1. A.Ayriyan 1, V.Ivanov 1, S.Lebedev 1,2, G.Ososkov 1 in collaboration with N.Chernov 3 1 st CBM Collaboration Meeting, JINR Duba,19-22 May 2009 1 JINR-LIT, Dubna, Russia 2 GSI, Darmstadt, Germany 3 The Univ. of Alabama at Birmingham, USA Email: ayriyan@jinr.ru

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

3. Why ellipse? 4 Introduction

4. http://cbm-wiki.gsi.de/cgi-bin/view/Public/PublicRich#Mirror 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. 5 Motivation

5. Goal Motivation 6 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. 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: 7 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. MinuitFitter

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

8. 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

9. Taubin method Taubin method is based on the following representation Now denominator is uniform for all points, this form is easier for practical minimization 10 TaubinFitter 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. Two steps to compare 1 st, both algorithms were compared on simulated data: a = 6.2; b = 5.6; x c = y c = 0.; σ x = 0.2; σ y = 0.2; e x = N(0, σ x ); e y = N(0, σ y ); 2 nd, both algorithms were compared on “real data”: 500 UrQMD events Au+Au at 25 AGeV +5e - + 5e +. Comparison 11

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

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

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

14. Summary AlgorithmEfficiency, % Number of Fakes per event Number of Clones per event Minuit Fitter 90.336.750.42 Taubin Fitter 93.025.990.70 Comparison 15

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