two methods for ellipse fitting in the cbm experiment

A.Ayriyan1, V.Ivanov1, S.Lebedev1,2, G.Ososkov1

in collaboration with N.Chernov3

1st CBM Collaboration Meeting, JINR Duba,19-22 May 2009

1 JINR-LIT, Dubna, Russia2 GSI, Darmstadt, Germany3 The Univ. of Alabama at Birmingham, USA

Email: ayriyan@jinr.ru

RICH in CBM at FAIR (Darmstadt, Germany)

http://www.gsi.de/fair/experiments/CBM/

3Introduction

Why ellipse?4Introduction

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.

5Motivation

GoalMotivation 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.

Algorithm based on the Minuit minimization

1 2 2d d a This method is based on Kepler’s ellipse equation

21 1 2 2 1 21

, , , , , , 2 minn

F F F F i i i ii

M x y x y a d x y d x y a

and minimization of the following function

using Minuit minimization with the following initial values:

1

1 2 2

5; 0.5 ;; 0.5 .

F C

F F C F C

a x x ay y y x x a

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

LSMLSM is based on minimization of

2, , minP X P X PS d

Classic LSM General LSM

How to calculate distances?

8TaubinFitter

Approximation of distanceDefine function (a conic section equation)

Take its Taylor expansion:

And normalize by its gradient to obtain the distance along the normal to our function

2 2XP Ax Bxy Cy Dx Ey F

2X X X X XTi i iP P P O d

X

Xi

ii

Pd

P

9TaubinFitter

2

2, ,X

P X P XXi

i

PS L

P

Taubin methodTaubin method is based on the following

representation

Now denominator is uniform for all points, this form is easier for practical minimization

2

2, , ,X

P X P X P XXi

i

PS L T

P

10TaubinFitter

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.

Two steps to compare1st, 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+.

Comparison 11

Accuracy12Comparison

0 10 20 30 40 50 60 70 80 900.000.01

0.020.03

0.040.05

0.060.070.08

0.090.10

, degree

||tr

ue -

mea

n|| 2

TTaubin; Npoints = 22 TMinuit; Npoints = 22

0 5 10 15 20 25 30 35 40 45 50 550.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

||tr

ue -

mea

n|| 2

Npoints

TTaubin; = 0 TMinuit; = 0 TTaubin; = /2 TMinuit; = /2

Mean error norm vs. theta (left) and number of points (right)

Time of calculation

0 5 10 15 20 25 30 35 40 45 50 550

102030405060708090

100110120130

T, s

econ

d

Npoints

TTaubin, = 0 TMinuit, = 0 TTaubin, = /2 TMinuit, = /2

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

100

, degree

T, s

econ

d TTaubin, Npoints = 22 TMinuit, Npoints = 22

13Comparison

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

Ring Finding efficiency vs. momentum14Comparison

Minuit Fitter

Taubin Fitter

Summary

Algorithm Efficiency, %

Number of Fakes per

event

Number of Clones per

event

Minuit Fitter

90.33 6.75 0.42

Taubin Fitter

93.02 5.99 0.70

Comparison 15

Conclusion16Conclusion

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