CBM RICH multiring recognition and

fitting

CBM Tracking Week, 24-28 January, 2005 at GSI, Darmstadt

B.Kostenko, E.Litvinenko, G.OsoskovJINR LIT

141980 Dubna, Russia

email: bkostenko@jinr.ru

1.Selective Hough transform for multi-ring recognition

Improvement of our previous algorithm:Simultaneous reconstruction of all rings belonging to a given cluster of hits is replaced by sequential recognizing and fitting the individual rings

Advantages:

a) Much faster,b) Total elimination of the false rings.

The first step: conformal mapping: Xu, Yv

From the circle

(X-Xc)2+(Y-Yc)2=R2

to a straight line:

u = 1/2 Xc – v Yc / Xc, where

u=X/(X2+Y2), v= Y/(X2+Y2)

Fig.1 Only one circle containing the principal hit with coordinates X0, Y0 is

recognized at any moment of time for a given cluster of

hits

Main idea of multi-ring recognition: if one (principal) hit is fixed, only hits belonging to the same ring give the contribution to the peak in the parameter space. Rings run over X0, Y0 and one or two hits taken from anther circle are seen as distributed background. High speed of calculation - due to a drastic reduction of the number of possible combination: C3

n C2n

The second step: Hough transformAfter conformal mapping the Hough transform is performed. The hits on the straight line will be converted into a region around a fixed point in the parameter space. The points in this region correspond to hits belonging to the same ring. Thus, the circle is recognized.

Fig.2 ( Notations of the axes should be replaced z u, x v )

Estimation of ring parameters in the selective Hough transform algorithm:

For pairs of points (ui,vi) and (uj,vj) in a fixed vicinity of the peak in the parameter space we estimate corresponding coordinates of the ring center,

XC (ij)= ½ (vi - vj)/(uj vi - ui vj), YC (ij)= - XC (ij) uj /vj + 1/2vj, and take their mean value,

XC <XC (ij)> , YC <YC (ij)>.

The ring radius is the averaged value of

R (i) ((Xi - XC )2 + (Yi - YC )2 )1/2,

where Xi and Yi are coordinates of hits corresponding the taken vicinity of the peak in the parameter space.

Why only ring fragments (arcs) can be fitted by selective Hough transform

1. Due to errors of the coordinates X0, Y0 of the principal hit, see Fig.1, instead of the straight line in the left side of Fig.2 we have, strictly speaking, a parabola:

v 1/2YC - u XC/YC – u2 δ (R/YC)3,

where δ = R – (XC

2 + YC2)1/2

is a parameter taking into account the errors of the principal hit coordinates X0, Y0 .

2. Because of the ring deformation in the detecting plane of the RICH (circles ellipses and ovals)

Corollary: Hough transform gives only an input data for further data processing.

2. Rigid ring recognition of deformed circles

Elastic net analogue algorithm for traveling salesman problem (R.Durbin, D.Willshaw) Why it is unsuitable for us: strong deformation of the net in the course of evolution.

Appropriate modification: sharp bends are hampered. Consequence: the net is close to the circle during all the time.

Idea: Firstly, the circle is placed along a part of hits (an arc) found by selective Hough transform. These hits should hold it in the right position and direct, due to its small flexibility, along the rest of the ring’s hits with a proper adjustment to a deformed ring.

elastic net.eps

Elastic Net

Xi, Yi – 2-D coordinates of cities (hits) and points on the net.Effective energy of interaction for points on the net with each other and and with the cities:

F= i ln [j exp{-(Xi - Yj)2/2K2}] + j (Yj+1 - Yj)2. (1)

K describes the attraction radius between cities and points on the net. Since it slowly decreases in the course of evolution of the net, It is possible to introduce the temperature T= 2K2 describing corresponding “annealing” procedure. Dynamics: gradient equation

dYj /dt = - K dF/d Yj ⇨ for t=1: Yj - K dF/d Yj. (2)

Local minimization of the energy F is guaranteed at every step by the formula

F j dF/ dYj Yj = - Kj (dF/ dYj )2 < 0. Global minimization is reached due to the “annealing” procedure in the limit of the infinitely slow temperature decrease.

For energy (1) the equation of evolution (2) gives the iteration scheme:

Yj i wij (Xi - Yj) + 2K (Yj+1 - 2Yj + Yj-1). (3)

Annealing schedule: K is decreased by 1% after 25 iteration of Yj and is changed from 0.2 to 0.01 - 0.02. Rigid ring dynamics Additional energy making difficult sharp turns of the net:

V= V0 exp ( b2), where the variable b= (k1 – k2)/(1 + k1 k2)describes the value of the net bend,

k1 = (2 - 1)/(2 - 1), k1 = (3 - 2)/(3 - 2),

Y1, Y2, Y3 = (1, 1), (2, 2), (3, 3) are some three successive points on the net.An additional term in r.h.s. part of the equation of motion (3):

2 j (bj-1Vj-1 dbj-1/dYj + bjVj dbj/dYj + bj+1Vj+1 dbj+1/dYj ). (4)

For derivatives db/dY in (4) there are simple explicit formulae.

3.Robust fitting

Necessity for the robust fitting:

1) Selective Hough transform determines circles parameters only approximately and gives an initial data for further robust fitting,

2) Rigid ring algorithm only matches hits belonging to deformed rings and does not define their parameters. After corresponding spatial rotation, deformed ring can take a form close to the circle and should be fitted by the robust fitting program.

Improvements of the robust fitting algorithm which are necessary to do:

1) To elaborate more accurate annealing procedure for increasing the accuracy of the program installed in cbmroot,

2) To elaborate specialized minimization algorithm instead of MINUIT for speeding-up the program (already done, acceleration about one order).

4.Simultaneous Fit of Two or more circlesTwo or more overlapping rings often occur and can cause lots of errorsTherefore a method is proposed to fit them at once. The key-word is ROBUST fit of data to a combined equation of those circles.It can be done by multiplying the corresponding number of the circle equations. For instance, for the forth order curve joining two circles it gives the six parameter equation.

The LSM estimation of all parameters requires the search for the global minimum of the non-linear functional

(5)We linearize it assuming that the distances of the initial values of our parameters from the exact values of corresponding parameters are so small that any of their products or squares are negligible.

Simultaneous Fit of Two or more circles (continued)Then we substitute in (1) each parameter by the sum of its initial value and corresponding delta-correction and omit all members of the second and higher order of smallness. Then the functional (5) is converted into the quadratic function of these corrections, so its minimization is not a complicate problem. Robust weights wi in (5) are calculated iteratively by formulae

where

Adding obtained corrections to the initial values of parameters we repeat the whole procedure iteratively until the corrections become less than a prescribed value or the number of iterations attains its limits. The search of parameters initial value is a hard problem. Its solution depends on the data peculiarity.

a bCERES/NA-45 results of fitting the simulated discretized signals of two circles, a) distance (in pad size) between circles =10, b) distance =1

Status of corresponding software:Fortran version of the program is under testing on CBM simulated data. C++ conversion in the CBM framework is planned in the 1st quarter 2005