4.2 Track finding
4.2.1 An overview of the track reconstruction methods
Conformal mapping
One of the standard global track reconstruction methods is Conformal map-ping [85], which is used in the presence of a uniform magnetic field. The main idea of the method is to simplify the event pattern by applying a map, which transforms the circular tracks of charged primary particles in a uniform magnetic
4.2 Track finding 75
Track reconstruction 7
Track finding: Conformal mapping
) y y ( y
y x ) x
u(
) y y ( ) x x ( r
r y y r
x u x
c c
c
0 0
0
2 0 2
0 2
2 0 2
0
2 1
;
− −
−
= −
− +
−
=
= −
= −
ϑ ϑ
ϑ
25 GeV Au+Au 7(>=3) station:
efficiency ∼95 % ghosts < 7 % clones < 10 %
Figure 4.5: The conformal mapping method for the track reconstruction task in CBM: orig-inal tracks in real space (top) and straight tracks after conformal transformation (bottom) [86].
field into straight lines in a conformal space (Fig. 4.5). The conformal transfor-mation used for this task transforms coordinates from measurement space (x, y) into conformal space (x0, y0):
x0 = x−x0
(x−x0)2+ (y−y0)2; (4.22) y0 = y−y0
(x−x0)2+ (y−y0)2. (4.23) Since reconstructing a straight line is much easier than searching for a circle, in the conformal space these lines can be easily found with histograming. How-ever, the form of the transformation immediately reveals the first obstacle of the method: the transformation requires knowing in advance the position of the pri-mary vertex (xpv, ypv). This problem is usually avoided by assuming a common vertex with coordinates (0;0), which will turn the transformation equations into:
x0 = x
x2+y2; (4.24)
y0 = y
x2+y2. (4.25)
However, this simplification does not help to generalize the method for the search of secondary tracks. Further simplification is done by assuming a uniform mag-netic field, which is not true for a general case.
Despite the disadvantages, the conformal mapping method can be successfully applied for the search of primary tracks in a simple event topologies, where it can have a simple and fast implementation in hardware.
Hough transform
One more example of a global track finder algorithm is the Hough Transform [87].
This method converts the measurements from a real space (x, y) into the param-eter space (a, b). Fig. 4.6 illustrates the example when the measurements of the curved track, described by two parameters — track curvature and emission angle, are plotted in the parameter space, providing as a result the intersection point of the initial track parameters. Let us consider the simplest case of a straight track:
y= a·x+b. (4.26)
In this case the transformation is:
b = −a·x+y. (4.27)
Thus, a certain position measurement in the detector plane (xi, yi) represents a straight line in the parameter space (a, b). Together several such measurements of one initial track are transformed into a set of lines in the parameter space intersecting in the cluster region, which is usually localized in the algorithm with a simple histograming.
Parameter space
Track curvature
Emission angle Real space
x y
x,y - u,v
Figure 4.6: The Hough transform method for the track reconstruction task: original track in the real space (left side) and straight lines in the parameter space, corresponding to certain data points on the initial trajectory (right side).
The obvious limitation of the method arises from the need of a global track model for the stage of transformation to the parameter space. However, even in the case where an analytic global track model exists, usually it is a simplifica-tion, since it cannot include the effects of multiple scattering or inhomogeneities of a magnetic field. Also it is hard to obtain track errors in the case of the transformation. When it comes to hardware implementation, the memory band-width usually limits the implementation due to the need for multidimentional histograming. In case of the CBM experiment the biggest obstacle with the Hough-transform-based track finder (Fig. 4.7) was the memory issue, since the algorithm required about 1.2 GB of RAM for the Hough-Space, otherwise it took 15 minutes due to memory swapping [88].
Track following
An example of a very intuitive local method of track reconstruction (Fig. 4.8) is track following [89]. As the name suggests, the main idea of the method is the prediction of the position of the initial track in the next consecutive detector plane, assuming a certain local model of track propagation in the detector volume.
Figure 13.3: One 2-dimensional Hough plane filled with transformed hits. A central plane processing the hits near the beam pipe is shown here. Planes more apart from the beam pipe contain less transformed hits. There are seven peaks in the histogram (black points) corresponding to seven found particle tracks. A peak is defined by more than three hits in consecutive detector layers. Six peaks can be assigned to certain MC tracks. The lower most peak corresponds to no real track, but accumulates a peak from five hits of different tracks.
Figure 13.4:Track finding efficiency for tracks with hits in at least three tracking stations as function of momen-tum using the Hough transform method for central Au+Au events at different beam momenta
rate is highest for low and high momenta. At low momenta, the track multiplicity is highest and thus combinatorial coincidences are more likely. Tracks at high momenta are less numerous, but correspond to almost straight lines near the beam pipe with high track densities. The mean numbers of efficiency and ghost rate for the three beam momenta are given in table 13.1. While the efficiency is almost the same, the ghost rate increases quickly with beam momentum.
Both the efficiency and the ghost rate depend on the multiplicity of the event, the number of detector
Figure 4.7: One 2-dimensional Hough plane filled with transformed hits [88]. A central plane processing the hits near the beam pipe is shown here. There are seven peaks in the histogram (black points), corresponding to the seven particle tracks found. A peak is defined by more than three hits in consecutive detector layers. Six peaks can be assigned to certain MC tracks.
The lower most peak does not correspond to any real track.
The prediction is checked by searching for registered hits in a certain corridor region around the predicted position. The width of the corridor is chosen with respect to the detector measurement precision and a possible multiple scattering effect due to detector material budget.
x
z z
y
1 2
3 4 5
6 1
2
3 4 5
Figure 4.8: The 3D track following method for CBM. Prediction and search in XoZ and Y oZ projection [88].
Most HEP experiments in the beginning needed an ideal Monte Carlo track finder, which could collect reconstructed hits into tracks using Monte Carlo in-formation. Based on such an ideal track finder it is very easy to implement a realistic track following.
However, there are certain limitations, which made the method not applicable
in certain cases. First of all, the algorithm is featured with an exponential growth of combinatorial combinations to be considered with increasing track multiplicity.
Thus, the algorithm can work efficiently within a reasonable time only up to a certain hit density. Another issue is raised, when it comes to implementation of the algorithm on the compute devices. The problem is that the algorithm real-izes a random memory access, while checking different next possible hits on the track. This random memory access usually becomes a bottleneck for a program implementation, due to the slow speed of such operations. Moreover, if one tries to examine the logic of the approach, it is easy to see that during the search the algorithm often has to repeat certain calculations several times, since some of the results get discarded.
In the early stages of the CBM experiment, when the STS detector was still planned as a pixel detector, the track following method was tested for the re-construction routine (Fig. 4.8). The track rere-construction procedure was accom-plished in 3D space on both x-z and y-z projections simultaneously [88]. The procedure alternated between both views, predicting a track position on the next station and searching for hits in the vicinity of the predicted position.
Starting from the middle of the target area, this point was sequentially con-nected with all hits in the first station in y-z view, where tracks were close to straight lines. The straight lines driven via these two points were prolonged to the plane of the second station. All hits in an asymmetrical corridor around the intersection point were then used for fitting a parabola in x-z view which is pro-longed to the next station. Since several prolongations could happen, corridors were set around each point predicted on the third station. A similar corridor was set in the y-z view on the third station. If hits were found within these limits, they were attached to the track.
When the STS detector was redesigned using double-sided strip detector mod-ules, most of the methods could not cope with the increased number of hits and, as a result, more intensive combinatorial search.
Cellular-automaton-based track finder
As was shown, one of the major obstacles to be solved by each track finder is a huge and growing fast with track density amount of combinatorial
combi-nations, which track finder has to consider in order to bind together one- or two-dimensional measurements into five-dimensional tracks. Unfortunately, the exponential growth of the combinatorial enumeration at high track densities usu-ally makes it impossible to consider all combinations within a reasonable time frame. The CBM experiment can serve as a proper illustration of this problem, since the experiment has tried different tracking approaches for the planed pixel version of the STS detector. Unfortunately, most of them could not work any-more due to increased combinatorial combinations after switching from pixel to double-sided strip version of the STS.
However, a solid solution for combinatorial optimization was provided by the CA track finder algorithm. The CA track finder can be regarded as a local version of the Hopfield neural network [90] and will be discussed in detail in the next section.