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New Approach to Cluster Finding in Muon Chambers

This study presents a new approach to cluster finding in cathode pad chambers of the muon spectrometer. The method utilizes Bayesian unfolding and Maximum Likelihood - Expectation Maximization algorithms to improve coordinate resolution and accurately estimate the number of track candidates. Results demonstrate the effectiveness of the new approach in enhancing muon cluster finding.

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New Approach to Cluster Finding in Muon Chambers

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  1. 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

  2. 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

  3. 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 HitReconstructionin 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

  4. 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

  5. General view of the Forward Muon Spectrometer A New Approach to Cluster Finding in Cathode Pad Chambers of the Muon Spectrometer A. Zinchenko ALICE Week, 16-20 September, 2002

  6. 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) 487If anybody observes a single eventE(effect), the probabilitythat it has been due to the ithcause of Ci, i=1,2,..,ncisproportional to the probability of causeP(Ci) times theprobability of the cause to produce the effectP(E|Ci). • (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

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

  8. A New Approach to Cluster Finding in Cathode Pad Chambers of the Muon Spectrometer Basic Ideas of Bayesian Unfolding (cont) 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 the nthiteration 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 A. Zinchenko ALICE Week, 16-20 September, 2002

  9. 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

  10. 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 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 maximais not sufficiently accurate and results in deterioration of coordinate resolution. A. Zinchenko ALICE Week, 16-20 September, 2002

  11. A New Approach to Cluster Finding in Cathode Pad Chambers of the Muon Spectrometer Application to Muon Cluster Finding • 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 A. Zinchenko ALICE Week, 16-20 September, 2002

  12. A New Approach to Cluster Finding in Cathode Pad Chambers of the Muon Spectrometer Application to Muon Cluster Finding(cont) • 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. A. Zinchenko ALICE Week, 16-20 September, 2002

  13. 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

  14. 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: • cluster_i-to-cluster_j coupling: A. Zinchenko ALICE Week, 16-20 September, 2002

  15. A New Approach to Cluster Finding in Cathode Pad Chambers of the Muon Spectrometer Fitting procedure (cont) • Minimized function: • old: • new: • with - pad charge, f - function (integral of Mathieson function), A. Zinchenko ALICE Week, 16-20 September, 2002

  16. 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

  17. A New Approach to Cluster Finding in Cathode Pad Chambers of the Muon Spectrometer Results: residuals in the non-bending plane A. Zinchenko ALICE Week, 16-20 September, 2002

  18. 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

  19. 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

  20. 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

  21. 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

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