Transcript A New Approach to Cluster Finding and Hit Reconstruction

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
for JINR (Dubna) – SUBATECH (Nantes) Collaboration
A New Approach to Cluster Finding in Cathode Pad
Chambers of the Muon Spectrometer
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
A. Zinchenko
ALICE Week, 16-20 September, 2002
A New Approach to Cluster Finding in Cathode Pad
Chambers of the Muon Spectrometer
Overview
The Muon Spectrometer
Basic Ideas of Bayesian Unfolding
Application to Muon Cluster Finding
Results
Conclusions
A. Zinchenko
ALICE Week, 16-20 September, 2002
A New Approach to Cluster Finding in Cathode Pad
Chambers of the Muon Spectrometer
General view of the
Forward Muon Spectrometer
A. Zinchenko
ALICE Week, 16-20 September, 2002
A New Approach to Cluster Finding in Cathode Pad
Chambers of the Muon Spectrometer
Basic Ideas of Bayesian Unfolding
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 Ci, i=1,2,..,nc is
proportional to the probability of cause P(Ci) times the
probability of the cause to produce the effect P(E|Ci).
P(Ci | E ) 
P( E | Ci ) P(Ci )
ni
 P( E | C ) P(C )
i 1
i
i
(Lucy-Richardson method: L.B. Lucy, Astronom. J. 79 (1974)
745; W.H. Richardson, J. Opt. Soc. Amer. A 62 (1972) 55)
A. Zinchenko
ALICE Week, 16-20 September, 2002
A New Approach to Cluster Finding in Cathode Pad
Chambers of the Muon Spectrometer
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
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
L( λ ) 

i 1,..., nt
j 1,..., n p
A. Zinchenko
e
ij
ij
nij!
nij
ALICE Week, 16-20 September, 2002
A New Approach to Cluster Finding in Cathode Pad
Chambers of the Muon Spectrometer
Basic Ideas of Bayesian Unfolding (cont)
where λ  (1,..., n ) - vector of pixel activities ,  ij - the mean number of
emissions from the pixel j contributing to the tube i, n ij is the number of photon
pairs counted in the tube i from the pixel j, nt is the number of detector tubes,
n p - number of pixels. ij  cij j and cij   is jth pixel visibility from the tube i.
p
The Expectation step of the EM algorithm is done by taking a conditional
n

expectation of log-likelihood with respect to parameter estimates
at the
nth iteration and measured detector counts ni   nij .
j
The Maximization step is to take partial derivatives  /  j of the expectation, set
them to 0 and solve.
The new value for the jth component of the vector

n 1
j

nj
c
ij
i
A. Zinchenko
 cij
i
ni
 cik nk
k
ALICE Week, 16-20 September, 2002
A New Approach to Cluster Finding in Cathode Pad
Chambers of the Muon Spectrometer
Some Examples From Medical Imaging
“Journey to the Center of a Rat’s Heart’’
(Sci. Fi., PG-13)
“Halloween-12’’ (Horror, R)
A. Zinchenko
ALICE Week, 16-20 September, 2002
A New Approach to Cluster Finding in Cathode Pad
Chambers of the Muon Spectrometer
Application to Muon Cluster Finding
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 twodimensional 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.
A. Zinchenko
ALICE Week, 16-20 September, 2002
A New Approach to Cluster Finding in Cathode Pad Chambers
of the Muon Spectrometer
Application to Muon Cluster Finding
¥
New method: transform the "twocathode 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
A. Zinchenko
ALICE Week, 16-20 September, 2002
A New Approach to Cluster Finding in Cathode Pad
Chambers of the Muon Spectrometer
Application to Muon Cluster Finding(cont)
New method (cont):
0
Iterative procedure: Track j has an initial value p j of
energy release . Usually all
p
k 1
j

p kj
N pads
N pads
c
i 1
p j 0 's are set to 1.
ij
qi
cij k

fi
i 1
with
fi 
k
N tra cks
k
c
p
 ij j
j 1
where f i k - the expected signal on pad i for track energy
k
releases p j at the kth iteration,
c ij - the track-to-pad coupling(signal on pad i from track j given
by Mathieson) and
the number of tracks in the
N- tracks
array.
A. Zinchenko
ALICE Week, 16-20 September, 2002
A New Approach to Cluster Finding in Cathode Pad
Chambers of the Muon Spectrometer
Application to Muon Cluster Finding(cont)
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)
A. Zinchenko
ALICE Week, 16-20 September, 2002
A New Approach to Cluster Finding in Cathode Pad
Chambers of the Muon Spectrometer
Fitting procedure (what and how to fit)
¥
Coupling definitions:
¥
cluster_i-to-pad_j coupling:
i
c
c p
ij
¥

N pix
c
k 1
kj
cluster_i-to-cluster_j coupling:
c
c c
ij

N pads
c c
k 1
A. Zinchenko
c p
c p
ik
jk
ALICE Week, 16-20 September, 2002
A New Approach to Cluster Finding in Cathode Pad
Chambers of the Muon Spectrometer
Fitting procedure (cont)
¥
Minimized function:
" " 
2
¥
old:
¥
new:
¥
with
q
2
qf 
 

 q 
N pads
i
i
i

N pads q  f
2
i


" "
qi q
i

2
qi - pad charge, f - function (integral of Mathieson function),
1
N pads
N pads
 qi
A. Zinchenko
i
ALICE Week, 16-20 September, 2002
A New Approach to Cluster Finding in Cathode Pad
Chambers of the Muon Spectrometer
The same problem from another angle of view
¥
¥
Cluster and hit finding - non-parametric approach;
Coordinate reconstruction - parametric approach
A. Zinchenko
ALICE Week, 16-20 September, 2002
A New Approach to Cluster Finding in Cathode Pad
Chambers of the Muon Spectrometer
Results:
A. Zinchenko
residuals in the non-bending plane
ALICE Week, 16-20 September, 2002
A New Approach to Cluster Finding in Cathode Pad
Chambers of the Muon Spectrometer
Results(cont): residuals in the bending plane
A. Zinchenko
ALICE Week, 16-20 September, 2002
A New Approach to Cluster Finding in Cathode Pad
Chambers of the Muon Spectrometer
Results(cont): residuals in the bending plane
A. Zinchenko
ALICE Week, 16-20 September, 2002
A New Approach to Cluster Finding in Cathode Pad
Chambers of the Muon Spectrometer
Results(cont): residuals in the bending plane
A. Zinchenko
ALICE Week, 16-20 September, 2002
A New Approach to Cluster Finding in Cathode Pad Chambers
of the Muon Spectrometer
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. Zinchenko
ALICE Week, 16-20 September, 2002