(green = true tracks, red = found tracks )
failure of finding
a track superimposed
to another

22. …. and in the usual XY plane
(green = true tracks, red = found tracks )

23. V U One more trick ….. In order to the fit to work efficiently all
tracklets are first rotated so that they become
almost parallel to the U axis. The result is rotated
back after the fit. V U

24. Result of the fit for an event with 10 tracks, each 10 GeV/c , in the
conformal plane (green = true tracks, red = found tracks )

25. … and in the XY plane (green = true tracks, red = found tracks )

26. 0  azimuthal angle at z = 0
Second step : find the remaining two parameters of the helix
using the skew straw hits.
I apply similar thechniques for finding the remaining two parameters of
the Helix with the use of the skew straws (0 and K)
5 parameters :
x0  abscissa of center of cylinder
y0  ordinate of center of cylinder
0  azimuthal angle at z = 0
R  radius of cylinder
K  rate of increase of 

27. First I change coordinates from x, y on the surface
of the trajectory cylinder to z,  of the cylindrical coordinates :
it is like ‘cutting’ the trajectory cylinder at  = 0 and spreading it
on a plane Z X Y
cut here
P(x ,y, z) R
(x0 ,y0)  27

28. Finding of the two remaining parameters of the Helix : K and 0
 = Kz + 0  2 z
The Helix trajectory is a sequence of straight lines

29. Finding of the two remaining parameters of the Helix : K and 0
Cylinder on which the
helix lies Y
Helix
equidrift cylinder of
the skewed straw Z
T = tangent to helix
N = perpendicular to equidrift cylinder X

30.  z Finding of the two remaining parameters of the Helix : K and 0
…. and the intersection between the skew straws equidrift cylinder
and the cylinder of the Helix trajectory now contains all the
mathematical ugliness since the dependence of  as a function of
Z is nasty BUT 
Helix trajectory
cylinder 2
Skew straws
intersection
Helix trajectory
Helix trajectory z
The Helix trajectory is a sequence of straight lines

31.  z Assume the intersection of the skew straws with the Helix
cylinder are ellipsis
I notice that due to the geometry and small inclination ( 3°) of the skew straws the tangency of the straight line essentilly occurs at the edges of the ellipses 
Helix trajectory
cylinder
Helix trajectory
skewed straw
intersection z
Assume the intersection such ellipsis with the Helix trajectory
occurs at the edges of the ellipsis

32.  z This is demonstrated further by this picture of all the skew
straws intersections projected onto the  z diagram 
skewed straw
intersections
MC true helix trajectory z

33. So the strategy for the skew straw hits is :
For each helix circle found in the XY plane in the previous step,
figure out which skew straws intersect the helix cylinder; this is
already a powerful selection for skew hits.
Then perform a fit of a straight line in the  Z plane, passing at the
edges of the ellipsis. The fit is performed with the same MILP
technique as before as the right-left ambiguity is solved again with
the aid of integer varibles; the Big M method is used again to allow
for the exclusion of hits too far away from the straight line.

34.  z Some examples of a fit with the skew hits : 10 track event,
0.3 GeV/c momentum each  z

35.  z Some examples of a fit with the skew hits : 1 track event,
5 GeV/c momentum 
green = true track, red = found track z

36.  z Some examples of a fit with the skew hits : 1 track event,
10 GeV momentum 
green = true track, red = found track z

37.  z Typical example of failure of the fit with the skew hits :
6 track event, 10 GeV momentum each 
green = true track, red = found track z

38. Summary table of Track Finder performance in terms of CPU time
CPU used : AMD Athlon 65 bits, 3000+, 2GHz clock
Total P (Gev/c)
# generated tracks
Cputime/track (sec)
0.3 1
0.1 6
0.17 10
0.21
5.0
0.08
0.26
10.0
0.27
0.4

39. Summary table of Track Finder performance in terms of hits and tracks
Total P
GeV/c
Generated
tracks
per event
Total # reaso-nable tracks
gene-rated
% of recon-struct-ed tracks
Ghost tracks found (%)
Total genera-ted hits paral-lel straws
% of found hits paral-lel straws
Wrong paral-lel hits associ-ated (%)
Total genera-ted hits in skew straws
% of found hits in skew straws
Wrong skew
hits associ-ated
(%)
0.3 1 19
100
328
170 6
114
1910 98 8
1198 95
11.7 10
189
3205 96 14
2353
19.8
5.0 18
292
98.6
151 76
2.6
1244
9.8
816
40.6
142
2.8
2299 13
1503 99
37.9
10.0
318
149 3 87
5.7
1433
98.8
774
98.7 17
125
5.6
2031
93.7 12
1273 47

40. Summary table of performance in terms of track parameters
Total P
(GeV/c)
Generated
tracks
per event
(found –true)/true
R (%)
 (%)
K (%)
0 (%)
0.3 1
2.1
1.1
Bad 6
2.4
2.5 10
0.9
5.0 26 9 37 53 76
1.6
10.0
bad
1.3
108 3
119 23

41. Conclusions and next tings to do
Abandoned Hough transform and adopt a more traditional approach
that works well overall.
Usage of MILP in fitting ensures reasonably small CPU time
consumption per track
Usage of MILP in fitting ensures reasonably small CPU time
consumption per track
Efficiency in finding the hits belonging to tracks is satisfactory with
all topologies.
Still some problems in wrong hits association for the skew hits due
to bad fitting ; this needs to be fixed
The values of the helix parameter found are bad for K and 0 and in
some cases also for R and . This may not be so important since
finding the helix parameters is not the job of the pattern recognition
anyway.