A New Approach to Cluster Finding in Cathode Pad Chambers of the Muon Spectrometer A. Zinchenko for...

A New Approach to Cluster Finding in Cathode Pad Chambers

of the Muon Spectrometer

A. Zinchenko

for JINR (Dubna) – SUBATECH (Nantes) Collaboration

ALICE Week

16-20 September, 2002

A "New" Approach to Cluster Finding in Cathode Pad Chambers

of the Muon Spectrometer

A. Zinchenko

for JINR (Dubna) – SUBATECH (Nantes) Collaboration

ALICE Week

16-20 September, 2002

A New Approach to Cluster Finding in Cathode Pad Chambers of the Muon Spectrometer

A. Zinchenko ALICE Week, 16-20 September, 2002

Ancestors and close relatives

A. Zinchenko, “A new approach to cluster finding in muon chambers” at Muon Group Meeting, Le Mont-Dore, 25-29 March, 2002

G. Chabratova, A. Zinchenko, “A New Approach to Cluster Finding

and Hit Reconstruction in Cathode Pad Chambers and Its Development for the Forward Muon Spectrometer of ALICE” at VIII Int. Workshop on Advanc. Comp. and Anal. Techniq. in Phys. Res. (ACAT’2002) 24-28 June, 2002, Moscow, Russia

E.A. Kolganova, E.L. Kosarev, G.A. Ososkov, “Superresolution algorithms for data analysis of discrete detectors in nuclear physics”, NIM A443 (2000) 464



Overview

The Muon Spectrometer

Basic Ideas of Bayesian Unfolding

Application to Muon Cluster Finding

Results

Conclusions

General view of the Forward Muon Spectrometer



Bayes' theorem: G. D’Agostini, NIM A362 (1995) 487 If anybody observes a single event E(effect), the probability that it has been due to the ith cause of C

i,

i=1,2,..,nc is proportional to the probability of cause P(C

i)

times the probability of the cause to produce the effect P(E|Ci).

(Lucy-Richardson method: L.B. Lucy, Astronom. J. 79 (1974)

745; W.H. Richardson, J. Opt. Soc. Amer. A 62 (1972) 55)



Basic Ideas of Bayesian Unfolding

in

iii

iii

CPCEP

CPCEPECP

1

)()|(

)()|()|(


Application to the positron emission tomography. Pairs of photons are registered in coincidence by pairs of detectors. The region, where the image is reconstructed, is subdivided into pixels and different pairs of detectors form tubes. Since the number of emissions inside pixels and number of of photon coincidences are described by the Poisson statistics, the likelihood function can be written

Basic Ideas of Bayesian Unfolding (cont) Maximum Likelihood - Expectation Maximization (MLEM):

L.A. Shepp and Y. Vardi, IEEE Trans. Med. Im., v.1 (1982) 113


p

t

ij

ij

njni ij

nij

neL

,...,1,...,1 !

)(λ


where - vector of pixel activities , - the mean number of emissions from the pixel j contributing to the tube i, is the number of photon pairs counted in the tube i from the pixel j, is the number of detector tubes, - number of pixels. and is jth pixel visibility from the tube i.

The Expectation step of the EM algorithm is done by taking a conditional expectation of log-likelihood with respect to parameter estimates at thenth iteration and measured detector counts .

The Maximization step is to take partial derivatives of the expectation, set them to 0 and solve.

The new value for the jth component of the vector

),...,( 1 pnλ


Basic Ideas of Bayesian Unfolding (cont)

n ij

ij

j

iji nn

ik

nkik

iij

iij

njn

j c

nc

c

1

tn

pn jijij c ijc

n

j /


“Journey to the Center of a Rat’s Heart’’ (Sci. Fi., PG-13)

“Halloween-12’’ (Horror, R)


Some Examples From Medical Imaging


Detector: Cathode Pad Chamber (CPC). Two cathodes with pad sizes varying from (0.4x0.6) cm2 to (10.x0.7143) cm2 .

Old method: The pad charge distribution is described by two-dimensional integral of the Mathieson function and coordinates of a single track are extracted from a fit of the pad charge distribution by this function. The number of track candidates is estimated from the number of local maxima in the pad charge distribution. Amount of events with significant overlapping of signal and background induced pad charges is not negligible. Estimation of the number of track candidates from the local maxima is not sufficiently accurate and results in deterioration of coordinate resolution.




New method: transform the "two-cathode fixed pad size" problem into the "one-anode plane variable pixel size" one. Find groups of adjacent pads on one

cathode and overlapping with them pads on the other cathode precluster

Precluster array of pixels in the anode plane with size defined by the overlap of pads on both cathodes.

It is assumed that each pixel contains a track




New method (cont):Iterative procedure: Track j has an initial value of energy release . Usually all 's are set to 1.

with

where - the expected signal on pad i for track energy releases at the kth iteration, - the track-to-pad coupling(signal on pad i from track j given by Mathieson) and - the number of tracks in the array.

ki

iN

iijN

iij

kjk

j f

qc

c

pp

pads

pads

1

1

1

Application to Muon Cluster Finding(cont)

p j0

p j0

f ik

p jk

cij


tracksN

j

kjij

ki pcf

1

N tracks


New method (cont): after several iterations (10-15) the pixel size is decreased by two;

track candidates with the lowest energy releases are removed if the total number of candidates exceeds the number of pads;

interactive procedure is repeated;

Algorithm stops when the pixel size becomes sufficiently small (1mm)

Application to Muon Cluster Finding(cont)



Coupling definitions:

cluster_i-to-pad_j coupling:

cluster_i-to-cluster_j coupling:

Fitting procedure (what and how to fit)


N

ipix

kkj

pc

ij cc1

N pads

k

pc

jk

pc

ik

cc

ij ccc1


Minimized function:

old:

new:

with - pad charge, f - function (integral of Mathieson function),

Fitting procedure (cont)


N

q

fqpads

ii

i

2

2""

N

qq

fqpads

i i

i

2

2""

N

qN

qpads

ii

pads

1

qi


Cluster and hit finding - non-parametric approach; Coordinate reconstruction - parametric approach

The same problem from another angle of view




Results: residuals in the non-bending plane



Results(cont): residuals in the bending plane



Conclusions:

• There is a nice method of cluster finding and hit reconstruction • It gives quite promising results for cathode pad chambers of the Forward Muon Spectrometer

Acknowledgements:

The work has been partly supported under the Protocol of International Cooperation SUBATECH-JINR 97-31 and INTAS grant 00-00538.

A New Approach to Cluster Finding in Cathode Pad Chambers of the Muon Spectrometer A. Zinchenko for...

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