Track reconstruction and pattern recognition V.V. Ivanov Laboratory of Information Technologies Joint Institute for Nuclear Research, Dubna, Russia CBM Collaboration Meeting GSI, Darmstadt, October 6-8, 2004

1. 3D track recognition by the track-following method. 2. Accurate momentum restoration based on orthogonal polynomial sets. 3. Application of Kalman filter for momentum restoration. 4. Polynomial approximation in a problem of accurate momenta reconstruction. 5. RICH ring guidance by the Hough transform approach. 6. Looking for the STS “secondary” tracks. 7. Fast algorithms for accurate charged particle tracing through inhomogeneous magnetic field. 8. Concluding remarks. Outline

1. STS track recognition by 3D track-following method Gennady Ososkov, A.Airiyan, A.Lebedev, S.Lebedev Laboratory of Information Technologies Joint Institute for Nuclear Research, Dubna, Russia home page:

Fast track fitting in non-uniform magnetic field Our goal : is to recognize all tracks of any STS events by hits measured on seven STS stations taking into account the magnetic field non-homogeneity 3D view of a typical STS simulated event

Stage 1. Starting from the middle of the target area, we connect this point sequentially with all points in the first station in Y-Z view, where tracks are close to straight lines. Each time we prolong the straight line driven via these two points to the plane of the second station. Then, we set aside from the crossing point an asymmetrical corridor and look for all points appeared in this corridor. Then, for each of these found points and two on sections previous we draw in X-Z projection a parabola which is prolonged to the next station. Since several prolongations can happened, we set aside corridors from each of point predicted on the third station. Then, coming to the Y-Z view, we set also a corridor aside of the corresponding prediction on the third station and look for hits in it. Prediction and search in YoZ view Prediction and search in XoZ view

The main trick, guaranteed the final track-finding efficiency on the level of % depending on the track momentum, consists of very careful preliminary calculations of the table with limits for those corridors depending on the station number and the sign of the particle charge. This table is calculated on the of many thousands simulated events. Since the conventional approach to use symmetrical 3σ corridors failed, we use the distributions of deviations between real hit positions and their predictions by prolonging straight line or parabola for all stations. Examples of those distributions for XoZ views on the 3-d and 6-th stations or for YoZ views on the 3-d and 5-th stations One can see their asymmetry and sizes growing from mkm to sm

Stage 2. Having tracks found at the first stage, we make a selection on the basis of the following criterion: each point can only belong to one track. So any track with one or more points belonging to some other track is rejected. Efficiency vs. momentum Average number of ghosts is 1.62 per event (we consider as a ghost a track that has at least one wrongly found point) Preliminary results Mean efficiency is 92.8%

As one can see from figures with corridor distributions, a corridor in the 4-th station is 25 times more narrow, then in the 5-th station and 30 times then in the 6-th station. That is due to (1) multiple scattering, (2) too simple track fitting model, (3) STS geometry is not optimal. Therefore, to decrease the corridors and improve prediction accuracy, we should apply, first, the Kalman filter technique and, then, more elaborated track model (say, Runge-Kutta solution for the real map of magnetic field). Vertex positioning improvement. Although the algorithm has shown its stability to minor vertex shifts, the option exists to estimate the vertex position after the first run on the basis of several reference tracks. Then in the second run all track-following process is recalculated with the improved vertex position. The Kalman filter applying for better vertex estimation is planned. Future plans

2. High accuracy momentum restoration applying orthogonal polynomial sets E.P.Akishina, V.V.Ivanov, E.I.Litvinenko Laboratory of Information Technologies Joint Institute for Nuclear Research Dubna, Russia

An algorithm, which provides a high accuracy momentum restoration of a charged particle from points of the trajectory through an inhomogeneous field of the CBM dipole magnet, is developed. A set of representative trajectories is used to construct a multidimensional function which gives the momentum in terms of observable quantities.

Uniform magnetic field approximation The momentum p is calculated by a formula: - constant - deflection angle

- point in the first STS - tangents of trajectory in this point We have to construct the inverse function: In inhomogeneous magnetic field φ is function: Method of accurate momentum restoration

The procedure includes two steps: 1.A representative set of all relevant trajectories through the magnet is computed, and the corresponding set of angles is formed. 2.This set is used to construct the explicit function, which gives the momentum in terms of observables. Each trajectory is determined by five variables: 1. - coordinates of a point in the first STS 2. - tangents of a particle trajectory in this point 3. - the particle momentum.

Method of accurate momentum restoration - range of i-th variable - normalization to [-1,1] - discrete points due to Tchebysheff distribution The set of determines the collection of fixed trajectories, which are traced through the magnetic field and the set of corresponding deflections is calculated.

Method of accurate momentum restoration Let the range of φ(·) be ; φ(·) is normalized to [- 1,1]: discrete number of is chosen Applying inverse interpolation, we can calculate corresponding values of

Method of accurate momentum restoration Let be in the form The coefficients are calculated using the formula: where T(x) is orthogonal Tchebysheff polynomial

Accuracy of momentum restoration - number of “points” for which trajectories were calculated - momentum values in the range 1-10 GeV/c Total number of : 5 x 5 x 5 x 5 x 7 = 4375 Total number of coefficients can be significantly decreased without accuracy loss. The significant coefficients are selected from the whole set using Fisher test: 89 significant coefficients.

Accuracy of momentum restoration Top plots (all 4375 coefficients): left plot is the distribution (in MeV/c) and right plot RMS = 0.147% Bottom plots (89 coefficients): left plot is the distribution (in MeV/c) and right plot

Accuracy of momentum restoration In order to estimate the accuracy of the method on data close to real data, we used the GEANT data: σ = 0.263% Distribution of (in MeV/c) for GEANT data (for positively charged particles). Distribution of for GEANT data (for positively charged particles).

Accuracy of momentum restoration Distribution of (in MeV/c) for GEANT data (for all tracks). Distribution of for GEANT data (for all tracks). σ = 0.342%

Conclusion Main results of this work can be summurized as follows: The possibility to reconstruct charged particle momenta registered by coordinate STS-detectors with a high accuracy is demonstrated: The algorithm for momenta restoration is elaborated. With the help of the later the estimation was realized both the accuracy of the mathematical method (which includes inaccuracies of the magnetic field) and the accuracy the momentum determination on the basis of the GEANT data.

Our next steps are as follows 1.To increase the accuracy of the algorithm by separate restoration of momenta for paricles with different charges. 2.To increase the accuracy of the algorithm by dividing the whole interval of momenta on a few subintervals. 3.To do the estimation of the momenta determination applying the procedure of fitting particle trajectories on the basis of Kalman filter.

3. Using Kalman filter for accurate momentum restoration Baginyan, V.V.Ivanov, P.V.Zrelov Laboratory of Information Technologies Joint Institute for Nuclear Research Dubna, Russia

Kalman filter for momentum reconstruction  The results on momentum reconstruction based on orthogonal polynomials (OP) using measurements of STS-detectors and magnetic field map and applying track fitting with Kalman filter are presented.  The accuracy of momentum restoration is estimated.

Kalman filter for momentum reconstruction (positive particles)(negative particles)

Kalman filter for momentum reconstruction (all particles)(all particles adding 456 values, hom.field)

Conclusion 1.A method of precise determination of momenta with using a Kalman filter and a method of orthogonal polynoms has been implemented. 2. Encouraging results have been obtained on determination of particle momentum with an average relative accuracy no more than 1%. 3. Use for part of tracks of approximation of a homogeneous field, when calculating the momentum, does not strongly change the results. 4. There is a difference in the accuracy of the method for positive and negative particles with a viewpoint of the accuracy of the obtained results. 5. There are ``reserves'' for improvement of the results. In particular, in the future other extrapolating procedures will be considered (excepting the parabolic model considered in this work) for obtaining coordinates and angles at the 7-th chamber.

4. Polynomial approximation in problem of accurate momenta reconstruction P.G.Akishin Laboratory of Information Technologies Joint Institute for Nuclear Research Dubna, Russia

Procedure of momentum reconstruction includes the following steps: 1.Input data preparation using trajectory hits obtained by a track-finder. 2. Fitting trajectory hits by different approximation methods: a) polynomial, b) cubic-spline, c) B-spline. 3. Momentum reconstruction by “attaching” the motion equations to the fitted trajectory.

Accuracy of momentum reconstruction: dp/p No multiple scatteringWith multiple scattering

5. CBM RICH ring guidance by the Hough transform approach B.Kostenko, G.Ososkov, A.Solovjev Laboratory of Information Technologies Joint Institute for Nuclear Research Dubna, Russia

CBM RICH ring recognition is complicated very much, due to low quality of the ring guidance information obtained by STS tracks extrapolation. Therefore, a preliminary search for ring centers and radii was demanded (P. Senger). A fragment of the simulated CBM RICH event with ring guidance points. They are distributed almost independently on the rings positions. Robust fitting corresponds to the blue circles. Two false circles are obtained. CBM RICH1 design

The Hough transform (HT) HT converts points from the measurement space to the parameter space and permits to find most “popular” places in it. Problem: huge number of parameters (about 1.7x10 8 possible centers for 1000 hits). Event 1, top mirror. Black points are hits, blue ones are centers. Its treatment requires sec CpuTime of 2.2 GHz computer. Many rings contain too many points necessary for their handling (excess information).

Inference: to reduce number of hits for recognition of more simple cases ( “thinning out” of data ), to take out hits corresponding these cases and to consider more difficult cases after that. Event 1, top mirror: 1/3 part of hits are taken. Its treatment requires ~ 80 sec CpuTime of 2.2 GHz computer.

Contraction mapping procedure: D(A,B)  D(A,B) / N, where N>1 “Attraction” and “merging” of centers Event 1, top mirror: distribution of ring centers after the first contraction mapping. Non-overlapping rings are recognized

Second contraction mapping: all rings are found, no false rings are present, non-overlapping rings are found with a good accuracy (compare with Robust fitting)

Further developments: - to test algorithm more thoroughly and to tune its parameters (only 7 events were considered up to now ), - to organize the interconnection between results of processing of different extractions from the data with the purpose to optimize the parameters of rings, - to mark out found circles and consider non-recognized rings.

6. Looking for the STS “secondary” tracks V.Ivanov, G.Ososkov, A.Airiyan Laboratory of Information Technologies Joint Intstitute for Nuclear Research, Dubna, Russia

Good observation: 90% of the secondary vertexes are spread out of the 100 micron area around the vertex, so there is a hope to distinguish them from the primary tracks. The problem is to find a distinguishing criterion. Two parameters are obvious: - track momentum; - track deviation from the vertex. Secondary vertex X-Z coordinate distribution An enlarged fragment of the previous figure

Variuos measures of difference between the primary and secondary tracks Then to intensify the difference we take logarithms on both axes (b) Log(DEV) vs. Log(P). only about 40-50% of secondaries can be reliably extracted by a carefully chosen cut 1. Deviations versus momentum Since the deviations (DEV) vary from microns to meters we take its logarithm. (a) Log(DEV) vs. P. Secondaries marked in blue

An example of the secondary track search by thresholding in 3D-space of Log(divX), Log(divY), Log(divZ) The cut-off -5.4 was found of the sample of hit tracks of them are secondary tracks 300 tracks were cut off. 250 of them –secondaries Thus, a straight-forward cut-off gives the efficiency about 25% 3D distribution of Log(divX), Log(divY), Log(divZ), primaries marked by blue, secondary tracks marked by red

2. Track bending in magnetic field can be measured in XoZ plane either by the angle α between momentum directions on the 1 st and the last stations or by the shift S generated by these directions ( see figure ) Result is about the same: the bulky part of the secondary tracks is again mixed up undistinguishable with the primary tracks The angle α versus Log(DEV/P)

Track stream crossections for one event in YoX planes of the STS stations 1,2,4,7 Pay attention to the growing scales The idea was to look for narrow maximum, which could mark possible jet with the vertex lying between the previous and the current station

The same crossections for the secondary tracks only

7. Fast algorithms for accurate charged particle tracing through inhomogeneous magnetic field P.G.Akishin Laboratory of Information Technologies Joint Institute for Nuclear Research Dubna, Russia

Charged particle tracing through inhomogeneous magnetic field The problem is to accurately prolong a charged particle trajectory from the 7-th STS to the position of the RICH detector. In work [1] there were developed fast numerical algorithms for fast solution of motion equations for a charged particle moving in inhomogeneous magnetic field. 1. P.G.Akishin: The numerical multidimensional approximation of the fast solution of equations describing a charged particle motion in non-homogeneous magnetic field, JINR preprint, P , Dubna, 2002 (in Russian).

General remarks Tracking Momentum reconstruction RICH algorithms Secondary vertices Tracing track trajectory out of the dipole magnet