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Track Reconstruction in the Timepix testbeam

Matthew Reid1st Year PhD

University of Warwick

Interested in pattern recognition for pixel detectors

New techniques that could be applicable to the upgrade

Aim– Increase the reconstruction efficiency of tracks in the Timepix test beam

Any questions just interrupt me!

Introduction

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Uses a window and where r=“reference cluster”, such that all points with respect to a reference cluster lie within the tolerance .

Only produces tracks with EXTREMELY small angle about z-axis,

The choice of cluster within the window is arbitrary and therefore chosen cluster may NOT be the closest.

New tracks then cannot be built because the true clusters incorporated in previous track.

Cause for ghost tracks as we increase

Original Code

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Reference cluster Cluster in detector i

𝜖𝜖

k-Nearest Neighbour (kNN) algorithm - FAST Uses a Euclidean distance equation

Uses the (x,y) coordinates of a seed point to find nearest neighbours in adjacent plane

Ideas behind new code – Pattern Recognition I

Tree structure of the kNN algorithm

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Based on a kNN algorithm and Kalman Filter Tree structure => lowest chi2 => best track If there are no degenerate neighbours then

have possible tracks

Plot of 3D points

for two iterations

Ideas behind new code – Pattern Recognition II

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SEED HIT

Candidate hitsafter 1st iteration

Candidate hitsafter 2nd iteration

Example for k=3, projections in both xz-plane and yz-plane: (Note degeneracy)

xz-plane yz-plane

Ideas behind new code – Pattern Recognition III

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zz

yx

SEED HIT SEED HIT

Candidate hits 1st iterCandidate hits 1st iter

Candidate hits 2nd iterCandidate hits 2nd iter

Kalman Filter gives best estimates for a linear system (see backup slides for detail)

Allows you to take into account multiple scattering based on Moliere formula for thick material to find better track fit

Measurement uncertainty based on pitch of pixel ~ 55µm

Ideas behind new code – Kalman Filter I

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Example taken using:bin/tpanal –c cond/Alignment507.dat –z /afs/cern.ch/lhcb/group/vertex/vol7/Timepix/ZSData/Run507.root –n 9

MAX number of tracks with current algorithm is the plane with the

lowest number of clusters?... 109? WRONG!!

Distribution of Clusters and Alignment - I

Detector Plane Number of Clusters

C03W0015 110

K05W0019 109

D09W0015 124

M06W0015 111

I02W0019 109

E05W0015 113

DUT D04W0015Bias the data set so ignore

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The maximum number of clusters depends on the alignment of detectors

Since the alignment is not perfect we are required to look in area shown in graph

Hence not all clusters can be used in the reconstruction.

Distribution of Clusters and Alignment - II

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Distribution of Clusters and Alignment - III

Detector Plane Number of Clusters

Number of Aligned Clusters

C03W0015 110 94

K05W0019 109 101

D09W0015 124 93

M06W0015 111 91

I02W0019 109 95

E05W0015 113 97

TOTAL 676 571

Reject clusters outside of these alignment conditions

and

Reconstruction performance - IOriginal Algorithm New Algorithm

Number of Tracks 51 75

Number of rejected tracks - 16

Efficiency 54% 79%

Global < > all tracks 0.0714282 ?? 1.3548 (0.0908276)

Projection of Tracks in xz

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z

x

z

xy

HENCE 1in5 chance that data set it as bad as thisGOOD!

Residuals are of a comparable order to the original code giving ~6µm, over all planes (unfortunately)

Reconstruction performance - II

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Highly dependent on the initial arrangement of hits as a seed and that the points are sufficiently spaced across detector (low occupancy).

Code design (there isn’t any..)

Speed and optimisation (currently very POOR! Not utilising the benefits of Kalman)

Rejecting tracks if the points have already been assigned to another track (no comparison via ) similar method to original code

Eventually => VERTEXING of lead bump bonds… Various methods to look into – Hopfield neural network,

Inverse Kalman filter.

Issues and things for the future

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If the fitting function accurately predicts the means of the parent distribution, then the estimated variance , should agree well with the variance of the parent distribution, , and their ratios should be close to 1

Old code has no denominator for one hence its small size

for (std::vector< TestBeamCluster* >::iterator j=clustersforalignment->begin(); j < clustersforalignment->end(); ++j){      float xresidual=(*j)->globalX()-x_inter_global;      float yresidual=(*j)->globalY()-y_inter_global;      chi2 += xresidual*xresidual + yresidual*yresidual;}

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Backup Slides – I - goodness of fit

Forward Geometry – gradients and State vector given as a 4 dimensional

vector

Propagation proceeds via

where is the noise term

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Backup Slides - II

Measurement error is dealt with by the following

Where is the measurement error (noise term)

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Backup Slides - III

Moliere Formula for Gaussian scattering () based on perpendicular planes to z –axis (we have 9 degree rotations).

Gives the Covariance matrix for the scattering in the Kalman filter

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Backup Slides IV