tracking with the calice si-w electromagnetic calorimeter ......the tracking capabilities of the...

CAN-023aJune 1, 2010

Tracking with the CALICE Si-Welectromagnetic calorimeter prototype

using the Hough transformThe CALICE collaboration ∗

∗ Corresponding author: Felix Fehr ([email protected])

This note contains preliminary CALICE results, and is for the use of members of theCALICE collaboration and others to whom permission has been given.

AbstractThe CALICE collaboration designs and tests highly granular calorimeters for

a prospective e+e− International Linear Collider. This note describes and char-acterises a robust tracking algorithm based on the Hough transform with whichthe tracking capabilities of the finely granulated silicon-tungsten electromagneticcalorimeter are addressed in an initial study.

List of Tables

Contents

1 Introduction 31.1 Highly granular calorimetry for the ILC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 The Si-W electromagnetic calorimeter prototype . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Motivation for this study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 The Hough transform 42.1 Definition of the Hough space for the detection of straight lines . . . . . . . . . . . . . . . . . . . . 52.2 Filtering and weighting of MIP compatible hits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Post-processing and peak detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Determination of the track parameters and compatibility checks . . . . . . . . . . . . . . . . . . . 8

3 Performance of the algorithm 83.1 Monte Carlo simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Track detection efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4 Hit selection efficiency and purity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Application of the algorithm to test beam data 13

5 Summary and conclusions 17

A Additional material 21

List of Figures1 Schematic of the CALICE Si-W ECAL prototype . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Definition of the Hough space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Illustration of the Gerig and Klein backmapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Simulated overlay event in the Si-W ECAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Distance between MIP and shower axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Fraction of common cells of MIP and shower vs. distance . . . . . . . . . . . . . . . . . . . . . . . 117 Resolution of the track reconstruction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 MIP detection efficiency as a function of common cells. . . . . . . . . . . . . . . . . . . . . . . . . 149 MIP detection efficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1410 Overall hit selection performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1511 Hit selection performance as a function of proximity. . . . . . . . . . . . . . . . . . . . . . . . . . 1512 Hit selection performance as a function of the distance MIP / shower. . . . . . . . . . . . . . . . . . 1513 Example events from the test beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1614 Čerenkov off sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1815 Čerenkov on. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1816 Čerenkov on, less than 50 hits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1917 Čerenkov on, more than 49 hits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1918 Čerenkov off, less than 50 hits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2019 Čerenkov off, more than 49 hits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2020 Operation characteristics of the algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

List of Tables1 Summary of efficiency and purity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Bonus scheme in the Hough transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2

1 Introduction1

1.1 Highly granular calorimetry for the ILC2The CALICE collaboration designs and tests highly granular calorimeters for a prospective e+e−3International Linear Collider (ILC). The ILC is destined for performing high-precision measure-4ments in order to validate the results anticipated from the Large Hadron Collider (LHC) and to5continue the high-energy-physics research programme in the forthcoming years. At the envis-6aged TeV energy scale, the most interesting physics signatures in the final states typically arise7from (multiple) jets originating from hadronic decays of the W± and Z bosons, accompanied8by charged leptons or missing energy due to undetectable neutrinos. A reliable reconstruction9of these complicated patterns involves precise measurement of jet energies (with a resolution of10about 30%/

√E [GeV]), allowing for the identification of the W± and Z bosons in their decay11

channels.12Highly granular calorimetry is one of the designated technical key concepts for achieving this13

goal. Therefore CALICE studies compact calorimeter prototypes with high longitudinal and14transversal segmentation for the integration in ILC detectors. A series of test-beam experiments15has already provided valuable information about the reliability and the performance of the pro-16totypes. Moreover, recorded data are used to verify physics models in Monte Carlo tools for17the optimisation of the ILC design. Several options for electromagnetic and hadronic calorime-18ters are studied by the CALICE collaboration. The present work focuses on the silicon-tungsten19(Si-W) electromagnetic calorimeter prototype described in the following section.20

1.2 The Si-W electromagnetic calorimeter prototype21

Figure 1: Schematic of the CALICE Si-W ECAL prototype

The CALICE Si-W electromagnetic calorimeter (ECAL)22prototype [4] consists of thirty layers composed of23525 µm thick 1×1 cm2 silicon PIN diode pads as active24material sandwiched between passive tungsten absorber25sheets of variable thickness. The silicon sensors are ar-26ranged in groups of 6× 6 pads implemented on 4 square27inch wafers. An active detection layer is composed of an28array of 3×3 wafers, thus the active area of the calorime-29ter totals to 18× 18 cm2. The tungsten thickness is cho-30sen to be 1.4 mm (corresponding to 0.4 radiation lengths31X0) in the first ten layers, 2.8 mm (0.8 X0) in the next ten32layers, and 4.2 mm (1.2 X0) in the last ten layers. Conse-33quently, when counting only the tungsten absorber sheets,34the total depth of the calorimeter corresponds to 24 ra-35diation lengths. A schematic of the ECAL prototype is36shown in Fig. 1. Two axes of the coordinate system used37in this note are also indicated in the figure. The z-axis points parallel to the nominal beam direc-38tion. The y-axis is directed upwards and the x-axis completes a right-handed coordinate system.39

3

2 The Hough transform

1.3 Motivation for this study40

The novel high granularity approach to calorimetry adopted by the CALICE collaboration opens41new possibilities for particle identification and tracking using calorimeters. Conventionally,42calorimeters are built to measure the energy contained in electromagnetic or hadronic showers,43providing themselves only sparse spatial information. With high granularity by contrast, more44precise tracking of particles inside calorimeters becomes possible and can be used to improve45the allocation of energy deposits to reconstructed particles, which in turn improves the overall46jet energy resolution of the instrument. However, since most often multiple signals caused by47different particles do at least partially overlap in the calorimeter, simple fits (e.g. straight line48χ2 fits) are infeasible in most cases. Instead, tracking of particles requires much more robust49algorithms.50

In the following, a robust, yet simple tracking algorithm for the Si-W ECAL based on the51Hough transform (HT) [1] will be developed. This note presents a Monte Carlo simulation study52in which the detection efficiency for tracks in close vicinity to electromagnetic showers is eval-53uated. This permits us to address the tracking capabilities of the electromagnetic calorimeter54prototype in an initial study, and thus provides a benchmark for subsequent investigations. In the55final section of this note, the algorithm is applied to CALICE test beam data, and it is demon-56strated that the algorithm is capable of providing valuable, complementary information about57particle identity.58

2 The Hough transform59

The Hough transform is a well-known technique that is widely used to detect structures such as60curves or surfaces in a given set of points in an arbitrary feature space. Its initial applications61have been in image processing, where the method has proven its robustness. Later, algorithms62based on the Hough transform have been integrated into particle physics where fast and reliable63methods for track finding are required. Technically, the method can be divided into two steps,64consisting of a transformation from an n-dimensional feature space X into an m-dimensional65parameter space Ω, also referred to as Hough space (HS), and the subsequent detection of clusters66in this space.67

The following discussion is restricted to the simplest case possible, with n = 2 and m = 2. The68transformation step and the peak identification are described in more detail in the following sub-69sections which points out how the Hough transform can be utilised for the tracking of minimum70ionising particles (MIPs) in the CALICE Si-W ECAL.71

As the calorimeter is not exposed to magnetic fields during the test beam experiments, MIPs72deposit signals (around thirty hits per through-going MIP) along straight lines. In order to iden-73tify tracks created by minimum ionising particles in the calorimeter, the projections of the hit74calorimeter cells onto the (x,z)- and the (y,z)-plane are analysed, and patterns resembling straight75lines are identified. As the HT is not limited to straight lines, but can also be used to identify76curved tracks, the methods developed in this note can be generalised for ILC detectors in which77the calorimeters are finally exposed to magnetic fields.78

4

2.1 Definition of the Hough space for the detection of straight lines

2.1 Definition of the Hough space for the detection of straight lines79

The detection of straight lines in a two-dimensional space (which could be an image or, in our80case, a projection of a signature in the calorimeter onto a two-dimensional plane parallel to the81beam axis) is one of the most frequent applications of the Hough transform. In this case, the82parameter space is commonly defined to be spanned by the line parameters ρ and θ giving the83distance of the foot of the normal n to the line from the origin respectively its direction, cf. Fig. 2.84From the geometrical interpretation it is obvious that ρ is restricted to positive values. For a point85whose radius vector r encloses an angle φ with the z-axis, it can be shown that possible θ values86lie in the interval [φ −90◦,φ + 90◦]. The parameters ρ and θ of lines crossing a common point87(z,u) in the feature space are not independent, but interrelated according to a sinusoidal function88ρ = usinθ + zcosθ . A proof for this relation is given in the description of Fig. 2.89

90

The Hough transform is essentially a mapping of the point (z,u) onto a function ρ(z,u)(θ).91The corresponding functions for two (or more) points on a given line will intersect in the Hough92space at the line parameters. Prior to the transformation an appropriate discretisation (binning) of93the parameter space has to be chosen according to the expected achievable resolution in the two94parameters. In the transformation step, the bins in the Hough space are populated for each point95in the feature space according to Eq. 1. The principle of the Hough transform relies on the fact96that points on a line in the feature space will contribute to the same bin in the parameter space.97As a consequence, the parameters describing the line can be determined by scanning the Hough98space for maxima. It is noteworthy that in contrast to the standard applications of the HT in our99case the feature space it itself discrete. The optimal quantisation scheme is therefore primarily100constrained by the calorimeter design (i.e. the chosen granularity of the calorimeter).101

102

In general (cf. [2]) the Hough transform can be written in terms of a kernel function p, defined103

PARAMETERISATION OF STRAIGHT LINES

z

u

θ

r

ρ

n

(z,u)Figure 2: In the HT straight lines (crossing the point r = (z,u))in the plane are conventionally parameterised by the distanceρ of the foot of the normal n to the origin and its direction θ .In our case, u can either be the x- or the y-axis.The parameters ρ and θ of lines crossing the point (z,u) in thefeature space are not independent, but interrelated accordingto a sinusoidal function

ρ = usinθ + zcosθ . (1)

To prove this equation we note that n ⊥ r− n and thereforen · (r−n) = 0 or n ·r = ρ2 from which (cosθ ,sinθ) · (z,u) = ρfollows.

5

2 The Hough transform

jointly on the feature space (with n points xi ∈ X) and the Hough space (ω0 ∈Ω),104

H(ω0) =n

∑i=1

p(xi;ω0). (2)

The kernel function is most often implicitly chosen through the parameterisation and the quanti-105sation (binning) of the HS. For p(zi,ui;θ ,ρ) = Rect(ui sinθ+zi cosθ−ρd ) one recovers the standard106HT as described above, where the parameter d specifies the quantisation scheme1. The notion107expressed through Eq. 2 is indeed very useful as it allows one to include weights in order to108amplify signals by exploiting certain features in the analysis. The following section expands on109this point.110

2.2 Filtering and weighting of MIP compatible hits111

For an efficient identification of MIP signals in the calorimeter, a preselection of signal cells can112be performed. In this step, only cells are considered which are compatible with MIP signals in113terms of deposited energy. MIP compatible hits are required to have an energy deposit of no114more than 1.5 times the average energy induced by a minimum ionising particle traversing the115cell. Furthermore, in order to reduce the impact of noise hits, a minimal energy of 0.5 MIPs is116required. In the following steps only hits fulfilling these critera are taken into account.117

Algorithms developed for tracking of charged particles in cylindrical central drift chambers118implemented in high-energy physics detectors usually proceed radially inwards from hits in the119outer region, which is less densely populated with signals, towards the interaction vertex so as to120mitigate combinatorial problems arising from the relatively higher hit density in the central re-121gion. Being a global method, the Hough transform is inherently less sensitive to such problems,122and a priori ignorant of the specific event topology. However, in order to improve the detection123efficiency, the individual hits in the Hough transform can be weighted according to the proba-124bility of being due to MIP. In this note, a more pragmatic approach is adopted in which signal125hits are amplified by heuristic weights determined depending on the topology of the hits. The126weights are chosen depending on the position of the cell as well as by taking signals in neigh-127bouring cells into account. Minimum ionising particles create well-localised, connected signal128hits distributed along straight lines (tracks) through the calorimeter. Isolated energy deposits in129cells, which do not have signals larger than a certain maximal energy in neighbouring cells (cho-130sen to be 2.0 MIP equivalents) in the same layer and/or are connected to isolated hits in adjacent131layers are therefore amplified by a bonus weight. An important task of the algorithm is the sep-132aration of MIP tracks and electromagnetic showers. Electromagnetic showers reach the shower133maximum, where most of the energy is observed only after a few radiation lengths and typically134decay before the end of the calorimeter. Therefore, hits in the first and the last few layers of the135calorimeter are rewarded with a higher weight.136

1The rectangle function Rect(x) = 1 for x ∈ [−0.5,0.5] and Rect(x) = 0 for x /∈ [−0.5,0.5].

6

2.3 Post-processing and peak detection

2.3 Post-processing and peak detection137

Depending on the complexity of the feature space (here given by the observed hit pattern) several138maxima will occur in the Hough space, and a number of random peaks leading to faked tracks139(so-called ghost tracks) may accrue. In order to facilitate the identification of signal-induced140maxima in the Hough space, a post-processing step has been developed [3] which allows for an141efficient cleaning of the Hough space. The basic idea of the so-called Gerig and Klein backmap-142ping algorithm is a single point (hit cell) should be associated with exactly only one feature143(track) in the image (projection of the calorimeter). The algorithm allows each hit to contribute144only one vote. Out of the points to which a given hit contributed (according to Eq. 1), the one145having the most votes is selected and the hit is associated with the corresponding parameter146values. To minimise the number of ghost tracks further, eligible candidate tracks are required147to have a certain number of votes. An example of the cleaning process (for the (y,z)-space) is148shown in Fig. 3.

POST-PROCESSING STEP

[rad]θ

-3 -2-1

0 12

3

[mm]ρ

020

4060

80100

120140

160180

200

sum

of

hit

wei

gh

ts

020

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160

(a) Hough space for a MIP before cleaning.

[rad]θ

-3 -2-1

0 12

3

[mm]ρ

020

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80100

120140

160180

200

sum

of

hit

wei

gh

ts

020

4060

80100

120140

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(b) Hough space for a MIP after cleaning.

Figure 3: Illustration of the Gerig and Klein backmapping process. The Hough space for a pure MIPevent in the ECAL is shown on the left. After the cleaning process, a single peak remains in the Houghspace corresponding to the line parameters of the MIP track in the (y,z)-plane.

149

For our analysis, this post-processing step has been generalised to combine (x,z) and (y,z)150information. In the generalised algorithm, the parameter values of the detected maxima with their151corresponding weight are registered in a four-dimensional space (ρx,θx,ρy,θy). This ensures152that each hit is associated with exactly only one particle track in three dimensions. As signal153hits in the (x,z) plane are also signal hits in the (y,z) plane, this approach leads to an improved154suppression of background hits. A quality threshold clustering algorithm [5] is employed to scan155the parameter space for clusters and to identify the final set of hits.156

7

3 Performance of the algorithm

2.4 Determination of the track parameters and compatibility checks157After the Hough transform and the post-processing step have been performed, the track param-158eters can be calculated from the line parameters in the Hough space. The intersection point159x0 = (x0,y0,0) of the track with the front face of the ECAL (at z = 0) and the track direction v160are related to the Hough parameters according to161

x(s) = x0 + sv =

ρx sinθxρy sinθy0

+ s√1+ tan2(θx−90◦)+ tan2(θy−90◦)

tanθx−90◦tanθy−90◦1

. (3)One should note that the time is not determined by the algorithm. The parameter s denotes162the length of the path traversed by the particle in the calorimeter when reaching the point x(s).163The parameterisation is restricted to tracks crossing the plane z = 0, and hence fails to describe164tracks orthogonal to the z-axis. This, however, is not a limitation in our study, as the beam is165parallel to the z-axis. One of the desired features of tracking algorithms is a low probability for166reporting false tracks. Since any combination of two points induces a straight line and thus a167track candidate, certain quality requirements have to be imposed for the Hough transform. In168this note the following variables have been studied.169

• The minimal number of hits on the candidate track.170

• The summed weights (assigned in the transformation step, see Sec. 2.2) of hits of the track171candidate, yielding a score of the candidate track.172

These variables are studied with the Monte Carlo samples discussed in the following section and173are adjusted to reduce the rate of faked tracks down to a minimum, while keeping most of the174signal-induced tracks. Since these two variables already allow reducing the rate of false tracks to175about 0.008 per event under challenging conditions (described in the next section) the following176discussion is restricted to just these two variables for the sake of simplicity. Further, potentially177useful variables include178

• the first layer which is hit by the track candidate,179

• the length of the longest track segment (given by the number of consecutive layers hit in180the segment),181

• a penalty function evaluating the gaps occurring on the track (e.g. as studied in [6]), and182

• the number of track segments with hits in at least 2,3, or 4 consecutive layers.183

3 Performance of the algorithm184

3.1 Monte Carlo simulation185Detailed simulations of the ECAL response to electrons and MIPs have been performed using186the Mokka simulation framework [7]. Three dedicated samples were prepared for the evalua-187tion of the HT algorithm. Each of the generated samples comprises 100.000 events. The first188

8

3.2 Resolution

sample consists of pure MIP events, this sample can be used to study the performance of the189algorithm under background free conditions. The second reference sample contains electromag-190netic showers induced by 30 GeV electrons. This provides the background sample for the study.191The benchmark sample is obtained by superimposing MIPs with electromagnetic showers. Each192overlay event contains a MIP track and an electron shower. This sample allows us to demon-193strate that the algorithm is capable of identifying tracks in the close vicinity of electromagnetic194showers and can thus be used to identify overlay events in the CALICE test beam data. The sim-195ulated particle beams are parallel and normally incident on the calorimeter front-face. The beam196profiles are Gaussian in x and y, having a generated width of σx = 14.5 mm and σy = 6.7 mm,197with their centres overlapping. This should reflect the conditions in the test beam experiments198for overlay-events. This is a conservative assumption since the distribution of halo particles is199probably broader than the electron beam. One example of a simulated overlay-event is shown in200Fig. 4.201

The performance of the tracking algorithm can be judged by the resolution of the track pa-202rameters (direction, point of intersection with the front face of the ECAL) achieved as well as203by its capabilities to detect tracks in noisy environments and by the efficiency and the purity of204the hit selection. Evidently, the quality of the track detection and reconstruction depends on the205separation of the muon track and the electromagnetic shower. Thus, it is advisable to study the206performance of the algorithm as a function of the separation. A convenient measure of the sepa-207ration is given by ratio of the number of cells with contributions from both, MIP and shower, to208the number of cells containing MIP signals,209

κ =number of cells with signal from MIP and shower

number of cells with MIP signal. (4)

This variable describes the proximity of the two signatures in a straightforward way. For κ = 0,210MIP and electromagnetic shower are completely separated, whereas for κ = 1 the minimum211ionising particle is completely hidden inside the shower.212

The distribution of the distance between electron shower axis and MIP track is show in Fig. 5213together with the beam profiles and the distribution of the proximity variable. The chosen beam214profiles result in an average distance of about 2 cm between shower axis and MIP. This has to215be compared to the effective Molière radius of the ECAL, ρeff ≈ 20 mm which contains 90% of216the shower energy. Considering a granularity of 1 cm the detection of MIPs in this sample seems217challenging.218

3.2 Resolution219

The full track parameters can be determined by the presented algorithm. The results of the recon-220struction can be compared with the true Monte Carlo parameters. The residuals (reconstructed221value minus true value) for the intersection of the track with the front face of the ECAL (z = 0)222are extracted from the MIP simulation and are shown in Fig. 7 as an example. In Fig. 7a the resid-223uals for the x-coordinate are displayed, in Fig. 7b the residuals for the y-coordinate are given. A224Gaussian fit is indicated in addition. The resolution, being the width of the fitted Gaussian, can225

9


OVERLAY EVENT IN THE ECAL

z [mm]

0 2040 60

80 100120 140

160 180200

y [mm]

−100−80

−60−40

−200

2040

60

x [

mm

]

−80

−60

−40

−20

0

20

40

60

80

100

Figure 4: Simulated overlay event in the Si-W ECAL containing an electromagnetic shower induced bya 30 GeV electron and a minimum ionising particle track. Squares denote MIP-induced hits, circlescorrespond to electron-induced hits, and stars mark cells with contributions from both, electromagneticshower and MIP. Hits, which are selected by the algorithm, are indicated in red. In this example thethrough-going MIP track parallel to the beam axis is nicely identified by the algorithm. Note that thepresented HT algorithm achieves a very high hit selection purity and a good selection efficiency. Only afew hits which do not fulfill the MIP condition (explained in section Sec. 2.2) or lie to far apart from thetrack are not recovered by the algorithm. The selection efficiency in this example is 76% with a purity of100%.

be extracted from the fit. The achieved resolution is σx = 5.8 mm and σy = 5.3 mm respec-226tively. The y-residual distribution is slightly shifted as the origin of the coordinate system does227not coincide with the beam centre.228

3.3 Track detection efficiency229In this section the capability of detecting tracks in the vicinity of electromagnetic showers is230investigated. The performance of the algorithm is studied using the overlay sample described231above. The detection efficiency is defined as232

E =number of detected MIPs

number of generated events. (5)

10

3.3 Track detection efficiency

CHARACTERISTICS OF THE BENCHMARK SAMPLE

(a) Distance between shower axis and MIP.

κfraction of common cells 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

frac

tio

n o

f ev

ents

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

CALICE preliminary

(b) Distribution of the proximity κ

Figure 5: (a) Distribution of the distance between simulated MIP track and electron shower axis in mil-limetres, normalised to unit area. The inset shows a scatter plot of the (x,y) beam profile: blue pointsmark electrons, red points MIPs. (b) Distribution of the fraction of common hits κ , as defined in the text.The distribution has been normalised to unit area. Note that κ is a fraction of integer numbers, this causesthe tiny structure visible at around 0.5 and 0.6.

distance MIP shower axis [mm]0 10 20 30 40 50 60

>κav

. fra

ctio

n o

f co

mm

on

cel

ls <

0

0.1

0.2

0.3

0.4

0.5

0.6

CALICE preliminary

Figure 6: Average proximity κ as a function of the distance between shower axis and MIP track inmillimetres. The transverse energy profile of electromagnetic showers exhibits a long exponential tail.This manifests itself in a halo of low energy hits stretching out over radial distances of several centimetresfrom the initial electron direction.

11


[mm]mc - xrecx-50 -40 -30 -20 -10 0 10 20 30 40 50

eve

nts

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14CALICE preliminary

(a) x0 residuals

[mm]mc

- yrec

y-50 -40 -30 -20 -10 0 10 20 30 40 50

eve

nts

0

0.02

0.04

0.06

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0.1

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CALICE preliminary

(b) y0 residuals

Figure 7: In this plot, the reconstructed coordinates of the intersection point of the track with the frontface of the ECAL are compared with the true values extracted from the MIP simulation. On the left sidethe residuals for the x-coordinate are depicted. The right side shows the distribution for the y-coordinate.Both distributions have been normalised to unit area. A Gaussian (red line) has been fitted to eachdistribution. See text for further information.

The detection efficiency is shown as a function of the separation in terms of the proximity vari-233able κ in Fig. 8 and as a function of the distance between the parallel shower axis and MIP in234Fig. 9. From Fig. 8 it can be concluded that the algorithm performs well, even under difficult235circumstances. At already substantial contamination (around κ = 0.5) detection efficiencies of236over 90% are still reached. As expected, the track detection efficiency approaches 100% if MIP237and shower are completely separated (κ → 0). As can be seen from Fig. 9, the detection effi-238ciency is already about 90% at distances of about 2 cm. It should be emphasised that the results239have been achieved using ECAL information only. The performance of the reconstruction can be240improved further by taking supplementary information from the tracking devices and the HCAL241into account.242

3.4 Hit selection efficiency and purity243The quality of the reconstruction can be described by the efficiency for selecting signal hits. The244event-wise hit selection efficiency is given by245

ε =number of identified MIP hits

number of true MIP hits. (6)

Another important quantity is the purity of the hit selection, defined as246

p =number of true MIP hitsnumber of identified hits

. (7)

The overall purity and efficiency of the hit selection is shown in Fig. 10. The mean efficiency247is 79% and the mean purity is 93%. For a reliable identification a good purity is indispensable.248

12

The results therefore been optimised in terms of purity. In Fig. 11, the hit selection purity and249efficiency is depicted as a function of the proximity κ of shower and MIP. Finally, in Fig. 12, the250selection quantities are plotted as a function of the distance between shower axis and MIP.251

The results are summarised in Tab. 1.

FIGURES OF MERITDetection efficiency Hit selection efficiency Hit selection purity

88% 79% 93%

Table 1: Summary of the detection efficiency and hit selection purity and efficiency of the algorithmwhen applied to the simulated overlay sample. The results demonstrate that the algorithm is capable ofresolving tracks in close vicinity to electromagnetic showers. High hit selection purities can be reachedwhile keeping good selection efficiencies. It should be emphasised that the results are achieved by usingECAL information only.

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4 Application of the algorithm to test beam data253The HT can be used to remove pion contamination in electron beams. The CALICE calorimeter254prototypes have been studied in dedicated experiments, in which they have been exposed to255beams of different particles species in a wide energy range. In particular, the response of the Si-256W ECAL to electrons has been investigated thoroughly. Since the test beam used for the electron257study contains a certain admixture of other particles, it is important to select a sufficiently clean258electron sample for the analysis. For this purpose Čerenkov counters are installed at the test259beam providing particle identity information. The pressure of the gas in the counters is adjusted260so that electrons radiate Čerenkov light, while heavier species do not. The signal of the Čerenkov261counter can therefore be used to select electrons and to discard muons and pions contained in the262beam.263

Since the Čerenkov counter information is not available for all of the CALICE test beam264runs and the Čerenkov may not be fully efficient, an independent, complementary method is265very valuable. The Hough transform tracking algorithm can be applied to the recorded data to266identify tracks from pions and muons in the calorimeter. In order to demonstrate that the HT267can be used to reject background in the electron sample we will look at an exemplary CALICE268run. We have purposely chosen a 15 GeV electron beam run (electrons normally incident on the269ECAL) with a high pion contamination, which gives us sufficient statistics for our discussion;270usually the background in the electron runs is smaller. The data have been processed with the271HT tracking algorithm.272

The distribution of the number of hits on the candidate track is plotted against the score of273the candidate track in Fig. 14 for the ’Čerenkov off’ (non electron) case, and in Fig. 15 for274the ’Čerenkov on’ events (electrons). In the ’Čerenkov off’ plot (Fig. 14), one can see a small275fraction of events (bottom left) with short (and mostly false) track candidates with a low score276(as defined in Sec. 2.4) and extended distribution of events containing longer tracks with a good277score. Events with more than 15 hits or a candidate score of more than 150 contain most likely278

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4 Application of the algorithm to test beam data

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Figure 9: MIP detection efficiency as a function of the distance (in millimetres) between the MIP and theshower axis.

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PROPERTIES OF THE HIT SELECTION

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Figure 10: Overall hit selection purity (left) and efficiency (right) for the generated overlay sample. Theshaded area in the last bin indicates the fraction of events with a purity (efficiency) of 100%.


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Figure 11: Average hit selection purity (left) and efficiency (right) as a function of the proximity κ ofelectromagnetic shower and MIP.


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Figure 12: Average hit selection purity (left) and efficiency (right) as a function of the distance betweenelectromagnetic shower and MIP.

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4 Application of the algorithm to test beam data

real tracks from pions or muons. The ’Čerenkov on’ plot (Fig. 15) shows a large fraction of279events in the bottom left part (no track events) and also a smaller fraction of events with good280candidate tracks. The latter are pion or muon events. By comparing the number of events with281good candidate tracks in the ’Čerenkov on’ and ’off’ case we can see that the Čerenkov counter282is quite efficient for this run, however, the efficiency is not 100% as illustrated by the band of283high hit/high score events in Fig. 15.284

The ’Čerenkov on’ event class is shown for events with less than 50 hits in the calorimeter285in Fig. 16 and for event with more than 49 hits in Fig. 17. From these plots we can see that286the inefficiency is larger in the case of low number of hits. This subsample is not crucial for287the electron analysis as it can be removed by a cut on the number of hits. The HT can be used288to argue in favour of this cut. As visible from Fig. 17, a tiny fraction of events with more hits289remains in the electron sample. The HT algorithm can be used to remove them and to provide an290even cleaner sample for the electron analysis. In Fig. 18 (for events with less than 50 hits) and291Fig. 19 (for events with more than 49 hits) the Čerenkov off distributions are shown. On the one292hand, we can see that most of the events with low number of hits have candidate tracks with a293high score. On the other hand, in Fig. 19 a small fraction of events flagged as non electrons by294the Čerenkov counter shows only a small score. These events can be pions interacting early in295the calorimeter and thus leaving only short tracks which cannot be unambiguously identified by296the HT without a detailed analysis.297

Most of the events in this run with ’Čerenkov on’ and a good track candidate are simply pion298events which are wrongly classified by the Čerenkov counter. Fig. 13a shows a very rare overlay299of two pions as an example. A tiny fraction of events shows electron events which are correctly300classified by the Čerenkov, but contain in addition signals from a second particle. Fig. 13b301depicts such an overlay event, containing an electron shower signature and a MIP track.302

APPLICATION TO TEST BEAM EVENTS

(a) Overlay of two pions. (b) Overlay of electron shower and MIP track.

Figure 13: Two example events from Run 300674 (with an e− beam at 15 GeV). The left plot shows anexceptionally rare overlay of most probably two pions. (Selected hits are marked by red and green points).The right plot is a more typical example showing an electron signature and a MIP track (red points).

16

As a result, we have demonstrated that the HT algorithm can be used to provide complemen-303tary particle identity information and can be used to check the quality of data recorded in the304CALICE test beam experiments.305

5 Summary and conclusions306In this note, we have developed a tracking algorithm for minimum ionising particles in the307CALICE Si-W calorimeter. The algorithm is based on the Hough transform of the (x,z)- and308(y,z)-projections of hit calorimeter cells. In a preselection step MIP compatible cells have been309selected according to the energy deposited. Typical MIP signals (isolated hits, hits on straight310lines, hits connected with hits on adjacent layers) are amplified by assigning heuristic weights311in the transformation step. We have introduced a new method for a combined analysis of the312Hough spaces, by using a generalised post-processing step in which the two two-dimensional313parameter spaces are combined. It has been shown that this method yields the full track param-314eters. The performance of the algorithm was studied with a dedicated Monte Carlo simulation.315In this way, it has been demonstrated that by using ECAL information only, even under diffi-316cult conditions such as a track in close vicinity to an electromagnetic shower, average detection317efficiencies better than 80% can be achieved. A detailed look at the performance / fraction of318common hits reveals that even up to contaminations of 50%, detection efficiencies of 90% can319be reached. The results obtained in this study can serve as a benchmark for further studies of320the tracking capabilities. The exemplary application of the algorithm to CALICE test beam data321has also demonstrated that the algorithm is capable of providing complementary particle identity322information.323

References324[1] P.V.C Hough, A method and means for recognising complex patterns, US Patent No. 3,069,654, 1962.325

[2] J.Princen, J. Illingworth, and J.Kittler, A formal definition of the Hough transform: Properties and326relationships, J.Math. Imaging Vision, vol. 1, pp. 153-168, 1992.327

[3] G. Gerig, and F. Klein, Fast contour identification through efficient Hough transform and simplified328interpretation strategy, in Proc. 8th Int. Conf. Pattern Recognition, vol. 1, pp. 498-500, 1986.329

[4] The CALICE collaboration, Design and electronics commissioning of the physics prototype of a Si-W330electromagnetic calorimeter for the International Linear Collider, JINST, 2008.331

[5] L.J. Heyer et al., Exploring expression data: Identification and analysis of coexpressed genes.,332Genome Research 9(11), 1106–1115, 1999.333

[6] L. Weuste, A Study of Track Segments within Hadronic Showers with a Highly Granular Hadronic334Calorimeter, Diploma thesis, LMU Munich, 2009335

[7] Mokka home page, http://polzope.in2p3.fr:8081/Mokka336

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http://polzope.in2p3.fr:8081/Mokka

References

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Figure 17: Number of hits on vs. score of candidate track for CALICE Run 300674 (electron beam withsubstantial pion contamination). This plot shows the distribution for events flagged as ’electron’ withmore than 49 cells hit in the ECAL.

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Figure 19: Number of hits on vs. score of candidate track for CALICE Run 300674 (electron beam withsubstantial pion contamination). This plot shows the distribution for events flagged as ’non electron’ withmore than 49 cells hit in the ECAL.

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A Additional material337

This appendix contains supplementary informationfor the use within the CALICE collaboration only.

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The weighting factors that are used in the HT algorithm are summarised in Tab. 2.339

Hit Type Layer 1-5 Layer 6-25 Layer 26-30isolated hit 3.0 2.0 3.0isolated and connected 3.5 2.5 3.5isolated and connected to isolated 4.0 3.0 4.0remaining hits 1 1 1

Table 2: Bonus scheme for hit weights in the Hough transform.

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Figure 20: This plot shows the operation characteristic of the algorithm when applied to the overlaysample. The detection efficiency for MIPs in the overlay sample and the fraction of reported ghost tracksper event in a clean electron sample are computed in dependence on a cut on the score of the trackcandidate, cf. Sec. 2.4. This plot gives a lower limit on the performance of the algorithm as includingmore quality variables improves the ratio.

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IntroductionHighly granular calorimetry for the ILCThe Si-W electromagnetic calorimeter prototypeMotivation for this study

The Hough transformDefinition of the Hough space for the detection of straight linesFiltering and weighting of MIP compatible hitsPost-processing and peak detectionDetermination of the track parameters and compatibility checks

Performance of the algorithmMonte Carlo simulationResolutionTrack detection efficiencyHit selection efficiency and purity

Application of the algorithm to test beam dataSummary and conclusionsAdditional material

tracking with the calice si-w electromagnetic calorimeter ......the tracking capabilities of the...

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