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: [email protected]
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