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Two methods for ellipse fitting in the CBM experiment

Two methods for ellipse fitting in the CBM experiment. A.Ayriyan 1 , V.Ivanov 1 , S.Lebedev 1,2 , G.Ososkov 1 in collaboration with N.Chernov 3. 1 JINR-LIT , Dubna , Russia 2 GSI, Darmstadt, Germany 3 The Univ. of Alabama at Birmingham, USA Email: ayriyan@jinr.ru.

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Two methods for ellipse fitting in the CBM experiment

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  1. Two methods for ellipse fitting in the CBM experiment A.Ayriyan1, V.Ivanov1, S.Lebedev1,2, G.Ososkov1 in collaboration withN.Chernov3 1JINR-LIT, Dubna, Russia 2 GSI, Darmstadt, Germany 3 The Univ. of Alabama at Birmingham, USA Email: ayriyan@jinr.ru 1st CBM Collaboration Meeting, JINR Duba,19-22 May 2009

  2. Introduction 3 RICH in CBM at FAIR (Darmstadt, Germany) http://www.gsi.de/fair/experiments/CBM/

  3. Introduction 4 Why ellipse?

  4. Motivation 5 Ellipse fitting is important for PID in RICH • For Ring Finder Ring Finder uses Ellipse Fitter algorithm. • For Electron Identification • Electron Identification • uses Ellipse Fitter • algorithm. http://cbm-wiki.gsi.de/cgi-bin/view/Public/PublicRich#Mirror

  5. Motivation 6 Goal The ellipse fitting algorithm currently implemented in the CBM Framework is based on the MINUIT minimization. We propose another algorithm based on the Taubin method. Our goal is to compare this algorithms in order to show advantages of the Taubin method for data analysis in the RICH detector.

  6. MinuitFitter 7 Algorithm based on the Minuit minimization This method is based on Kepler’s ellipse equation and minimization of the following function using Minuit minimization with the following initial values: Although Minuit Fitter shows admissible accuracy and, therefore, it is used currently as a default method in the CBM Framework, this algorithm doesn't give statistically optimal estimators of ellipse parameters.

  7. TaubinFitter 8 LSM • LSM is based on minimization of How to calculate distances? Classic LSM General LSM

  8. TaubinFitter 9 Approximation of distance Define function (a conic section equation) Take its Taylor expansion: And normalize by its gradient to obtain the distance along the normal to our function

  9. TaubinFitter 10 Taubin method • Taubin method is based on the following representation • Now denominator is uniform for all points, this form is easier for practical minimization Actually, Taubin method also doesn’t give completely optimal estimates from the statistical point of view, but the proposed approximation is a rational function whose minimum is easy to calculate.

  10. Comparison 11 Two steps to compare 1st, both algorithms were compared on simulated data: a=6.2;b=5.6;xc=yc=0.; σx=0.2;σy=0.2;ex = N(0,σx); ey = N(0,σy); 2nd, both algorithms were compared on “real data”: 500 UrQMD events Au+Au at 25 AGeV +5e-+ 5e+.

  11. Comparison 12 Accuracy Mean error norm vs. theta (left) and number of points (right)

  12. Comparison 13 Time of calculation Time per 100k ellipses vs. theta (left) and number of points (right)

  13. Comparison 14 Ring Finding efficiency vs. momentum Minuit Fitter Taubin Fitter

  14. Comparison 15 Summary

  15. Conclusion 16 Conclusion Taubin Fitter is 10~30 times faster than Minuit Fitter; moreover Taubin Fitter is practically independent of the number of points Ring Finder shows better efficiency with Taubin Fitter than with Minuit one Taubin method is statistically more accurate than the method based on Minuit minimization Taubin method is not iterative and doesn’t need a starting value; this is important in data analysis with RICH

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